This is joint work with Eric Friedlander, Julia Pevtsova and Andrei Suslin. We consider modules over an elementary abelian group on which every element in the radical, but not the square of the radical, has the same Jordan canonical form. Such modules can be used to define bundles on projective spaces and Grassmanians. They have many interesting properties. We can get them as submodules of any module of the group algebra. In this talk I will discuss some of the constructions and their generalizations.

Note for Attendees

There will be tea and cookies in the PIMS 1st floor lounge at approximately 2:45pm.

This talk presents joint work with Kee Lam of UBC and Nancy Cardim, Maria Herminia de Mello, and Mario Olivero da Silva in Rio de Janeiro, Brazil. Given the total space of any orthogonal sphere bundle over a sphere, we determine its stable span and span.

Abstract: Given a reasonable filtration of spaces F_0, F_1,..., F_n, and a homology theory H, we define what it means to split the filtration with respect to H and then give a criterion for when this is possible. We strongly refine this criterion
to decide when the filtration splits stably (and hence splits with respect to any homology theory). Many examples will be
discussed (like Steenrod splitting, snaith splitting, configuration spaces, commuting tuples in lie groups, Miller
splitting of unitary groups, etc). This work is by and with Stylian Zanos (Lille).

Abstract: Cocycle categories give a simple, flexible way to describe morphisms in a homotopy category, provided that the underlying model structure is sufficiently well behaved. "Well behaved" model structures include simplicial sets, spectra, simplicial presheaves and presheaves of spectra, together with all good localizations such as the motivic model structure of Morel and Voevodsky

Abstract: Suppose that H is an algebraic group which is defined over a field k, and let L be the algebraic closure of k. The canonical stalk for the etale topology on k induces a simplicial set map from the classifying space B(H-tors) of the groupoid of H-torsors (aka. principal H-bundles) to the space BH(L). The homotopy fibres of this map are groupoids of pointed torsors, suitably defined. These fibres can be analyzed with cocycle techniques: their path components are representations of the "absolute Galois groupoid" in H, and each path component is contractible. The arguments for these results are relatively simple, and applications will be displayed.

Abstract: Conjecture: For any map f: E \to S^1 from a closed 4-manifold to a circle whose homotopy fiber has the homotopy type of a 3-manifold, there exists a fiber bundle \bar f : \bar E \to S^1 where \bar E is a 4-manifold homotopy equivalent to E.

Theorem (joint with Shmuel Weinberger) The conjecture is true when the 3-manifold is a lens space with odd order fundamental group.

The proof involves a surgery theoretic argument which involves a lemma of Gauss used in his third proof of the law of quadratic reciprocity.

This theorem answers a question of Jonathan Hillman, asked in the context of 4-dimensional geometries:

Theorem: Any 4-manifold with Euler characteristic zero and fundamental group a semidirect product where Z acts on Z/odd is homotopy equivalent to a self isometry of a lens space.

Abstract: We give a model for the moduli space of Riemann surfaces with one or more boundary curves using harmonic functions and canonical tesselations. The resulting simplicial complex is homeomorphic
to a flat vector bundle over the moduli space.

Abstract: The simpicial complex of talk I consists of pieces of the classifying spaces of symmetric groups. We use this to investigate the homology of moduli spaces. At the end, we discuss generalizations, where the symmetric groups are replaced by other families of Coxeter groups

Abstract: We give an example of two finitely generated quasi-isometric
groups that are not bilipschitz equivalent. The proof involves
structure of quasi-isometries from rigidity theorems, analysis of
bilipschitz maps of the n-adics and uniformly finite homology

Abstract: We study and determine a homotopy type of
the moduli space of all generalized Morse functions on d-manifolds for given d.
This moduli space is closely connected to the moduli space of all Morse functions
studied by Madsen and Weiss and classifying space of the corresponding cobordism
category.

Abstract:Equivariant Lefschetz invariants have already appeared in algebraic topology. Here I will show how to approach them using the so-called equivariant KK-theory of Kasparov — the main tool of the new field of noncommutative geometry. I will sketch the construction of Lefschetz invariants for equivariant self-maps of a G-space, where G is a
discrete group, and then define them for more general objects than just
self-maps, called correspondences. There are always many interesting
equivariant self-correspondences of a space with a group action, even if the
group is not discrete. The case of compact connected groups seems in particular quite interesting. We state an equivariant version of the Lefschetz fixed-point formula for this situation. In the resulting formula, the geometric side is based on equivariant index theory of elliptic operators, while the global algebraic side involves the module trace of Hattori and Stallings. Computing the relevant traces seems to be a problem belonging properly to algebraic geometry

Abstract: In the 1960s, Wall developed a theory of finiteness obstructions for CW-complexes. We extend this result to diagrams of spaces and use it to investigate which homotopy G-spheres can be realized on a finite complex. We will conclude with some new examples of finite (non-linear, non-free) G spheres for some small groups G

Abstract: In stable homotopy theory the thick subcategory theorem of Hopkins and Smith classifies thick subcategories of the triangulated category of p-torsion finite spectra. Unstably Bousfield classified nullity classes of p-torsion finite suspensions. We look at analagous results in the derived category of a commutative noetherian ring D(R), and some of its subcategories satisfying suitable finiteness conditions. Hopkins and later Neeman proved that thick subcategories of D_{perf}(R) can be classified by their supports, which are subsets of Spec(R). In analogy to Bousfield's result, we show that nullity classes in D^b_{fg}(R) can be classified by certain increasing functions from Z into Spec(R). By an observation of Keller and Vossieck, it turns out that t-structures are just nullity classes together with a right adjoint of the inclusion. From this we derive an extra condition that the increasing function must satisfy to correspond to a t-structure. When R has a dualizing complex, applying a construction of Deligne and Bezrukavnikov thengives a classification of all the t-sctructures in D^b_{fg}(R)

Abstract: Motivated by basic questions from robotics and molecular biology, we consider certain configuration spaces, and some associated maps to two dimensional Euclidian space. We are able to understand the singular sets for a natural subset of these maps.
For this subset, we apply variants of Bott-Morse theory to determine the structure of inverse images - in the configuration spaces - of curves and points in the plane.
In turn, these results answer key questions about the structure of spaces of length preserving immersions of polygons into Euclidean space and provide insights into the process of protein folding

Abstract: Fully irreducible outer automorphisms of a free group are analogous to loxodromic isometries of hyperbolic space, or to pseudo-Anosov elements of the mapping class group of a surface. We develop methods for constructing customized fully irreducible elements of a free group F of rank k. For example, there exists for any matrix A in GL(k,Z) a non-geometric fully irreducible element inducing the action of A on the non-abelian free group of rank k. This is an analogue of a well-known theorem for the mapping class group. This is joint work with Matt Clay.

Abstract: Suppose that \Sigma is a hyperbolic surface of finite type and let \Map(\Sigma) be its mapping class group. It is due to Morita that the canonical homomorphism \Diff(\Sigma)\to\Map(\Sigma) does not split. A first goal of this talk is to give a very simple proof of this fact showing in fact that this also remains true when restricted to some rather small subgroups of the mapping class group. After having proved this, I will show that while \Map(\Sigma) admits a natural Lipschitz action on the unit tangent bundle T^1\Sigma of \Sigma, this action is not homotopic to any smooth action. This is partly joint work with Mladen Bestvina and Tom Church.

It is well know that for p = 2, the K(1)-localization of KO is EO_1,
and for p = 2; 3, the K(2)-localization of TMF is
EO_2. When does the K(n)-localization of TAF contain a factor of EO_n?
We will provide a complete answer. This is joint work with Mike
Hopkins.

Abstract: A Bers slice is a naturally embedded copy of the Teichmuller
space in the SL(2,C) character variety of a surface. We prove that Bers
slices are never algebraic. A corollary is that Thurston's skinning map
is never constant. The proof involves the theory of complex projective
structures and a little algebraic geometry.

In recent work of Baas-Dundas-Richter-Rognes, the authors prove that the classifying space of 2-vector bundles, K(Vect) is equivalent to the algebraic K-theory of the connective K-theory spectrum ku. In this talk we will show that K(Vect) is the group completion of the classifying space of the 2-category of 2-vector spaces, which is a symmetric monoidal 2-category. We will explain how to use the symmetric monoidal structure to produce a $\Gamma$-2-category, which will give an infinite loop space structure on K(Vect). Then we will show that the equivalence of BDRR is a map of infinite loop spaces.

Abstract
The "colored Tverberg problem" asks for a smallest size of the color
classes in a (d+1)-colored point set C in R^d that forces
the existence of an intersecting family of r "rainbow" simplices with
disjoint, multicolored vertex sets from C. Using equivariant topology
applied to a modified problem, we prove the optimal lower bound
conjectured by Barany and Larman (1992) for the case of partition into
r parts, if r+1 is a prime.
The modified problem has a "unifying" Tverberg-Vrecica type
generalization, which implies Tverberg's theorem as well as the ham
sandwich theorem.
This is joint work with Pavle V. Blagojevic and Gunter M. Ziegler.

The Torelli group is the subgroup of the mapping class group of a surface
which acts trivially on the surface's first homology group. Despite the
pioneering work of Birman, Johnson, and many others numerous basic
questions about it remain open. I will begin by describing some history
and background, and then I will discuss a new (infinite) presentation of
the Torelli group whose generators and relations have simple topological
interpretations.

In joint work with Adam Clay, we establish a necessary condition that an automorphism of an orderable group can preserve an
ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications will be given to knot theory and to the fundamental groups of fibred spaces. An example: if surgery on a fibred knot in $S^3$ (or in a homology 3-sphere) produces a 3-manifold whose fundamental group is orderable, then the surgery must be longitudinal (0-framed) and the Alexander polynomial of the knot must have a positive real root.

The braid group has a left-invariant total ordering, called the Dehornoy ordering. In this talk, I introduce the Dehornoy floor of braids, which measures a complexity of braids by using the Dehornoy ordering.

I will give a new lower bound of knot genus by using the Dehornoy floor.

I also discuss other applications of the Dehornoy floor to knot theory.

Abstract: I will describe a new operad (the "splicing operad") that acts
on a fairly broad class of embedding spaces. Previously I constructed an
action of the operad of little (j+1)-cubes on the space of framed long
embeddings of R^j in R^n. This operad action can be seen an extension of
the cubes action that allows for a general type of splicing operation. The
space of long embeddings of R into R^3 was described as a free 2-cubes
object over the subspace of prime long knots. With respect to the
splicing operad, long knots in R^3 are again free, but rather than being
free on the prime long knot subspace, the generating subspace is the (much
smaller) torus and hyperbolic knot subspace. Moreover, the splicing
operad has a particularly pleasant homotopy-type from the point of view of
its structure maps.

Abstract: I discuss a scheme of fault-tolerant quantum computation which is driven by local projective measurements on an entangled quantum state of many qubits, a so-called cluster state. There are two
fundamentally different ways of evolving quantum states, namely unitary evolution and projective measurement. Both can be used to realize quantum computation. The approach discussed in this talk uses the
latter. The constructions involved in making cluster state quantum computation robust against decoherence (=quantum-mechanical error) are in large part topological. In particular, Z_2 relative homology plays an
important role.

I begin with a short introduction to quantum computation, and explain the notions of "universality" and "fault-tolerance" in quantum computation. A very brief introduction to the field of quantum error-correction
will be included. Then I turn to the main subject of my talk, cluster state quantum computation. After a brief discussion its universality, I will turn to the question of how to make cluster state quantum computation
fault-tolerant. At that point, elements of topology will come into the picture.

Abstract: I will talk about work with Dan Isaksen on the motivic
homotopy groups of spheres, focusing on the story of the Hopf
maps. In classical algebraic topology the Hopf maps generate a
very small and easily computed subring of the stable homotopy ring.
The motivic Hopf maps generate a larger ring, and we don't yet
know explicity what it is. In the talk I'll describe some of what we do know.

ABSTRACT: Let f be the obvious covering map from Euclidean n-space to the n-torus. It is well known that if L is any straight line in n-space, then the closure of f(L) is a very nice submanifold of the n-torus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is "unipotent." We will discuss the statement of Ratner's Theorem, and a few of its important consequences. Topological and geometric aspects will be emphasized, while algebraic technicalities will be pushed to the background.

We will discuss our motivation for understanding what the analogue of DeRham
cohomology should be for E_{\infty}-algebras in connection with algebraic
K-theory and topological cyclic homology

The calculus of functors provides a framework for analyzing
functors of spaces, and more generally, model categories, in
terms of polynomial-like approximations to the functors. In this
talk, I will give an introduction to the calculus of functors
and discuss recent work with Kristine Bauer and Randy McCarthy
aimed at using calculus to better understand DeRham cohomology
in new contexts.

In the 1990's, Greg Arone gave a description of the Snaith splitting of spaces of the form $\Omega^m \Sigma^m X$. His method extended to give a kind of functorial filtration of any space of the form Maps(K, X) where K is a finite complex. In good cases, this leads to a stable splitting of these mapping spaces. We will describe Arone's result with an eye toward applying this to other mapping space functors.

Abstract: This will be a mostly expository talk on the new 3-manifold invariant, Heegaard-Floer homology, developed in the last decade in a series of remarkable papers by Ozsvath and Szabo. Emphasis will be on the remarkable applications of the theory.

Abstract: In this talk, we'll use classic definitions from calculus of real variables to motivate what it means to be a polynomial functor of degree n, the construction of the derivative of a functor and the basics of the `Taylor Series' of a functor as well as do a few simple calculations.

Abstract: Two G-spaces X and Y are said to be hG-equivalent if their Borel constructions are equivalent over BG. In this talk we introduce an obstruction which determines when a G-sphere X is hG-equivalent to a finite G-sphere Y, at a prime p dividing the order of G. We will also discuss how to generalize this to a global finiteness obstruction, and, in the case of G-spheres, relate the finiteness obstruction to the dimension function of a G-sphere.

A Galois descent theorem for n-types will be displayed and explained. This result is used, together with an appropriate version of the homotopy theory of pro objects, to give a descent criterion for diagrams of spaces which are defined on the etale site of a field. The need for such a criterion first arose in connection with the algebraic K-theory of fields.

Gedrich and Gruenberg(1987)and Ikenaga(1984)studied the algebraic
invariants silpZG,the supremum of the injective lengths of the projective
ZG-modules,and spli,the supremum of the projective lengths of the
injective ZG-modules,in connection with the existence of complete
cohomological functors for the group G and showed that these are related
to the virtual cohomological dimension of G. We will show that silpZG and
spliZG are related to the Gorenstein dimension of ZG and to the existence
of a finite dimensional model for E_G, the classifying space for proper
actions

In these talks we describe our proofs of the Hanna Neumann Conjecture. This
conjecture of the late 1950's can be described both as a problem in group
theory or as one in graph theory. Our first proof is longer and interprets the
problem using homology of sheaves on graphs; our second proof is very short and
uses only simple graph theory, but was inspired from the type of induction used
in our first proof. Both proofs demonstrate the strengthened form of the
conjecture formulated by Walter Neumann, and both proofs use earlier resolved
cases of the conjecture.

A crucial idea of the proof is to express the seemingly awkward notion of
"reduced cyclicity" of a graph in simpler terms, involving limits over covering
maps. This "limit homology theory" may be of independent interest, and is
related to the Atiyah Conjecture; our theory requires some curious linear
algebra that also may be of independent interest.

Our talks will not assume any previous knowledge of sheaf theory.

In these talks we describe our proofs of the Hanna Neumann Conjecture. This
conjecture of the late 1950's can be described both as a problem in group
theory or as one in graph theory. Our first proof is longer and interprets the
problem using homology of sheaves on graphs; our second proof is very short and
uses only simple graph theory, but was inspired from the type of induction used
in our first proof. Both proofs demonstrate the strengthened form of the
conjecture formulated by Walter Neumann, and both proofs use earlier resolved
cases of the conjecture.

A crucial idea of the proof is to express the seemingly awkward notion of
"reduced cyclicity" of a graph in simpler terms, involving limits over covering
maps. This "limit homology theory" may be of independent interest, and is
related to the Atiyah Conjecture; our theory requires some curious linear
algebra that also may be of independent interest.

Our talks will not assume any previous knowledge of sheaf theory.

In this talk we will describe the underlying construction of the main
theorem of Galatius' paper. This main theorem is a generalization of the
Pontryagin-Thom construction and establishes a weak homotopy equivalence
between the classifying space of a cobordism category and the infinite loop
space of the spectrum MTSO(n). We then compute the rational cohomology of
this infinite loop space and describe how the generalized MMM-classes come
from it.

The recent solution of M. Hill, M. Hopkins and D. Ravanel of the Kervaire invariant problem is likely to lead to a new area in the study of homotopy groups of spheres. In this talk I will explain that problem, and indicate what the above three authors have achieved. I’ll discuss the impact of their result on the study of vector fields and mention some problems which now open up in this “Post Kervaire-Milnor” era

We study the space M_g of isometry classes (or conformal equivalence classes) of smooth manifolds, diffeomorphic to #^g(S^d \times S^d), the connected sum of g copies of S^d \times S^d. For 2d=2, this is essentially the moduli space of Riemann surfaces. There is a variant M_{g,1} where we consider moduli of manifolds with an embedded D^{2d}; connected sum with S^d \times S^d gives a map M_{g,1} \to M_{g+1,1}, and we can form the direct limit M_{\infty,1}. The work of Madsen and Weiss on Mumford's conjecture determines the homology of M_{\infty,1} in the case 2d=2. We give a similar description of the homology of M_{\infty,1} in higher dimensions (2d \geq 6). This is joint work with Oscar Randal-Williams.

There is a general cohomology defined by Sweedler for
co-commutative Hopf algebras, generalizing the usual cohomology of a
group or a Lie algebra. Recently it was discovered that
low-dimensional groups could be defined without the co-commutativity
requirement. In joint work with Christian Kassel, we have given the
first few examples of computations with these, in the case of algebras
of functions on groups. These turn out to be related to torsors in
algebraic geometry, and Drinfeld twists in quantum groups theory

I am going to explain how representations of the braid
groups endowed with a compatible symplectic form give rise to link
invariants with values in the Witt ring of the field considered. The
construction makes use of Maslov indices. In the end, using the Burau
representation, we get one invariant which "contains" many others:
signatures, Jones metaplectic invariants, and a polynomial which is
almost the one by Alexander-Conway.

(Joint work with C. Procesi and M.Vergne) Let G be a compact
Lie group with Lie algebra g. Given a G-manifold M with a G equivari-
ant one form w we consider the zeroes M^{0} of the corresponding moment
map and define a map, called innitesimal index, of S[g*]^{G}-modules
from the equivariant cohomology of M^{0} with compact support to the
space of invariant distributions on g*.
In the case in which G is a torus, N is a linear complex representation
of G, M = T*N with tautological one form we are going to explain
how this is used to compute the equivariant cohomology of M^{0} with
compact support using certain spaces of polynomial which appear in
approximation theory.

To each finite group G one associates an algebraic invariant rk(G), called
the rank of G, and a topological invariant hrk(G), called the homotopy rank
of G. The number hrk(G) is defined in terms of free G-actions on products
of spheres. In this talk we define the two invariants and discuss the rank
conjecture, which states that rk(G)=hrk(G).

Multiplicative differential forms are relevant whenever considering an object with a smooth groupoid of symmetries. One can ask what is the corresponding infinitesimal object, and in fact some of the most important examples arise from this direction. The geometric structures of Hamiltonian mechanics - Poisson manifolds, Dirac structures, etc. - can be viewed as infinitesimal data which, when integrated, yield multiplicative 2-forms on Lie groupoids.

I will explore the relationship between multiplicative structures on Lie groupoids and their infinitesimal counterparts on Lie algebroids.

The goal of this talk is to describe the sheaf model for the infinite loop space of the Thom spectrum. Since the main theorem of Galatius, Madsen, Tillmann, Weiss is that this sheaf model is homotopy equivalent to the classifying space of the cobordism category understanding this sheaf model is an important step in proving the Mumford conjecture.

Toric schemes admit a combinatorial description which, in turn, allows
one to describe sheaves of modules by certain diagram categories. I
will explain these basic constructions in some detail, and then give a
non-standard approach to constructing the derived category of a
regular toric scheme; in technical language, the derived category will
appear as the homotpy category of a colocalisation of a simple diagram
category.

After talking about how a sheaf of categories gives rise to a topological category, I will finish the proof of the main theorem of Galatius, Madsen, Weiss and Tillmann which states that the classifying space of the cobordism category is weakly homotopy equivalent to the infinite loop space of the Thom spectrum.

In the first half of the talk we will provide a global picture of how all the pieces (that we constructed throughout this series) fit together to prove Mumford's conjecture following the approach in Galatius, Madsen, Tillman and Weiss. In the second half we outline the proof of the last piece of information needed to complete the proof.

In this talk we present an algebraic context for knot theory. Knotted
trivalent graphs (KTGs) along with standard operations defined on them
form a finitely presented algebraic structure which includes knots, and in
which many topological knot properties are defineable using simple
formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way
to study knots. We present a construction for such an invariant: the
starting point is extending the Kontsevich integral of knots to KTGs. This
was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki
in the late 90's using the theory of associators. We present an elementary
construction building on Kontsevich's original definition, and discuss the
homomorphic properties of the invariant, which, as it turns out,
intertwines all the standard KTG operations except for one, called the
edge unzip. We prove that in fact no universal finite type invariant of
KTGs can intertwine all the standard operations at once, and present an
alternative construction of the space of KTGs on which a homomorphic
universal finite type invariant exists. This space retains all the good
properties of the original KTGs: it is finitely presented, includes knots,
and is closely related to Drinfel'd associators. Partly joint work with
Dror Bar-Natan.

I will first give a gentle introduction to LS category and
then review some of the progress that has been made over the last 10
years in connection with Iwase's counterexample to Ganea's conjecture.
This will include many problems that are still open. At the end I will
describe some recent work with Rodriguez.

This is a first talk of a series about Hochschild (co)homology and cyclic (co)homology.
We will start with the basic definitions and later we will see some applications of these
theories to stringy topology

As part of Smale's program for smooth dynamics, David Ruelle gave a definition of a Smale space as (roughly) a topological dynamical system which has a local product structure of contracting and expanding directions for the dynamics. A special case is certain symbolic dynamical systems called shifts of finite type. In the late 1970's, Wolfgang Krieger, motivated by ideas from C*-algebra theory and K-theory, provided a beautiful algebraic invariant for shifts of finite type. The aim of this talk is to show how this invariant may be extended to the class of all Smale spaces as a kind of homology theory which provides a Lefschetz formula. Such a theory was conjectured by Bowen. (I will attempt to define and give examples of all the dynamical concepts: Smale space, shifts of finite type, etc.)

Let G be a compact connected Lie group act on a topological space X
in such a way that all isotropy subgroups are connected and of maximal
rank. In this talk we provide a criterion to determine when K_{G}^{*}(X) is
free over the representation ring R(G).

In this talk we will discuss properties of spaces of homomorphisms
Hom(Q,G) where Q is a discrete group and G a Lie group. The example
given by the ordered commuting n-tuples in a compact Lie group will be
explained in some detail. We will discuss how spaces of homomorphisms
and the descending central series of the free groups can be used to
construct a filtration of the classifying space BG. Homotopy properties
of these constructions will be given for finite groups, and cohomology
calculations provided for compact Lie groups. We will also describe
results on understanding both the number and stable homotopy type of
the components of related spaces of representations.

We consider the space L of long knots, i.e. of smooth embedding of the real line into a d-dimensional vector space R^d with a fixed behaviour at infinity. This space is almost the same as the space of all smooth embeddings of the circle into a d-dimensional sphere, Emb(S^1,S^d).
The fact that there is no knot in codimension > 2 implies that L is connected when d>3, but actually this space L is not contractible. In this talk I will explain how we can actually describe the homotopy type of that space. I will in particular explain that the rational homology of that space L is computable as the homology of some explicit combinatorial chain complex of generalized chord diagrams. Another way of saying it is that the Vassiliev spectral sequence computing the rational homology of L collapses at the E2-page. This fact is closely related to the fact that the
operad of little disks is formal.
Joint work with Victor Turchin and Ismar Volic

In the talk I will survey some results, techniques, and open problems, concerning finite group actions on (i) spheres, (ii) products of spheres and (iii) non-compact space forms modelled on indefinite quadratic forms.

I will introduce the concept of p-local compact groups, which
are combinatorial structures that generalize p-local ﬁnite groups and p-compact
groups, along with some examples. Then I will describe some partial results
towards showing the existence of embeddings of these objects in p-completed
classifying spaces of unitary groups.

Using a group of automorphisms of the free group F_2 isomorphic to the braid group B_3, we construct a non-abelian one-cocycle P: F_2 --> F_2, which turns out to take palindromic values and to be continuous for the profinite topology on F_2. We characterize the elements of the image of the map P and use B_3 to express the relation between elements having the same image. (Joint work with Christophe Reutenauer, UQAM.)

There is a growing body of work that supports a connection between L-spaces and 3-manifolds with non-left-orderable fundamental group, in fact a Seifert fibred manifold is an L-space if and only if its fundamental group is not left-orderable. In this talk I'll provide evidence for a connection that extends beyond the class of Seifert fibred manifolds, by showing that L-spaces behave similarly to non-left-orderability with respect to the operation of Dehn surgery on a manifold. It is with this goal in mind one is led to define a decayed knot; decayed knots have the property that sufficiently large surgery always yields a manifold with non-left-orderable fundamental group. Moreover, cables of decayed knots are also decayed, as long as the ratio of the cabling coefficients is chosen to be large enough. I'll show how both of these properties mirror the behaviour of knots which admit L-space surgeries, and outline some questions for future research. This is joint work with Liam Watson.

We describe some recent results about isometry groups of aspherical Riemannian manifolds, and also isometry groups of their universal covers. For instance, we show that on an irreducible locally symmetric space of dimension > 2, no metric has more symmetry than the locally symmetric metric.

We prove that every closed, smooth manifold of at least dimension 3 admits a sequence of Riemannian metrics with pinched curvature, volume tending to infinity but whose first eigenvalue of the Laplacian remains bounded away from 0. As a consequence we construct sequences of hyperbolic knots whose complements have again volume tending to infinity and whose Cheeger constant is uniformly bounded away from 0. This is joint work with Marc Lackenby.

It seemed quite difficult to determined crossing number of a link until Jones discovered a new polynomial invariant. Important results about crossing number of a link were shown after Kauffman gave a method for calculating Jones polynomial.

I will survey the relation between the reduced degree of Jones polynomial of a link and its crossing number and discuss Jones polynomial of a pretzel link. I will introduce an adequate diagram and discuss minimality of a diagram from the viewpoint of it. I will discuss the difference between the reduced degrees of Jones polynomials of pretzel links and their crossing numbers, and whether Jones polynomial is a complete invariant on alternating pretzel knots.

In this talk I will prove that in the presence of bounds for the rank of the fundamental group and the injectivity radius, the kth eigenvalue of the Laplacian of a closed hyperbolic 3-manifold M is bounded from above and below by a multiple of vol(M)^{-2}.

The Torelli group is the subgroup of the mapping class group which acts trivially on the homology of the surface. It is the first term of the Johnson filtration, the sequence of subgroups which act trivially on the surface group modulo some term of its lower central series. We prove that the abstract commensurator of each of these subgroups is the full mapping class group. This is joint work with Martin Bridson and Juan Souto.

I will try to give a short survey of some of the major results in the study of Kleinian groups in recent years. We will concentrate on the proof of the Bers' Density Conjecture and I will try to give an outline of the proof based on a joint work with J. Souto.

Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected and of maximal rank. Let T be a maximal torus with Weyl group W. We derive conditions on the induced action of W on the fixed-point set of T which imply that the equivariant K-theory of X is a free module over the representation ring of G. This can be applied to compute the equivariant K-theory of spaces of ordered commuting elements in certain compact Lie groups. This is joint work with J.M.Gomez.

Diffeomorphisms that preserve a symplectic structure have unexpected rigidity properties. In particular, many manifold have subsets that cannot be displaced (i.e. moved to a disjoint position) by a symplectic isotopy though they can be smoothly displaced. Toric manifolds provide a good setting in which to study these questions because they have a purely combinatorial description.

This talk will describe some recent progress in understanding which toric fibers can be displaced. I will try to make the subject accessible to those who do not know toric or symplectic geometry.

Associated to a tree T in the boundary of Outer space is a symbolic dynamical system called the dual lamination of T, denoted L^2(T). We develop a two-part inductive procedure for studying L^2(T). One part is known: it is a slight generalization of the Rips machine as developed by Coulbois-Hilion; the other part is new: it is a generalization of the classical Rauzy-Veech induction. As an application we characterize trees T for which L^2(T) is minimal. As a further application we give a description of the Gromov boundary of the complex of free factors: it is the space of measure classes of arational trees. (Jointly with T. Coulbois and A. Hilion.)

I will present the ideas of a version of a theorem of Hedlund for compact laminations (or foliations). More precisely: If L is a compact minimal Riemannian lamination by surfaces of negative curvature, we give a sufficient condition for the horocycle flow on its unit tangent bundle to be minimal, in other words every orbit of the flow is dense.

The study of finite-sheeted covering spaces of 3-manifolds has been invigorated in recent years by the resolution of several long-standing conjectures by Kahn-Markovic, Agol and Wise. In this talk, I will discuss how using this work one can reformulate some of the central open questions in the field in terms of objects called solenoids. These objects are formed by taking inverse limits of families of finite-sheeted covering spaces of a compact manifold M, and they can be thought of as pro-finite analogues of covering spaces of M. While such an object can in general be quite complicated, I will show in this talk that if M is a compact aspherical 3-manifold, then the solenoid given by taking the inverse limit of the family of all finite-sheeted connected covering spaces of M looks like a disk from the perspective of Cech cohomology with coefficients in any finite module. I will then talk about the relevance of this result to elementary questions about finite-sheeted covers.

The "group" a knot in 3-space is by definition the fundamental group of its complement; it is one of the oldest algebraic tools used to study knots. Only recently was it discovered that all knot groups can be endowed with a left-invariant ordering. Some even have two-sided invariant orderings, while others do not. This talk will discuss the current state of the art on this subject, and why it is interesting. It will be accessible to grad students.

Stratified flops show up in the birational geometry of symplectic varieties such as moduli spaces of sheaves. Varieties related by such flops are often derived equivalent (meaning that there is an equivalence between their derived categories of coherent sheaves). After recalling a bit about the geometry of flops I will discuss a general method for constructing such equivalences and illustrate with some examples and applications.

I will give a survey of results concerning the actions of a mapping class on the homology of various finite covers to which it lifts. I will draw connections to 3-manifold theory, especially largeness, growth of torsion homology and Alexander polynomials.

L-spaces are 3-manifolds with simplest possible Heegaard Floer homology. These arise naturally in many applications of Heegaard Floer theory, and as a result it has been asked if there is an alternate characterization of this class of 3-manifolds. A recent conjecture suggests the following: A 3-manifold is an L-space if and only if its fundamental group is not left-orderable. This talk will attempt to put this conjecture in context and describe some of the evidence supporting it.

Over the past few years there has been considerable activity in exploiting the power of algebraic topology to investigate areas outside of mathematics. The phrase 'applied algebraic topology'
is no longer an oxymoron!
Even more recently the intrinsically random nature of the world is beginning to bring statistical and probabilistic tools to these problems, leading to the birth
of a new area of 'random algebraic topology'.
In this talk I will discuss some of the few results in random algebraic topology, including the persistence homology of the sub-level sets of Gaussian processes
over manifolds, and limit theorems for the Betti numbers of random complexes built over random point processes.
Since this is to be a joint Probability/Topology seminar, I shall assume no prior knowledge in either area.

Given a lie group G and a finitely generated group P we can give a topology to the set of group homomorphisms Hom(P,G) as a subset of G^k. There has

been an increasing interest in understanding these spaces, and particularly their connected components, for their relevance in bundle theory. In particular when P=Z^k the space Hom(Z^k,G) is identified with the set of commuting k-tuples in G. We will present some of the generalities of these spaces and a possible systematic approach to their study. Then we will use that approach applied to the particular case when G=O(n), the group of orthogonal matrices, and compute the number of components of Hom(Z^k,O(n)).

I will talk on work in progress (with Albert Ruiz) on Kac-Moody groups over finite fields from a topological point of view, including some explicit cohomological computations (at non-characteristic primes) as well as some (conjectural) general properties.

Studying Gromov-Hausdorff limits of sequences of Riemannian manifolds (M_i) satisfying suitable conditions on their local geometry is an extremely fruitful idea. However, in the most interesting case that the diameter of M_i grows without bounds, one is forced to choose base points p_i\in M_i and consider limits of the pointed spaces (M_i,p_i) in the pointed Gromov-Hausdorff topology. The choice of the base points p_i influences enormously the obtained limits. Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs; this basically amounts to "choosing the base point by random". In this talk I will describe the distributional limits of sequences (M_i) of manifolds with uniformly pinched curvature and satisfying a certain condition of quasi-conformal nature. I will also explain how these results yield a modest extension of Benjamini's and Schramm's original result. This is joint work with Hossein Namazi and Pekka Pankka.

Let G be a finitely presented group. If the process of iteratively passing to vertex groups in a maximal graph of groups decomposition of G over finite subgroups, and then to vertex groups in maximal decompositions of the factors over two-ended subgroups, terminates, we say that G is strongly accessible. Delzant and Potyagailo argue that this process always terminates for certain types of splittings of finitely presented groups, in particular hyperbolic groups without two-torsion. I will give an example showing that their proof cannot be correct, and sketch a new proof that (relatively) hyperbolic groups without two-torsion are strongly accessible. This is joint work with N. Touikan.

In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the degree k rational cohomology of P\Sigma_n tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n -- that is, in some precise sense, the description of the decomposition of these cohomology groups into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.

Every transverse knot in a contact 3-manifold is represented as a closed braid in an open book. In this talk, based on a new technique called an open book foliation, we give a formula of self-linking number in terms of braids and open books. Surprisingly, our self-linking number formula essentially uses Johnson's homomorphism. This is a joint work with Keiko Kawamuro (Univ. Iowa).

A famous result by Steven Smale states that we can turn the sphere inside-out through immersions: this is called the eversion of the sphere. We will explain this result and the strategy of its proof which is a "cut-and-paste" strategy quite standard in algebraic topology. This approach allows us to understand globally the space of all immersions of a given manifold in another one, like the space of all immersion of the sphere in R^3 in the case of Smale's eversion. This theory has been enhanced by Goodwillie in the 1990's to understand spaces of embeddings. We will explain how this can be applied to understand spaces of knots, that is the spaces of all embeddings of a circle into a fixed euclidean space.

This is joint work with Srikanth Iyengar on results that connects work of Mike Hopkins in homotopy theory and commutative algebra with a theorem of Dave Benson, Jeremy Rickard and myself on group representations. I will spend most of the lecture talking about what the words mean and why we are interested in the results.

Note for Attendees

Please note coffee and cookies will be served beforehand at 2:45 p.m.

It was a result of Greenlees and Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality, which was proved by showing the contractibility of the corresponding Tate spectrum. In this series of two talks, I will explain the proof, discuss its generalization to quotient orbifolds and consequences with examples. Some background in equivariant stable homotopy theory will be given. If time permits, I will also explain why the duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks.

This is joint work with Tetsuya Ito. Fractional Dehn twist coefficient (FDTC), defined by Honda-Kazez-Matic, is an invariant of mapping classes. In this talk we study FDTC by using open book foliation method, then obtain results in topology, geometry, and contact geometry of the open-book-manifold of a mapping class.

It was a result of Greenlees and Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality, which was proved by showing the contractibility of the corresponding Tate spectrum. In this series of two talks, I will explain the proof, discuss its generalization to quotient orbifolds and consequences with examples. Some background in equivariant stable homotopy theory will be given. If time permits, I will also explain why the duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks.

Note for Attendees

The topology seminar is rescheduled to Tuesday for this week.

A closed geodesic on a surface can also be viewed as a closed orbit of the geodesic flow on the unit tangent bundle of the surface. In this talk I will discuss the main tool for studying the knot-properties of closed orbits of (three dimensional) flows. This tool is called a template, first defined and used by Birman and Williams to study the well known Lorenzbutterfly.

The theory of templates was first used for geodesic flows by Ghys. I will discuss questions related to his extraordinary result, that the closed geodesics on the modular surface equal the closed orbits on the Lorenz butterfly, and will discuss some generalization of his methods to other hyperbolic surfaces.

The talk will be an extended advertisement of a program to apply topological ideas to computation of essential dimension of groups.

The essential dimension of a group G measures to what extent every generically free action of G on an algebraic variety can be "compressed". For the symmetric group S_n, the essential dimension is directly related to the classical question how much an algebraic equation of degree n can be simplified by a rational change of variables.

I will introduce a topological approach to obtaining lower bounds on essential dimension. I will then survey some (non-topological) advances in theory of essential dimension and discuss some parallels in topology. Finally I will speculate on the possibility to relate results in the theory of topological group actions and results on essential dimension in a way that might benefit both fields.

Let G be a finite group, we consider the homotopy colimit of classifying spaces of abelian subgroups of G. This space is a K(\pi,1) for certain finite groups, but there are examples when the space has non-vanishing higher homotopy groups. I will also talk about the complex K-theory of this space and give some examples.

The study of central simple algebras over a field is a venerable topic in ring theory. There is a generalization of central simple algebras to schemes in the étale topology (in fact to arbitrary ringed sites) due to Grothendieck. The group of equivalence classes of Azumaya algebras over X is known as the Brauer group of X. By comparing the étale topology on a smooth complex variety X with the classical topology, we are able to use results from classical obstruction theory in order to obstruct the existence of certain Azumaya algebras. After giving an introduction to Azumaya algebras and the Brauer group, we shall present one such result, which furnishes lower bounds on the ranks of Azumaya algebras on spaces of low cohomological dimension.

We discuss the general problem of computing the cohomology and topological K-theory for classifying spaces of crystallographic groups. Integral computations will be provided for groups with prime order holonomy.

We discuss the general problem of computing the cohomology and topological K-theory for classifying spaces of crystallographic groups. Integral computations will be provided for groups with prime order holonomy.

Consider the moduli space M_{g,s} of Riemann surfaces of genus g with s marked points as an orbifold. In this talk I will determine all (non-constant) holomorphic maps M_{g,s}\to M_{g',s'} if g\ge 6 and g'\le 2g-2. This is joint work with Javier Aramayona.

Topological Complexity (TC) of a space is a concept motivated by the motion planning problem in Robotics. This turns out to be a homotopy invariant, closely related to Lusternik-Schnirelmann category (LS-cat). In fact, TC has proved more delicate than LS-cat given its relationship with some difficult problems in Algebraic Topology such as the Immersion Problem for Projective Spaces.

In this talk we will discuss basic properties and examples of TC. We will also discuss some recent progress on the computation of TC of some homogeneous spaces, and talk about some new techniques to compute TC based on Hopf-like invariants.

Consider the variety \Hom(\Gamma,G) where \Gamma is a finitely generated nilpotent group, and G a semisimple Lie group. I will discuss joint work with Juan Souto on homotopy retractions from this variety to the representation variety in K, a maximal compact subgroup of G.

The moduli space of Riemann surfaces M_g parametrizes bundles of genus g surfaces. A classical theorem of J. Harer implies that the homology H_k(M_g) is independent of g, as long as g is large compared to k. In joint work with Oscar Randal-Williams, we establish an analogue of this result for manifolds of higher dimension: The role of the genus g surface is played by the connected sum of g copies of S^n \times S^n.

Every knot can be presented on the union of finitely many half planes which have a common boundary line, so that each half plane contains a single arc of the knot. Such a presentation is called an arc presentation of the knot. The arc index of a knot is the minimal number of half planes needed in its arc presentations.

A grid diagram of a knot is a knot diagram constructed by finitely many vertical line segments and the same number of horizontal line segments such that at each crossing a vertical segment crosses over a horizontal segment. A grid diagram with n vertical segments is easily converted to an arc presentation on n half planes, and vice versa.

Grid diagrams are useful in several ways. A slight modification of a grid diagram gives a front projection of its Legendrian imbedding. Grid diagrams are used to compute Heegaard Floer homology and Khovanov homology.

In this work, we've tabulated prime knots by arc index up to arc index twelve. This is achieved by generating grid diagrams of prime knots up to arc index twelve. This tabulation contains all prime alternating knots up to ten crossings and all prime non alternating knots up to twelve crossings. The largest crossing number among prime knots with arc index twelve is twenty eight.

The notion of relative twisting of curves on a surface, the special case of subsurface projection to an annulus, is an important tool in the theory of mapping class groups. We develop an analogue for the outer automorphisms of a free group. This is joint work with Matt Clay.

Algebra and topology are old friends. Many topological problems are solved by applying algebraic methods. But sometimes the relationship can work the other way. My talk will discuss how the topological viewpoint can be used to establish the basic facts regarding orderability of groups.

For each of the sequences of groups in the title, the k-th rational cohomology is independent of n in a linear range n >= c*k, and this "stable cohomology" has been explicitly computed in each case. In contrast, very little is known about the unstable cohomology, which lies outside this range.

I will explain a conjecture on a new kind of stability in the unstable cohomology of these groups, in a range near the "top dimension" (the virtual cohomological dimension). For SL_n(Z) the conjecture implies that the unstable cohomology actually vanishes in that range. One key ingredient is a version of Poincare duality for these groups based on the topology of the curve complex and the algebra of modular symbols. Based on joint work with Benson Farb and Andrew Putman.

Symplectic topology can be thought as the mathematical versant of String theory: they were born independently at the same time, the second one as a fantastic enterprise to unify large-scale and low-scale physics, and the first one to solve classical dynamical problems on periodic orbits of physical problems, the famous Arnold conjectures. In the 80's, Gromov's revolutionary work opened a new perspective by presenting symplectic topology as an almost Kähler geometry (a concept that he defined), and constructing the corresponding theory which is entirely covariant (whereas algebraic geometry is entirely contravariant). A few years later, Floer and Hofer established the bridge between the two interpretations of Symplectic topology, the one as a dynamical theory and the one as a Kähler theory. This bridge was confirmed for the first time by Lalonde-McDuff who related explicitly the first theory to the second by showing that Gromov's Non-Squeezing Theorem is equivalent to Hofer's energy-capacity inequality. Nowadays, Symplectic Topology is a very vibrant subject, and there is perhaps no other subject that produces new and deep moduli spaces at such a pace ! More recent results will also be presented.

We apply the techniques of formal Demazure operators to obtain an algebraic model for the T-equivariant oriented cohomology of the variety of Borel subgroups of a linear algebraic group. This is a joint project with B. Calm\`es and C. Zhong.

Scalar curvature measures the asymptotic volume growth of small balls in Riemannian manifolds. In the case of positive scalar curvature the growth rate is smaller than in the flat, euclidean case. Typical examples are round spheres of dimension at least two.

We will discuss the interplay of analytic, geometric and topological methods for the investigation of manifolds of positive scalar curvature.

A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r ≤ k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non-simply connected spaces.

Homology stability for families of discrete groups such as the symmetric groups, linear groups, braid groups and mapping class groups are well-known. Extensions to diffeomorphism groups of manifolds more generally have only been proved recently in a few special cases. We will revisit some classical results on configuration spaces, extend them to the equivariant setting, and prove homology stability for so-called symmetric diffeomorphism groups for arbitrary manifolds.

We compute the integral cohomology ring of configuration spaces of two distinct points on a given real projective space. As an application, we obtain the symmetric topological complexity of real projective spaces of dimensions 5 and 6.

Let M be a connected, orientable, piecewise linear manifold of dimension n and let B be a closed submanifold of M. Let PL(M, B) be the group of orientation preserving PL homeomorphisms of M which are pointwise fixed on B. The group operation is composition of functions.

In joint work with Danny Calegari we show that if B has codimension zero or one, the group PL(M,B) is locally indicable. This means that every finitely-generated subgroup has the integers as a quotient. It follows that PL(M,B) is left-orderable and therefore has no elements of finite order.

I will discuss the topology of the space Hom(N,G) of homomorphisms from a finitely generated group N into a reductive complex linear algebraic group G (e.g. a special linear group). When K is a maximal compact subgroup of G (e.g. the subgroup of special unitary matrices), Hom(N,K) is a subspace of Hom(N,G). Although in general these topological spaces are quite different, I will show that when N is nilpotent there is a strong deformation retraction of Hom(N,G) onto Hom(N,K).

Theorem: If a finite group G acts freely and homologically trivially on a finite complex K which has the homotopy type of a product of k-spheres, then G acts freely and homologically trivially on a product of (k+1)-spheres.

Corollary: Any group acts freely and homologically trivially on a product of spheres.

Given a finite group G, homotopy colimit of the classifying spaces of its abelian subgroups capture information about the commutativity in the group. For the class of extraspecial 2-groups of rank greater than 2 these colimits are not of the homotopy type of a K(\pi,1) space. The main ingreadient in the proof is the calculation of the fundamental group. Another natural question is the complex K-theory of these homotopy colimits, which can be computed modulo torsion.

Budney recently constructed an operad which encodes splicing of knots and proved a theorem decomposing the space of (long) knots over this operad. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we have constructed a colored operad that encodes this infection operation. This operad captures all the relations in the 2-string link monoid. We can also show that a certain subspace of 2-string links is freely generated over a suboperad of our infection colored operad by its subspace of prime links.

Traditionally, we use mirror symmetry to map a difficult problem (A-model) to an easier problem (B-model). Recently, there is a great deal of activities in mathematics to understand the modularity properties of Gromov-Witten theory, a phenomenon suggested by BCOV almost twenty years ago. Mirror symmetry is again used in a crucial way. However, the new usage of mirror does not map a difficult problem to easy problem. Instead, we make both side of mirror symmetry to work together in a deep way. I will explain this interesting phenomenon in the talk. First we will give an overview of the entire story and then we will focus on the appearance of quasi-modularity.

In an Exposé published in 1968, Alexander Grothendieck generalized the notion of a central simple algebra over a field by defining Azumaya algebras in locally ringed topoi. Specific examples of Azumaya algebras in locally ringed topoi include (up to isomorphism) principal PUn or POn bundles on CW complexes, as well as Azumaya algebras over commutative rings. If the topos is connected, it is possible to define two invariants of an Azumaya algebra, the period & the index, which measure the nontriviality of the algebra. It is a classical theorem that the period & index have the same prime divisors in the case of central simple algebras over a field, and this is also known in the case of PUn bundles over CW complexes. In this talk, I will show that in the case of any locally ringed topos, the period & index have the same prime divisors.

In an Exposé published in 1968, Alexander Grothendieck generalized the notion of a central simple algebra over a field by defining Azumaya algebras in locally ringed topoi. Specific examples of Azumaya algebras in locally ringed topoi include (up to isomorphism) principal PUn or POn bundles on CW complexes, as well as Azumaya algebras over commutative rings. If the topos is connected, it is possible to define two invariants of an Azumaya algebra, the period & the index, which measure the nontriviality of the algebra. It is a classical theorem that the period & index have the same prime divisors in the case of central simple algebras over a field, and this is also known in the case of PUn bundles over CW complexes. In this talk, I will show that in the case of any locally ringed topos, the period & index have the same prime divisors.

The curve complex is a finite dimensional, locally infinite, unbounded and 17-hyperbolic simplicial
complex associated with surfaces. The intricate relationship between the curve complex of surfaces
embedded in 3-manifolds and the topology and geometry of the manifolds will be discussed in the talk.

The Masur criterion for Teichmuller geodesics relates the geometry of the Teichmuller space and random walks on the mapping class group of a surface to dynamical properties of vertical foliations of quadratic differentials. A major problem in the study of outer automorphism group OUT(F) of a nonabelian free group has been to find an analog for the Masur criterion. We discuss difficulties and explain our approach to this problem. We also mention applications of this result particularly in describing the space of random walks on the group of OUT(F). This is joint work with Alexandra Pettet and Patrick Reynolds.

This talk will discuss a version in A^1 homotopy theory of the classical EHP sequence of James and Toda using the simplicial suspension map. This is joint work in progress with Ben Williams.

There are many analogies between the outer automorphism group of a free group Out(F) and the mapping class group of a surface Mod(S). I'll explain how each of these groups contains many right-angled Artin subgroups and how these subgroups can be used to understand the structure of both Mod(S) and Out(F). Interestingly, attempting to understand the properties of elements in right-angled Artin subgroups also reveals some major differences between Out(F) and Mod(S). I'll explain these differences and how they affect the study of Out(F).

I will discuss some types of problems where techniques from algebraic topology have led to successful resolutions of open problems in algebraic geometry. Then, I will outline several future directions where better knowledge of the topology of classifying spaces of compact Lie groups could lead to more results in algebra.

In this talk we will define a space built out of all the commuting n-tuples in a Lie group and discuss its role as a classifying space for commutativity. Applied to the unitary groups this gives rise to an infinite loop space and the notion of commutative K-theory. We will also provide computations for the rational cohomology in terms of mult-symmetric invariants. This is joint work with Jose Gomez.

The unit spheres in orthogonal representations of finite groups give examples of group actions on spheres. We investigate non-linear actions by studying chain complexes over the orbit category, and constructing finite G-CW complexes. This leads to new examples of homotopy representations with isotropy of rank one. This project is joint with Ergun Yalcin (Bilkent University, Ankara).

I will describe a project to classify all smooth 4-dimensional manifolds triangulable with 6 or less 4-dimensional simplices. In the process we have found many simple triangulated 2-knot exteriors, forming a strong analogy with 3-manifold theory.

We discuss a result which shows that every right-angled Artin group quasi-isometrically embeds in a planar pure braid group. As a consequence, we obtain examples of quasi-isometrically embedded closed hyperbolic manifold subgroups of pure braid groups in all dimensions. We also give some applications to decision problems in braid group theory. This represents joint work with Sang-hyun Kim.

The classification of p-compact groups was a great achievement towards the understanding of finite H-spaces. In this talk we will talk about the history of finite H-spaces beginning from compact Lie groups. Then we will introduce the notions of p-compact group and finite p-local H-space and give some examples. We will finish with some open questions in the area.

Laminations and free group actions on R-trees are useful tools for studying the structure of elements of and subgroups of outer automorphisms groups of free groups, denoted Out(F). After introducing laminations and R-trees in the context of Out(F), we will mention some recent progress on various topics related to Out(F).

For a free group F, there is a natural action of Out(F) on the space of representations Hom(F,PSL(2,C)). Minsky introduced an open subset PS(F) (for primitive stable representations) on which Out(F) acts properly discontinuously which is expected to be maximal. I will explain the definition of this set PS(F) and introduce examples and properties of primitive stable representations with discrete image.

Discovered (or invented?) by Richard Laver in the 1990s, the tables that are now known as Laver tables are finite structures obeying the self-distributivity law x(yz)=(xy)(xz). Although their construction is totally explicit, some of their combinatorial properties are (so far) established only using unprovable set theoretical axioms, a quite unusual and paradoxical situation. We shall explain the construction of Laver tables, their connection with set theory, and their potential applications in low-dimensional topology via the recent computation of some associated cocycles.

Analogous to string topology of manifolds, string topology of classifying spaces studies the rich algebraic structure admitted by the homology groups of free loop spaces of classifying spaces of compact Lie groups. In this talk, I will discuss my recent joint work with Richard Hepworth where we extend the previously available structure in string topology of classifying spaces into a novel kind of field theory which includes operations parameterized by homology groups of automorphism groups of free groups with boundaries in addition to operations parameterized by homology groups of mapping class groups of surfaces. This work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds.
Preprint: http://arxiv.org/abs/1308.6169

Chern classes of complex manifolds play a key role in geometry and topology. In this talk we shall discuss how these classes extend to singular varieties. In fact there are various possible extensions, depending on which properties of Chern classes you want to preserve. This is closely related to asking who plays the role of the tangent bundle at the singular points (where there is no tangent bundle).

In this talk I will mention some conjectures about group rings (Idempotent conjecture, unit conjecture) and mention their stable versions. Those involve algebraic K-theory. I will explain how the Farrell-Jones conjecture implies those stable versions. I will then finish with the status of the Farrell-Jones conjecture. The cheapest way to prove it for a certain group is to use its inheritance properties, but this also touches very interesting questions in (geometric) group theory.

In the beginning of this talk I will use the Farrell-Jones conjecture to express the K-theory of R[Z^2] in Terms of the K-theory of R. Geometric conditions on a Group that imply the conjecture will be mentioned . The class of Groups for which the conjecture is known is quite large I will define it and mention some interesting open cases.

Let G be a Lie group. The space Hom(Z^n,G) of commuting n-tuples in G was extensively studied over the last few years. Two related generalizations of this space are the space of almost commuting tuples, in the sense that the commutator of every pair of elements in each tuple is in the center of G, and the space of homomorphisms Hom(Gamma, G) from a central extension Gamma of a free abelian group by a free abelian group to G.

In this talk, I will describe the structures of these spaces and the relations between them in the case G=U(m). I will also discuss questions such as the number of path components and the rational homotopy type of these spaces.

Classical knot groups, that is fundamental groups of knot complements in 3-space, are known to be torsion-free. However, we show that for many knots, their groups contain generalized torsion: a nontrivial element such that some product of conjugates of that element equals the identity. One example (the hyperbilic knot 5_2) was discovered with the aid of a Python program written by the USRA student Geoff Naylor. Other examples include torus knots, algebraic knots in the sense of Milnor (arising from singularities of complex curves) and satellites of knots whose groups contain generalized torsion. Although all knot groups are left-orderable, the existence of generalized torsion is an obstruction to their being bi-orderable.

While an aspherical complex is determined up to homotopy by its
fundamental group, there are many geometrically different aspherical
manifolds with the same fundamental group. For instance, the punctured
torus and the pair of pants look quite different, but both have the same
fundamental group F_2. I will discuss constructions of aspherical
manifolds for a given fundamental group, talk about the smallest dimension
of such a manifold for a given group and describe some geometric
invariants that distinguish different aspherical manifolds with the same
fundamental group.
I will discuss this for right angled Artin groups (joint work with Mike
Davis, Boris Okun and Kevin Schreve) and possibly also for duality groups.

Differentiable stacks are generalizations of smooth manifolds suitable for modelling poor quotients, such as quotients by non-free Lie group actions. In this talk, we will define differentiable stacks and explain how they can also be used to model the leaf space of a foliation. In the following week, we will explain some recent results of ours about a nice subclass of differentiable stacks, called etale differentiable stacks, and explain some applications to foliation theory.

We will introduce infinity-topoi as generalized topological spaces, and show how using this language unifies the notion of manifold with that of etale differentiable stacks (generalized orbifolds) and their higher-categorical analogues. We will then give a completely categorical description of etale stacks in terms of a representability theorem. This theorem gives a recipe for constructing moduli stacks of geometric structures, and we will explain some examples of how this produces moduli-stacks presented by Lie groupoids that have been well studied in the foliation theory literature. Finally, we will explain how a generalization of Segal's theorem follows which describes the homotopy type of certain classifying spaces, and will explain the connection to the classification of foliations with transverse structures.

The formalism of six operations encodes the functorial behavior of (co)homology theories. It was first introduced by Grothendieck for the l-adic cohomology of schemes, and was later developed in a variety of other geometric contexts: D-modules on schemes, spectra parametrized by topological spaces, motivic spectra parametrized by schemes, etc. Equivariant homotopy theory is also best understood as a formalism of six operations for topological stacks.
In this talk I will discuss the basics and the significance of this formalism, and I will then describe an extension of motivic homotopy theory to algebraic stacks.

We are interested in classifying all finite groups which can act on a finite CW-complex homotopy equivalent to a sphere, such that all isotropy subgroups are rank one groups, i.e., they do not include Z/pxZ/p for any prime p. The equivalent question for free actions (all isotropy subgroups are trivial) has been answered completely by the works of P.A. Smith and R. Swan. For actions with rank one isotropy, we give a list of group theoretical conditions which guarantee the existence of such actions. Some of these conditions are necessary conditions depending on assumptions on fixed point subspaces. This is a joint work with Ian Hambleton.

This is a joint work with Susumu Hirose. We consider pseudo-Anosov elements of the mapping class groups on orientable surfaces. We are interested in a numerical invariant of pseudo-Anosovs, called the dilatation. The logarithm of the dilatation of a pseudo-Anosov mapping class is called the entropy. If we fix a surface, then the set of dilatations of pseudo-Anosovs defined on the surface is closed and discrete. In particular we can talk about a minimum of any subset of dilatations defined on the surface in question.

Penner proved that the minimal entropy of pseudo-Anosovs defined on a closed surface of genus g behaves like 1/g. Later Hironaka proved that the minimal entropy of pseudo-Anosovs in the handlebody subgroup on a closed surface of genus g also behaves like 1/g. We prove that the the minimal entropy of the hyperelliptic handlebody sugbroup of genus g has the same asymptotic behavior. (Our examples of pseudo-Anosovs improve the upper bound of the minimal entropy of the handlebody sugbroup given by Hironaka.) To do this, we study the spherical Hilden subgroup of the mapping class group defined on a sphere with 2n punctures, and we construct a sequence of pseudo-Anosovs with small dilatations in the spherical Hilden subgroups.

Given a hyperbolic surface S, consider any closed geodesic gamma on S. gamma is naturally embedded as a knot in the unit tangent bundle of S, and the complement of gamma is almost always a hyperbolic three manifold and thus has an intrinsic volume. In this talk I will describe a way to obtain an upper bound for this volume, linear with respect to the length of gamma. The proof goes through careful analysis of volumes for geodesics on the modular surface. This is joint work with Maxime Bergeron and Lior Silberman.

The existence of interesting multiplicative cohomology theories for orbifolds was first suggested by string theorists, and orbifold products have been intensely studied by mathematicians for the last fifteen years. My work with S. Scherotzke focuses on the virtual orbifold product introduced by Lupercio et al. (2007). We construct a categorification of the virtual orbifold product that leverages the geometry of derived loop stacks. By work of Ben-Zvi Francis Nadler, this reveals connections between virtual orbifold products and Drinfeld centers of monoidal categories, thus answering a question of Hinich.

Finite cubical complexes are abstract models for parallel processing systems. The vertices of a complex K are the states of the system, and the execution paths are morphisms of the corresponding path category P(K).

The theory of path categories and path 2-categories for finite oriented cubical and simplicial complexes will be reviewed. There is an algorithm for computing the path category P(K) of a finite complex K which is based on its path 2-category. This 2-category algorithm will be displayed, and complexity reduction methods for the algorithm will be discussed.

The 2-category algorithm works well only for toy examples. The size of the path category P(K) of a complex K can be an exponential function of the size of K. The algorithm has so far resisted parallelization.

One wants combinatorial local to global methods for addressing examples that are effectively infinite.

We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.

(This is a continuation of the talk of 23 Septemer) We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.

I will discuss classification results for subgroups of Out(Fn) (analogous to Ivanov's classification of subgroups of mapping class groups of surfaces), and more generally of automorphism groups of free products. In particular, I will present a version of the Tits alternative for the automorphism group of a free product. This is partly joint work with Vincent Guirardel.

The Hopf map eta is nilpotent in the stable homotopy groups of spheres. This is not so for the motivic Hopf map, considered as an element of the motivic stable homotopy groups of spheres. This suggests that the eta-local part of motivic stable homotopy theory is an interesting object of study. We will describe this for the base fields C and R.

In Balmer's framework of tensor triangular geometry, the prime thick tensor ideals in a tensor triangulated category C form a space which admits a continuous map to the Zariski spectrum Spec^h(End_u(1)) of homogeneous prime ideals in the graded endomorphism ring of the unit object. (Here the grading is induced by an element u of the Picard group of C.) If C is the stable motivic homotopy category and u is the punctured affine line, then this endomorphism ring is the Milnor-Witt K-theory ring of the base field. I will describe work by my student, Riley Thornton, which completely determines the homogeneous Zariski spectrum of Milnor-Witt K-theory in terms of the orderings on the base field. I will then comment on work in progress which uses the structure of this spectrum to study the thick subcategories of the stable motivic homotopy category.

A stable spherical fibration is classified by a map X → BGL₁(S) and Lewis showed that if this map is an infinite loop map or an n-fold loop map then the Thom spectrum is an E_∞- or E_n-ring spectrum, respectively. Ando, Blumberg, Hopkins, Gepner and Rezk introduced a new approach to Thom spectra using the language of ∞-categories. Using their approach, we will explain how to apply some simple (∞-)category theory to study multiplicative structures on Thom spectra, proving a generalization of Lewis's theorem and moreover characterizing the ring structure by a universal property. As an application I'll discuss a new (slightly simpler) proof of a remarkable theorem of Mahowald's realizing the Eilenberg-MacLane spectrum HF₂ as a Thom spectrum of a double loop map.

Braids represent mapping classes of the punctured disk,and hence braids induce automorphisms of the fundamental group of the punctured disk, i.e, automorphisms of the free groups. It is known that the free groups are bi-orderable. We consider which braid preserves some bi-ordering of the free group. Once we know a given braid preserves some biordering of the free group, the fundamental group of the mapping torus by the braid monodromy is bi-orderable. By using a criterion by Perron-Rolfsen together with a technique on the disk twists, we give new examples of links in the 3-sphere whose fundamental groups of the link exteriors are bi-orderable, for example, the Whitehead link, the minimally twisted 4- and 5- chain links. We also give an infinite sequence of pseudo-Anosov braids which do not preserve any bi-orderings of the free groups. As a corollary, it follows that the fundamental group of the Whitehead sister link (i.e, (-2,3,8)-pretzel link) exterior is not bi-orderable.

In these talks we review basic facts about finite group actions and how they can be extended using homotopical methods. In the second talk we will describe some joint work with J.Grodal on constructing exotic group actions for certain rank two finite groups.

In these talks we review basic facts about finite group actions and how they can be extended using homotopical methods. In the second talk we will describe some joint work with J.Grodal on constructing exotic group actions for certain rank two finite groups.

In this talk I will give a introduction in a very useful tool – Controlled Algebra.

Controlled Algebra is a way of building new additive categories with better properties out of given ones. One application is the Definition of negative K-theory. Another application is a description of K-theoretic assembly maps as maps induced by additive functors.

As explained last week, Controlled Algebra is a way of building new additive categories with better properties out of given ones. This time we will use it to describe K-theoretic assembly maps as maps induced by additive functors.

Note for Attendees

This week the seminar is moved to 4:15pm, to allow people to attend the Colloquium.

In this talk I will give some relations between spaces of homomorphisms when the target group G is a real linear algebraic group, through homotopy stable decompositions of simplicial spaces. To obtain a simplicial space Hom(L,G) out of spaces of homomorphisms we think of L, a (suitable) family of finitely generated groups, as a cosimplicial group.

Also, if G=U, the colimit of the unitary groups U(m), I will show when the geometric realization of Hom(L,U) has an "E-infinity-ring-space" structure.

We will introduce basic notions of quantum mechanics, mostly employed in condensed matter physics, such as a separable Hilbert space, Bloch's theorem and Fermi surfaces. Then we will describe the problem of stability of Fermi surfaces and relate it to the mathematical concepts of Fredholm operators and homotopy classes. Equipped with these concepts we show that our proposed scheme yields a classification of topologically stable Fermi surfaces by K^{-1}(X), where X is the Brillouin zone and K^{-1} is a well known functor in K-theory. We will show an explicit example when X = S^{1}, known as the spectral flow and its relation to quantum anomalies. This is work in progress joint with Alejandro Adem and Gordon W. Semenoff.

Anyone can have a theorem but only a select few mathematicians have a Calculus named after them! One such is Tom Goodwillie who constructed a theory analogous to differential calculus for functors from spaces to spaces. I will give an introduction to the Goodwillie Calculus and explain why the identity functor has such an interesting "Taylor series".

Let C be a triangulated category. The Grothendieck group K_0(C) is defined as the abelian group generated by symbols [X], where X is an object of C, moduli the relation [X] = [X'] + [X''] for every exact triangle X' -> X -> X'' in C.
A simple consequence of this relation is that the double suspension functor X -> X[2] induces the identity map from K_0(C) to itself. In this talk, I will explain how this observation can be seen as the shadow of a certain rotation-invariance phenomenon in algebraic K-theory, and describe the connection of this phenomenon with the theory of "topological Fukaya categories" introduced by Dyckerhoff and Kapranov.

The classical EHP sequence is a partial answer to the question of how far the unit map of the loop-suspension adjunction fails to be a weak equivalence. It can be used to move information from stable to unstable homotopy theory. I will explain why there is an EHP sequence in A^{1} algebraic topology, and some implications this has for the unstable A^{1} homotopy groups of spheres.

For any non-negative integer n there exist n-cusped hyperbolic 3-manifolds of minimal possible volume. They are sometimes not unique. For example there are exactly two distinct minimal 1-cusped examples: the figure eight complement, and another which is not a knot complement. Similarly there are distinct 2-cusped examples. I will show how these examples differ in terms of properties of their fundamental groups. In particular, in the pairs of examples in the 1 or 2 cusped case, one has bi-orderable fundamental group while the other’s group is not orderable. This is a preliminary announcement of work in progress with Eiko Kin (Osaka).

The cohomology of certain p-adic Lie groups plays a central role in chromatic homotopy theory. In this talk I will discuss these groups and discuss some of their elementary group theoretical structure. Then I will explain some recent group cohomology calculations relevant for stable homotopy theory. In particular I will highlight the result of some highly non-trivial calculation resulting in a surprising isomorphism which cries out for a conceptual explanation.

The classifying functor from categories to topological spaces provides a way of constructing spaces with certain properties or structure from categories with similar properties of structure. An important example of this is the construction of infinite loop spaces from symmetric monoidal categories. The particular kinds of extra structure can typically be encoded by monads on the category of small categories. In order to provide more flexibility in the kinds of morphisms allowed, one can work with the associated 2-monad in the 2-category of categories, functors, and natural transformations. In this talk I will give the categorical setup required, and I will give examples of interest to homotopy theorists. I will also outline how this method of working can give general statements about strictifications and comparisons of homotopy theories. This is partially based on work with two different sets of collaborators: Nick Gurski, Niles Johnson, and Marc Stephan; Bert Guillou, Peter May, and Mona Merling.

Using some ideas of Atiyah and Segal and a pushforward map defined using deformation groupoids we explain how to endow to the twisted geometric K-homology groups of a discrete group with an external product. Using the Baum-Connes assembly maps one can transfer this product to the twisted K-theory groups of the reduced group C*-algebra. This is a joint work with Noé Barcenas and Paulo Carrillo.

Mixing Voevodsky's filtration in motivic homotopy and Dugger's in C_2-equivariant homotopy theory leads to an interesting filtration on the C_2-equivariant motivic homotopy category. In this talk, I'll introduce these slice filtrations and talk about some joint work with P. A. Ostvaer, where we compute the resulting zero slice of the equivariant motivic sphere spectrum.

In this talk, we will discuss a new proof of the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber for semi-simple perverse sheaves of geometric origin on complex algebraic varieties. This proof follows from rather formal considerations of higher algebra, stable motivic homotopy theory and Grothendieck's six functors, avoiding both the positive-characteristic methods of the original proof, and the delicate analysis of degenerations of mixed Hodge structures involved in M. Saito's proof.

Carlsson conjectured that if a finite complex admits a free action by an elementary abelian p-group of rank n, then the sum of its mod-p Betti numbers is at least 2^n. For the prime p=2, he reduced the conjecture to an algebraic problem which he solved for low n. In this talk, we will retrace Carlsson's journey through homological and commutative algebra. The following week, I will report on joint work in progress with Jeremiah Heller with the goal of extending Carlsson's methods to all primes.

Carlsson conjectured that if a finite complex admits a free action by an elementary abelian p-group of rank n, then the sum of its mod-p Betti numbers is at least 2^n. For the prime p=2, he reduced the conjecture to an algebraic problem which he solved for low n. In this talk, I will report on joint work in progress with Jeremiah Heller with the goal of extending Carlsson's methods to all primes. The crucial ingredient is a new notion of Koszul p-complexes.

For this survey talk I will bring a bestiary of algebraic structures that are often less well known than they deserve to be. As I will explain, these are all related to symmetries and topology, and they have interesting symmetries of their own. This observation leads to new homology computations, and those will be the results that I will present along the way. Part of this work is joint with N. Wahl.

The "nonpositive immersion" property is a condition on a 2-complex X that generalizes being a surface. When X has this property, itsfundamental group appears to have has some very nice properties which I will discuss. I will spend the remainder of the talk outlining a proof that the nonpositive immersion property holds for a 2-complex obtained by attaching a single 2-cell to a graph. This was proven recently with Joseph Helfer and also independently by Lars Louder and Henry Wilton.

Given a fiber sequence of n-fold loop spaces X-->Y-->Z, and morphism of n-fold loop spaces Y-->Pic(R) for R an E_{n+1}-ring spectrum, we describe a method of producing a new morphism of (n-1)-fold loop spaces Z-->Pic(MX), where MX is the Thom spectrum associated to the composition X-->Y-->Pic(R). This new morphism has associated Thom spectrum MY, but constructed directly as an MX-module. In particular this induces a relative Thom isomorphism for MY over MX: MY⊗_{MX} MY = MY⊗Z. We will see a rough description of this construction as well as many examples allowing us to find equivalent forms of relative smash products of spectra like MString, MSpin, HZ/2, X(n) and many others. We also describe a way to use this construction to identify certain obstructions to giving a complex orientation on an associative ring spectrum.

Associated to a fibration E --> B with homotopy finite fiber is a stable wrong way map LB --> LE of free loop spaces coming from the transfer map in THH. This transfer is defined under the same hypotheses as the Becker-Gottlieb transfer, but on different objects. I will use duality in bicategories to explain why the THH transfer contains the Becker-Gottlieb transfer as a direct summand. The corresponding result for the A-theory transfer may then be deduced as a corollary. When the fibration is a smooth fiber bundle, the same methods give a three step description of the THH transfer in terms of explicit geometry over the free loop space. (Joint work with C. Malkiewich)

An ordered group (G,<) is a group G together with a strict total ordering < of its elements which is invariant under left- and right-multiplication. If such an ordering exists for a group, the group is said to be orderable. It is easy to see that if G and H are orderable, then so is their direct product. In 1949, A. A. Vinogradov proved that if G and H are orderable groups, then the free product G*H is also orderable. I’ll show that such an ordering can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms, using an algebraic trick due to G. Bergman. This was motivated by a certain question in the theory of the braid groups B_n and the Artin representation of B_n in the automorphism group Aut(F_n) of a free group.

Topology is full of ineffective arguments constructing objects and equivalences by algebra.

One of the great early achievements of algebraic topology was the work of Thom, followed by Milnor and Wall, on cobordism theory, which describes when a compact smooth (oriented) manifold is the boundary of some compact manifold with boundary. This method is typical of the problems that arise in the use of algebraic methods and is an early example of one of the dominant philosophies of geometric topology. The question we study is to what extent the complexity of a manifold can be used to bound, when it exists, the minimum necessary complexity of something that it bounds.

The goal of this talk is to explain generally some of the issues of making topology less ineffective.

We shall show that there are polynomial size nullcobordisms in a suitable sense. This is joint work with Greg Chambers, Dominic Dotterer and Fedor Manin.

We present mod-two cohomology of both symmetric and alternating groups as (almost) Hopf rings.

As is being seen in subjects such as representation stability, in these settings one gains substantial insight by considering all cases together, binding them with some structure. For these group cohomologies, that structure is a transfer or induction product, akin to taking external tensor product of S_n and S_m representations and inducing up to S_{n+m}. This product along with cup product and a standard coproduct together define a ring object in the category of coalgebras in the setting of symmetric groups, and are close enough to such an object for practical purposes in the setting of symmetric groups.

Settings such as general linear groups over finite fields now beg for investigation, as do a number of questions internal to topology (e.g. Margolis homology, working towards Morava K-theory) and at the interface with algebra (e.g. how do analogous structures interface with current understanding of modular representation theory of symmetric groups).

We will aim for the talk to be accessible to algebraists. While topology will be mentioned throughout, it will be presented in a supporting role, familiar to topologists but treatable as a black box when necessary to algebraists.

In recent years, important families of symmetric group representations have come to be better understood through the perspective of representation stability, a viewpoint introduced and developed by Thomas Church, Jordan Ellenberg, and Benson Farb, among others. A fundamental example of representation stability is the S_n-module structure for the $i$-th cohomology of the configuration space of n distinct, labeled points in the plane, or more generally in a connected, orientable manifold, as $i$ is held fixed and n grows. For the plane, this translates to Whitney homology of the partition lattice via an S_n-equivariant version of the Goresky-MacPherson formula. This talk will survey the combinatorial literature regarding the partition lattice and discuss what new things this can tell us about representation stability for configuration spaces. In particular, we deduce new, sharp stability bounds and representation theoretic structure through a combination of symmetric function technology and poset topology. This is a joint work with Vic Reiner.

A classical approach to understanding spaces of homomorphisms is to describe its connected components. We mainly focus in the spaces Hom(N_{n,q},SU(2)), where N_{n,q} denotes the free q-nilpotent group on n-generators. We show that the connected components arising from non-commuting q-nilpotent n-tuples in SU(2) are homeomorphic to RP^3 and we give the exact number of these. We prove it by showing a seemingly unknown result about SU(2) that states: all non-abelian nilpotent subgroups are conjugated to the quaternion group Q_8 or to the generalized quaternion groups Q_{2^q}, of order 2^q. Some applications of this result are the stable homotopy type of Hom(N_{n,q},SU(2)); a homotopy description of the classifying spaces B(q,SU(2)) of transitionally q-nilpotent principal SU(2)-bundles, and its derived versions for SO(3) and U(2). If time permits I'll also show some cohomology calculations for the spaces B(r,Q_{2^q}) for low values of r.
This is joint work with Omar Antolín Camarena.

A group is left-orderable if it has a strict total ordering that is invariant under multiplication from the left. For countable groups, this is equivalent to acting on the real line by order-preserving homeomorphisms. A group being circularly orderable has a slightly trickier algebraic definition than left-orderability, but in the countable case boils down, as expected, to the existence of a orientation-preserving action by homeomorphisms on the circle.

The set of all left-orderings of a group forms a topological space, and similarly, so does the set of all circular orderings. I will provide an introduction to these spaces, and discuss recent progress towards understanding the structure of groups whose spaces of circular orderings are “degenerate”, in the sense that they consist simply of a finite set of points with the discrete topology. This is joint work with Cristobal Rivas and Kathryn Mann.

I will discuss finite group actions on integral or rational homology 3-spheres. The main examples for this talk are the Brieskorn integral homology 3-spheres M(p,q,r) arising from isolated singularities, which bound smooth 4-manifolds with definite intersection forms. In addition, there are special infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. We ask whether the free periodic actions on Brieskorn spheres extend to smooth actions with isolated fixed points on one of these associated 4-manifolds.

The fact that |xy| = |x| |y| for real, complex, quaternion and octonion numbers x and y leads to the open question of finding all possible maps f: R^r X R^s ---> R^n satisfying |f(x,y)| = |x| |y| for each x in R^r and y in R^s. For example, that no such f can exist when r=s=n=16 was a celebrated result in classical algebra.
When the components of f(x,y) are required to be bilinear forms in the components of x and y with integer coefficients, one can crudely encode f by a colored matrix M of r rows and s columns, using n colors but avoiding certain "forbidden configurations". In 1981, S. Yuzvinsky conjectured that the chromatic number for this kind of coloring of M should be given by a certain function of r and s already familiar to topologists. In this talk I shall prove his conjecture for a majority of values of r and s, including the case of square matrices r=s. I shall explain how each step in my combinatorial proof was indeed suggested by topological considerations.
This talk is dedicated to the memory of professor Erhard Luft (1933-2017).

The Goodwillie derivatives of the identity functor on pointed spaces form an operad in spectra that is very closely related to the Lie operad. I will describe the mod 2 homology operations for algebras over this operad. This talk will not assume prior knowledge of either operads or Goodwillie calculus. Sadly, due to time constraints, it also won't explain any Goodwillie calculus! (Instead I will start from a combinatorial description of the derivatives of the identity functor.)

A saturated fusion system associated to a finite group G encodes the p-structure of the group as the Sylow p-subgroup enriched with additional conjugation. The fusion system contains just the right amount of algebraic information to for instance reconstruct the p-completion of BG, but not BG itself. Abstract saturated fusion systems F without ambient groups exist, and these have (p-completed) classifying spaces BF as well.

In a joint project with Tomer Schlank and Nat Stapleton, we combine the theory of abstract fusion systems with the work by Hopkins-Kuhn-Ravenel and Stapleton on transchromatic character maps, and we generalize several results from finite groups to fusion systems.

A main ingredient of this project is studying the free loop spaces L(BG) and L(BF) for groups and fusion systems, and constructing transfer maps from L(BG) to L(BH) when H is a subgroup of G.

Simplicial commutative rings are one of the first steps into "derived" rings that one can take. Many constructions for general E_infty-ring spectra or even Z-algebras are simpler in the world ofsimplicial commutative rings; however, from a purelyhomotopy-theoretic or categorical picture they are slightlymysterious. I will explain ongoing work with Bhargav Bhatt on anextended theory of "generalized rings" which extends this category toallow nonconnective objects. Many "equational" constructions whichcannot work with E_infty-rings extend well to generalized rings.

The study of spaces of homomorphisms from a discrete group to a compact Lie group has led to the definition of a new cohomology theory, called commutative K-theory. This theory, which was first introduced by Adem and Gomez, is a refinement of classical topological K-theory. It is defined using vector bundles which can be represented by commuting cocycles. I will begin the talk by discussing some general properties of the "classifying space for commutativity in a Lie group". Specialising to the unitary groups, I will show that the classifying space for commutative complex K-theory is precisely the E-Infinity ring space underlying the ku-group ring of BU(1). If time permits, I will mention some results about the real variant of commutative K-theory.

Hopf invariants, a basic construction in homotopy theory, are closely related to Lusternik–Schnirelmann category which, in turn, can be defined as the sectional category of a certain evaluation map. In this talk I'll introduce the notion of Hopf invariants for general fibrations and exhibit a connection between the Hopf invariants for a product fibration and those for the factors. Applications will be drawn to the motion planning problem.

I will explain some new techniques for computing the homology of braid groups with coefficients in a certain class of exponential representations that arise in a natural way from braided monoidal categories. Surprisingly (at least to me), these techniques are related to fundamental objects — Nichols algebras — in the theory of quantum groups and the classification theory of Hopf algebras. These techniques can be used to establish part of a function field analogue of Malle’s conjecture on the distribution of Galois groups. I will not discuss this application much in the topology seminar, but will focus on it in the colloquium. This is joint work with Jordan Ellenberg and TriThang Tran.

The local zeta function of a variety X over a finite field F_q is defined to be Z(X,t) = \exp\sum_{n > 0}\frac{|X(F_{q^n})|}{n}. This invariant depends only on the point counts of X over extensions of F_q. We discuss how Z(X,t) can be considered as a group homomorphism of K-groups and show how to lift it to a map between K-theory spectra.

The mod pSteenrod algebra is the (Hopf) algebra of stable operations on mod pcohomology, and in part measures the subtle behavior of p-local homotopy theory(as opposed rational homotopy theory, which is much simpler).A classical theorem of Dold-Thom tells us that the infinitesymmetric power of the n-dimensional sphere is the Eilenberg-Maclanespace K(Z, n),and one can use an appropriate modification of thisconstruction to compute the dual Steenrod algebra. The infinitesymmetric power of the sphere spectrum has a filtration whosek-thcofiber miraculously turns out to be the Steinbergsummand(from modular representation theory of GL_k(F_p)) of theclassifying space of (Z/p)^k. This opens the door for slickcomputations - for example, the Milnorindecomposables canbe picked out as explicit cells.

In this talk, I will introduce the concepts and resultschronologically. I will also include hands-on homotopy theory computations as time permits.

In episode 2 of the series, I will turn my attention to the setting of G-equivariant stable homotopy theory, where G is an abelianp-group. Analogous to the classical case, we can use symmetric powers of the equivariant sphere to filter H\underline{\F}_p, and the cofibers are Steinbergsummands of equivariant classifying spaces. We then study howthe cells of these spaces split after smashing with H\underline{\F}_pin the case G=C_p. When p=2, the result is a decomposition of H\underline{\F}_2 \sm H\underline{\F}_2 whose generators correspond torepresentation spheres, while at odd primes, we see something more unusual.

Basic questions in analytic number theory concern the density of one set in another (e.g. square-free integers in all integers). Motivated by Weil's number field/function field dictionary, we introduce a topological analogue measuring the “homological density” of one space in another. In arithmetic, Euler products can be used to show that many seemingly different densities coincide in the limit. By combining methods from manifold topology and algebraic combinatorics, we discover analogous coincidences for limiting homological densities arising from spaces of 0-cycles (e.g. configuration spaces of points) on smooth manifolds and complex varieties. We do not yet understand why these topological coincidences occur. This is joint work with Benson Farb and Melanie Wood.

In the 1970s, work of Adams, Baird, Bousfield, and Ravenel gave a description of the orders of the KU[1/2]-local stable homotopy groups of spheres as the denominators of special values of the Riemann zeta-function. Meanwhile, Lichtenbaum conjectured a formula, ultimately proven 30 years later as a consequence of the Iwasawa main theorem and the norm residue theorem, relating the orders of the algebraic K-groups of totally real number rings to special values of their Dedekind zeta-functions. In this talk I will describe two general approaches, an analytic approach and an algebraic approach, to a general kind of number theory that arises in any tensor triangulated category: this is a general framework for the above results and gneralizations of them, and which aims to describe the orders of Bousfield-localized stable homotopy groups of finite spectra in terms of special values of L-functions. Then I'll show off some new results in this framework, in particular, a "universal" description of the KU-local homotopy groups of the Moore spectrum S/p in terms of L-values, and as a consequence, a proof of a certain (infinite) family of cases of Leopoldt's conjecture, by counting orders of homotopy groups.

The algebraic K-theory, due to Quillen, of a field is related to a theory defined by Milnor called Milnor K-theory and denoted K^M. In the 1980s, Andrei Suslin constructed a map K_n(F) -> K^M_n(F), and conjectured that the image was the subgroup (n-1)! K^M_n(F). He also proved the conjecture for n<=3. For n=5, we reinterpret the construction as a construction in the A1 homotopy groups of spheres and BGL, and by calculating these groups, show that the conjecture is true in this case as well. This represents part of a joint project with Aravind Asok, Jean Fasel and Kirsten Wickelgren.

In this talk I will show how to use parametrized Morse theory to construct a map from the infinite loopspace of certain Thom spectrum, MTSpin(d), into the space of positive scalar curvature metrics on a closed spin manifold of dimension d > 4. My main novel construction is a cobordism category consisting of cobordisms equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. My first result identifies the homotopy type of the classifying space of this topological category with the infinite loopspace of another Thom spectrum related that is related to MTSpin(d), and built out of the space of Morse jets on Euclidean space. The result can viewed as an analogue of the well known theorem of Galatius, Madsen, Tillmann, and Weiss, for manifolds equipped with the extra geometric structure of a choice of admissible Morse function.

In the second part of the talk I will show how to use this cobordism category to probe the homotopy type of the space of positive scalar curvature metrics, R^{+}(M), on a closed, spin manifold M when dim(M) > 4. This uses a parametrized version of the Gromov-Lawson construction developed by Walsh and Chernysh. Our main result detects many non-trivial homotopy groups in the space of positive scalar curvature metrics R^{+}(M). It in particular gives an alternative proof and extension of a recent breakthrough theorem of Botvinnik, Ebert, and Randal-Williams.

The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. I'll set up this machinery and describe some of the applications that follow. This is joint work with Jonathan Hanselman and Jake Rasmussen.

The Torelli subgroups of mapping class groups are fundamental objects in low-dimensional topology, through some basic questions about their structure remain open. In this talk I will describe these groups, and how to use tools from representation theory to establish patterns their homology. This project is joint with Jeremy Miller and Peter Patzt. These "representation stability" results are an application of advances in a general algebraic framework for studying sequences of group representations.

Equivariant formality, a notion in equivariant topology introduced by Goresky-Kottwitz-Macpherson, is a desirable property of spaces with group actions. Examples of equivariantly formal spaces include compact symplectic manifolds equipped with Hamiltonian compact Lie group actions and projective varieties equipped with linear algebraic torus actions. Less is known about compact homogeneous spaces G/K equipped with the isotropy action of K, which is not necessarily of maximal rank. In this talk we will review previous attempts of characterizing equivariant formality of G/K, and present our recent results on this problem using an analogue of equivariant formality in K-theory. Part of the work presented in this talk is joint with Jeffrey Carlson.

One of the ways we understand Teichmuller space, endowed with the Teichmuller metric, is by studying Teichmuller discs. They exist in great abundance: There is a Teichmuller disc through any point and in any direction. Typically, their projection to moduli space is dense, and yet infinitely often their projection is a closed subvariety of moduli space called a Teichmuller curve. Recently, in joint work with Eskin, McMullen, and Mukamel, we discovered the first non-trivial examples of higher dimensional analogues of Teichmuller discs, namely totally geodesic submanifolds.

In this talk, we will explain that such higher dimensional totally geodesic submanifolds are much more rigid and rare than Teichmuller discs: Each must cover a closed subvariety of moduli space, and only finitely many such subvarieties exist in each moduli space. This result is an application of joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. One of the goals of the talk will be to explain what this cocycle is and why it lies at the heart of Teichmuller dynamics.

In this talk I will explain how to endow the moduli space of objects in certain wrapped Fukaya categories with a cluster structure. This will be achieved by counts of holomorphic disks between Lagrangians which are described by trivalent graphs. In particular, we will be recovering the Fock-Goncharov coordinates on the moduli space of flat connections on a surface and provide a symplectic interpretation for the non-Abelianization procedure via spectral networks.

Given a group G the space of n-tuples of pairwise commuting elements can be assembled into a subspace Bcom(G) of the classifying space BG. This space classifies certain types of principal bundles, but unlike BG it is not always aspherical for discrete groups.

The universal cover can be studied using posets, and when G is an extraspecial p-group, I will show that it has the homotopy type of a wedge of spheres. Extraspecial groups appear as the basic observables in quantum computation, and I will also talk about some recent applications of Bcom(G).

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres. This is joint work with I. Hambleton.

Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C_2, the cyclic group of order two. In this talk I will present a structure theorem for RO(C_2)-graded cohomology with constant Z/2 coefficients that substantially simplifies computations. The structure theorem says the cohomology of any finite C_2-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will give some examples and sketch the proof, which depends on a Toda bracket calculation.

We show that Lubin--Tate spectra at the prime 2 are Real oriented and Real Landweber exact. The proof is by application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for E_n with its C_2-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these C_2-fixed points.

The symmetry group of a knot in the three-sphere is the mapping class group of the knot’s exterior. Elements of order two with fixed point set meeting the boundary of the knot exterior are called strong inversions, and a pair (K,h) is called a strongly invertible knot when h is a strong inversion in the symmetry group of K. Studying the equivalence of strongly invertible knots amounts to studying conjugacy classes of strong inversions. I will discuss how to construct invariants of strongly invertible knots using Khovanov homology. Some of this is joint work with Mike Snape and, time permitting, I will also discuss work in progress with Andrew Lobb.

Contextuality is a property of quantum mechanics that sets it apart from classical physics. Recently, it has been established as a necessary ingredient that any quantum computation must have in order to provide a speedup over conventional classical computation [1], [2]. It has thus become a resource.

In my talk, I will first review the notion of quantum contextuality, and then explain how it is a useful commodity in quantum computation---for the models of quantum computation with magic states and measurement-based quantum computation (MBQC).

I’ll end with a cohomological picture underlying MBQC and contextuality [3], which is a focus of current research in my group.

[1] M. Howard et al., Nature (London) 510, 351 (2014).

[2] R. Raussendorf, Phys. Rev. A 88, 022322 (2013).

[3] C. Okay et al., Quantum Information and Computation 17, 1135-1166 (2017).

We start with a general introduction to pro-objects in topological and geometric settings, with descriptions of well-known examples, including Postnikov towers, absolute Galois groups, and etale homotopy types.

This is followed by a brief discussion of homotopy theories of pro-objects.

Homotopy theoretic techniques are applied to give the first correct comparison of finite descent and Galois cohomological descent for simplicial presheaves on the etale site of a field. Examples arise in the study of algebraic K-theory with finite coefficients over a field.

Any cobordism spectrum M associated to a classifying space BG is naturally equipped with a so-called Thom diagonal M->M ^ BG+ and Thom isomorphism M ^ M -> M ^ BG+. In the homotopy category, these maps give M the structure of a cotorsor for the coalgebra BG+. I'll describe how this structure can be lifted to the derived setting by proving a more general theorem about derived or "stacky" quotients of En-rings in ∞-categories. This theorem states that, given an action by an n-fold loop space G on an En-ring R, the stacky or derived quotient R//G is naturally a BG+-cotorsor (the fact that this specializes to a result for cobordism spectra follows from the work of Ando, Blumberg, Gepner, Hopkins and Rezk on Thom spectra and orientations). This result can be reinterpreted in the setting of noncommutative spectral algebraic geometry as saying that Spec(R//G) is a principal Spec(BG+)-bundle (or torsor) over Spec(R) in the category of affine En-varieties. If there is time, I'll speculate wildly about applications to geometry and noncommutative algebra.

Note for Attendees

The time of this seminar has changed. It now starts at 2:45, and precedes PIMS tea (which is now at a later time).

We will describe a geometric interpretation of Khovanov homology as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-dimensional sphere. The main ingredient is a construction which translates Khovanov (or Bar-Natan) invariant of a 4-ended tangle to an immersed curve. It is inspired by a result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted cobordism algebra) into the Fukaya category of the 4-punctured sphere. The main tool we will use is a category of peculiar modules, introduced by Zibrowius, which is a model for the Fukaya category of 2-sphere with 4 discs removed. This is joint work with Liam Watson and Claudius Zibrowius.

Church-Ellenberg-Farb introduced the theory of FI-modules to explain the phenomenon of representation stability of the cohomology of configuration spaces. I will explain the basics of how this story goes, and then explain how to extend their analysis to two generalized types of configuration spaces. Furthermore, I will explain how the Grothendieck-Lefschetz formula connects these topological stability phenomena to stabilization of statistics for polynomials and rational maps over finite fields.

The Heegaard Floer theory of a 4-ended tangle takes the form of an immersed curve (with possibly non-trivial local system) on the boundary of the tangle minus the tangle ends. The Heegaard Floer homology of a link can be computed as the Lagrangian intersection theory of the Heegaard Floer homologies of two 4-ended tangles obtained by splitting the link along an embedded 2-sphere.

I will outline the construction of the tangle invariant, with particular focus on the action of the mapping class group of the 4-punctured sphere. I will then discuss the current state of symmetry properties for this invariant in the light of the mutation conjecture.

Given two homotopy equivalent manifolds with different dimensions, it is natural to ask if the smaller one embeds in the larger one. We will discuss this problem in the case of four-manifolds homotopy equivalent to surfaces.

The main goal of this talk is to introduce transitionally commutative principal G-bundles and to show that they can be classified homotopically using a space B_comG that is called the classifying space for commutativity in G. In the second half of the talk I provide some examples of bundles that are trivial as principal G-bundles but not as transitionally commutative bundles.

The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll use branched surfaces to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. No background in Heegaard-Floer or foliation theories will be assumed.

Algebraic K-theory is an invariant defined on categories that records how object in the category are related by exact sequences --- it is a homotopical version of the classical Euler characteristic. However, there are many categories of interest that do not have exact sequences, but instead have cutting and pasting operations. For example, the category of varieties or the category of polytopes. I'll describe how to define a higher algebraic K-theory for categories like this, and show that it's not so different from the case of more algebraic categories. Even better, theorems like Quillen's Devissage and Localization can be proved internal to these structures. Time permitting, I'll describe how the cutting and pasting of polytopes is intimately related to the weight filtration on the algebraic K-theory of fields.

A Smith inequality refers to a rank inequality between the homology of a space with a G action and the homology of its fixed set. In the case of G = Z/2, I will discuss an analog of this statement for the Knot Floer homology of periodic knots, including a conjectural filtered refinement. These inequalities appear to give new restrictions on the Alexander polynomials of periodic alternating and periodic L-space knots.

To a 3-manifold with torus boundary, we can associate an element of the Fukaya category of the punctured torus—that is, a collection of immersed curves in the torus, decorated with local systems—such that when two such manifolds are glued the Heegaard Floer homology of resulting 3-manifold is recovered from Floer homology of the corresponding curves. These curves are a reformulation of the bordered Heegaard Floer defined by Lipshitz, Ozsvath, and Thurston, but their geometric nature makes them more user friendly. We will discuss some properties of these curves and some applications, including invariance of Heegaard Floer homology under genus one mutation and a rank inequality for Heegaard Floer homology of toroidal manifolds. This is joint work with J. Rasmussen and L. Watson.

In early 90's, there are two mathematical theories motivated by physics, Verlinde theory counting the number of generalized theta function on a curve and quantum cohomology counting the number of holomorphic maps. Early explicit computation shows that the level l GL_n Verlinde algebra is isomorphic to quantum cohomology ring of Grassmanian G(n, n+l). In 1993, Witten gave a conceptual explanation of this isomorphism, by proposing an equivalence between the quantum field theories which govern the level-l GL Verlinde algebra and the quantum cohomology of the Grassmannian. His physical derivation of the equivalence naturally leads to a mathematical problem: these two objects are conceptually isomorphic (without referring to detailed computations). In the talk, I will explain a K-theoretic version of Witten's conjecture motivated by our observation that Verlinde invariants is a K-theoretic invariant and hence should be compared to the quantum K-theory of Grassmanian. The main content of the talk is a reformation of the conjecture as the consequence of wall-crossing problem of GLSM and the resolution of the conjecture in rank 2. This is a joint work with Ming Zhang.

In joint work with Hans-Joachim Baues, we study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of Baues on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.

Using bordered Floer theory, we construct an invariant for 3-orbifolds with singular set a knot that generalizes the hat flavor of Heegaard Floer homology. We show that for a large class of 3-orbifolds the orbifold invariant behaves like HF-hat in that the orbifold invariant, together with a relative Z_2-grading, categorifies the order of H_1^orb. When the 3-orbifold arises as Dehn surgery on an integer-framed knot in S^3, we use the {-1,0,1}-valued knot invariant epsilon to determine the relationship between the orbifold invariant and HF-hat of the 3-manifold underlying the 3-orbifold.

It is an open problem whether there exists an exotic smooth structure on S^1 × S^3. Motivated by this problem, Akbulut asked if there is a simply connected homology cobordism from a homology sphere with non-trivial Rochlin invariant to itself. The main progress toward this question was made by Taubes who used his celebrated technique of gauge theory on manifolds with periodic ends to give a negative answer to this question for various homology spheres including the Poincare homology sphere. In this talk, I will present a different technique which can be used to generalize Taubes' result in various directions. In particular, we address Akbulut's question for a family of Brieskorn homology spheres, which is not covered by Taubes' theorem.

This is a joint work with G.Garkusha. In the talk I will discuss the constructionof fibrant replacements for spectra consisting of Thom spaces (suspension spectra of varieties and algebraic cobordism MGL being the motivating examples) that uses the theory of framed correspondences. As a consequence we get a description of the infinite loop space of MGL in terms of Hilbert schemes.

We introduce the notion of a spoke algebra, which encodes the data of a C_p-equivariant cohomology equipped with a coherent system of norms. This is a generalization of the notion of an ``\mathbb{E}_{\sigma}-algebra" for the group C_2 and the sign representation \sigma. We explain several naturally occurring examples (such as certain mapping spaces) and then describe several applications. The first is a method for delooping a suitably structured space by an irreducible, 2-dimensional C_p-representation in two steps. The second is the construction of certain interesting C_p-spectra related to the odd primary Kervaire invariant problem using a version of Koszul duality for spoke algebras. This is joint work with Jeremy Hahn.

The deformation space of a rational map is a submanifold of Teichmueller space parameterizing maps that are combinatorially and dynamically equivalent. S. Koch and I recently showed that this space need not be connected, and can have infinitely many components. In this talk we describe some tools from algebraic geometry, and the topology of coverings that reveal the obstructions to connectedness, and give some evidence that connectedness may be achieved by passing to augmented Teichmueller space.

The Fukaya category of a Weinstein manifolds W algebraically package all Lagrangian submanifolds of W into an A-infinity category. In this talk I will motivate why studying Lagragian submanifolds in symplectic manifolds is relevant and then I will give an overview of the definition of wrapped Fukaya categories. I will explain a theorem stating that for Weinstein manifolds a particular finite collection of Lagrangians generates this category. Finally we will see some element of the proof and applications of this generation criterion. This is joint work with G. Dimitroglou-Rizell, P. Ghiggini and R. Golovko.

I will discuss joint work with Lior Silberman in which we describe the topology of spaces of representations of Baumslag-Solitar groups into complex reductive algebraic groups.

The Grassmannian manifold of k-planes in \mathbb{R}^n has a group action if \mathbb{R}^n is taken to be a real representation of the group. When the group is C_2, the Schubert cell construction of the Grassmannian generalizes to an equivariant representation-cell structure. However, this generalization is not unique; an identification of representation with \mathbb{R}^n must be chosen.

I am interested in computing the RO(C_2)-graded Bredon cohomology of these spaces. Although a theorem of Kronholm dictates that this must be free, determining the degrees of the generators is nontrivial. The ambiguity introduced by the choice mentioned above turns out to be an asset for this task. Using a computation by Dan Dugger of the cohomology of an infinite equivariant Grassmannian, and some theorems about equivariant flag manifolds, I will present a way to succeed in finding the cohomologies of several infinite families of finite-dimensional equivariant Grassmannians.

We shall give an overview of what Khovanov cohomology and its relatives have told us about knot concordance over the past 15 years. No knowledge required. Includes joint work with Lukas Lewark and Peter Feller.

I will describe how to use the recent theory of prismatic cohomology of Bhatt--Morrow--Scholze to say something about the de Rham cohomology of classifying spaces of algebraic groups such as PGL_p in characteristic p. This leads to interesting new phenomena, including new obstruction classes to period=index for p-torsion classes in characteristic p.

This is a preliminary report on an ongoing project with Jack Morava on the braid groups and categorification.

The braid groups can be seen as the fundamental group of the space of monic square-free complex polynomials.
I will discuss the role of Sylvester's discriminant to understand the strange fact that the braid groups abelianize
to the integers, while the subgroup of pure braids has a much larger abelianization.

When a group acts faithfully by orientation-preserving homeomorphisms on S^{^1}, one can sometimes use the action on S^{^1} to prove the existence of a faithful order-preserving action by homeomorphisms on the real line. This can be reworded in algebraic terms by using circular orderings and left-orderings of groups: a circularly orderable group may secretly admit a left-ordering, though the existence of such an ordering may not be apparent.

In this talk I will review some classical results that use cohomology and circular orderings of a group to detect the existence of left-orderings. I'll also present one new technique for determining when a circularly orderable group admits a left-ordering. As a bonus, when a group is circularly orderable but NOT left-orderable, our new approach determines a subset of the natural numbers that precisely encodes the obstruction to the existence of a left-ordering. This is joint work with Ty Ghaswala and Jason Bell.

There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity $\alpha \in [1,\infty)$.

In an article published in 2009, David Benson described, for a finite group G, the mod p homology of the space \Omega(BG^{\wedge}_p) -- the loop space of the p-completion of BG -- in purely algebraic terms. In joint work with Carlos Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. Among other things, we showed that when \mathcal{C} is a small category, |\mathcal{C}| its geometric realization, R a commutative ring, and |\mathcal{C}|_R^+ a plus construction of |\mathcal{C}| with respect to homology with coefficients in R, then H_*(\Omega(|\mathcal{C}|_R^+);R) is the homology of any chain complex of projective R\mathcal{C}-modules that satisfy certain conditions. Benson's theorem is then the special case where \mathcal{C} is the category associated to a finite group G and R=\mathbb{F}_p, so that p-completion appears as a special case of the plus construction.

Carlsson conjectured that if a finite CW complex admits a free action by an elementary abelian p-group G of rank n, then the sum of its mod-p Betti numbers is at least 2^n. In 2017, Iyengar and Walker constructed equivariant chain complexes that are counterexamples to an algebraic version of this conjecture. Their work raised the question if these chain complexes can be realized topologically by free G-spaces to produce counterexamples to Carlsson’s conjecture. In this talk, I will explain multiplicative properties of the spectral sequence obtained by filtering the mod-p cochains of a space with a free p-group action by powers of the augmentation ideal and deduce that the counterexamples can not be realized topologically. This is joint work with Henrik Rüping.

The classifying space of the nth projective unitary group, BPUn, is a fundamental mathematical object. However, unlike its close cousins BUn, BSUn, etc., surprisingly little has been known to its cohomology. I will briefly introduce the roles of BPUn in the study of the topological period-index problem and anomalies in theoretical physics, and present recent progresses on the computation of its cohomology.

In the context of algebraic geometry, BPGLn, the classifying stack of the projective linear group PGLn, is an analog of BPUn, while the theory of Chow rings plays a similar role as singular cohomology. I will point out where the above computation pass to the Chow ring of BPGLn.

These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. We describe recent progress in the study of such surfaces including smooth examples, the fake projective planes. We also discuss open questions on Montgomery-Yang problem.

Take n objects and put them in a container. What is the configuration space of all the ways they can fit in the container without intersecting? How does the topology of that configuration space change depending on the size of the objects and the size of the container? We will look at configurations of segments in a disk, of squares in a rectangle, and of disks in an infinite strip. In the latter two cases, the configuration space is homotopy equivalent to a polyhedral cell complex that can be studied combinatorially.

Note for Attendees

The Topology seminar has been moved to ESB 4127 (the Math Bio seminar is now in ESB 4133)

The classifying space for commutativity, denoted by B_\text{com} G, of a Lie group G is assembled from commuting tuples in G as a subspace of the usual classifying space BG. The resulting space classifies principal G-bundles whose transition functions generate an abelian subgroup of G whenever they are simultaneously defined. The relationship between the homotopy type of G and the space B_\text{com} G is much more interesting, and non-trivial compared to the case of BG. In this talk, I will present a work, joint with Ben Williams, where we study the mod-\ell homotopy type of B_\text{com} G at a prime \ell. The techniques involve a homotopy colimit decomposition over a topological category generalizing the construction of Adem-Gomez and application of results on mapping spaces between classifying spaces of compact Lie groups due to Dwyer-Wilkerson. We show that for a connected compact Lie group the mod-\ell homotopy type of B_\text{com}G depends on the mod-\ell homotopy type of BG.

Conway mutation is an operation on links that is notoriously difficult to detect: it preserves many link invariants such as the signature, the Alexander polynomial or, more generally, the HOMFLY polynomial. Baldwin and Levine conjectured that δ-graded link Floer homology also belongs in this list—despite the fact that *bigraded* link Floer homology can distinguish some mutant knots such as the famous Kinoshita-Terasaka and Conway knots.

In [arXiv:1909.04267], I proved Baldwin and Levine's conjecture by studying symmetry properties of peculiar modules, an immersed curve invariant of 4-ended tangles. In this talk, I will sketch this proof.

Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e is greater than k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards a couple of interesting conjectures which I'll discuss. This is joint work with Duncan McCoy.

In Section 11.5 of their book, Freedman and Quinn showed that an aspherical 4-manifold M with polycyclic fundamental group pi is determined by pi and the boundary of M. We revisit this result, with a shift in emphasis. We show that if Mis aspherical and pi is elementary amenable then pi is 1, ZZ, a Baumslag-Solitar group BS(1,m) or (torsion free) polycyclic of Hirsch length 3 or 4. We characterise the possible boundaries, except for the cases with pi=BS(1,m) for some m.

I will discuss how to see symmetries of an alternating knot from any reduced alternating diagram. As a consequence we will see that the quotient of an alternating periodic knot is alternating, and that any alternating knot which is p and q periodic for odd primes p and q is also p*q periodic (c.f. the trefoil which is 2 and 3 but not 6 periodic). The main tool for this is the study of flypes, but we will also mention an approach using spanning surfaces.

We will attempt, in a colloquium-like fashion, to introduce the basic concepts of quantum mechanics and functional analysis necessary to derive the connection between twisted equivariant K-theory of a d-1 dimensional torus and quasi-adiabatic stability of Fermi surfaces of non-interacting fermions on a crystalline half-solid with a gapped bulk condition. This is part of the speaker's PhD thesis and is an extension of joint work with A.Adem, O.A. Camarena and G.W. Semenoff.

‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.

A group is said to be torsion-free if it has no elements of finite order. An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}. In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is invariant under left-multiplication.. If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.

I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary. In the PL category, this group is left-orderable, but not bi-orderable, for all n>1. Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.

Lattice cohomology is a combinatorial invariant of isolated singularities of complex surfaces introduced by Ozsvath Stipsicz and Szabo, and independently by Nemethi. The construction, inspired by Heegaard Floer homology, was recently employed by Nemethi to prove that a complex surface singularity is rational if and only if the fundamental group of its link is left orderable. I will give an introduction to the topic and discuss a few conjectures/new results. Some of these speculations are ongoing work in collaboration with Irving Dai.

In this talk, I will discuss a surprising connection between finite groups of Lie type and string topology of classifying spaces of compact connected Lie groups recently discovered by Jesper Grodal and myself: the cohomology of a finite group of Lie type is a module over the cohomology of the free loop space of the classifying space of the corresponding compact Lie group when the latter cohomology groups are equipped with a string topological multiplication. This module structure provides in particular a new perspective towards the Tezuka conjecture asserting that under certain conditions, the cohomologies of the two objects are isomorphic.

To a finite group G and an integer q > 0 one can associate the coset-poset of nilpotent subgroups of G of nilpotency class less than q. The nerve of this poset is a simplicial complex denoted E(q,G). It is closely related to a filtration of the classifying space BG studied by Adem, Cohen and Torres-Giese. In this talk I will address a question of C. Okay which asks for the relationship between simple connectivity of E(q,G) and nilpotency of G. This is ongoing work with Bernardo Villarreal.

I will describe why the trivial knot S^{2}-->S^{4} has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S^{1} in S^{1}xS^{n}^{ }(for n>2), a group structure on the isotopy-classes of reducing discs of S^{1}xD^{n}, and the action of the diffeomorphism group Diff(S^{1}xS^{n}) on the embedding space Emb(S^{1}, S^{1}xS^{n}). Roughly speaking, these results say there is no direct translation from dimension 3 to 4, for the Hatcher-Ivanov theorems on spaces of incompressible surfaces. Or said another way, isotopy in dimension 4 is more closely analogous to isotopy in high dimensions.

Looming in the background of my project, like some spooky mountain, is the L-space conjecture. Let’s just say that L-spaces are homology Lens spaces (with respect to Heegaard Floer homology) and there is a difficult question concerning them. One way to get a handle on L-spaces is to construct them by doing Dehn surgery on knots. In this talk, I will explain what (1,1) knots are, how to compute their Heegaard Floer homology and I will present the simple method due to Greene, Lewallen and Vafaee for checking almost instantly whether a (1,1) knot admits a Dehn surgery to an L-space.

Looming in the background of my project, like some spooky mountain, is the L-space conjecture. Let’s just say that L-spaces are homology Lens spaces (with respect to Heegaard Floer homology) and there is a difficult question concerning them. One way to get a handle on L-spaces is to construct them by doing Dehn surgery on knots. In this talk, I will explain what (1,1) knots are, how to compute their Heegaard Floer homology and I will present the simple method due to Greene, Lewallen and Vafaee for checking almost instantly whether a (1,1) knot admits a Dehn surgery to an L-space.

In the past four years, a number of immersed curve invariants have emerged in low-dimensional topology. Hanselman, Rasmussen and Watson used their immersed curve invariant for 3-manifolds with torus boundary to give an elegant proof of the fact that the space of L-space fillings of such 3-manifolds is always an interval. I will discuss some progress towards analogous results for 4-ended tangles in Heegaard Floer, Bar-Natan and Khovanov homology. This is joint work with Liam Watson and Artem Kotelskiy.

Plumbed three-manifolds are those three-manifolds that can be be realized as links of isolated complex surface singularities. Inspired by Heegaard Floer theory Nemethi introduced a combinatorial invariant of complex surface singularities (lattice cohomology) that is conjectured to be isomorphic to Heegaard Floer homology. I will expose some recent work in collaboration with John Baldwin, Irving Dai, and Steven Sivek showing that the lattice cohomology of an almost-rational singularity is isomorphic to the framed instanton Floer homology of its link. The proof goes through lattice cohomology and makes use of the decomposition along characteristic vectors of the instanton cobordism maps recently found by Baldwin and Sivek.

I will introduce deformations of lattice cohomology corresponding to the knot homologies found by Ozsváth, Stipsicz and Szabó in https://arxiv.org/abs/1407.1795. If time permits I will show that by means of holomorphic triangles counting, one can prove equivalence with the analytic theory in the case of algebraic knots sitting in graph manifolds with at most one bad point. This yields to some interesting combinatorial formulae for the upsilon invariant of the latter knots.

A variety of cut-and-paste techniques is being developed to study Khovanov, Heegaard Floer and instanton homologies. We will describe one of such techniques, centered around immersed curves in surfaces. First, we will give an overview of existing curve-invariants. Next, a criterion for when a bordered invariant can be viewed as an immersed curve will be given. Lastly, we will interpret knot Floer homology as an immersed curve in the twice-punctured disc, and describe how it is related to the immersed curve associated to the knot exterior. The talk is based on a joint work with Liam Watson and Claudius Zibrowius.

We prove that the L-space conjecture holds for those L-spaces obtained from Dehn surgery on knots which are closures of iterated 1-bridge braids, i.e., the braids obtained from satellite operations on 1-bridge braids. This statement generalizes Tran's result on non-left-orderability surgeries on L-space twisted torus knots of form T_{p,kp \pm 1}^{l,m}.

We will describe several appearances of Milnor’s invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.

Let U_n^r denote the space of r-tuples of nxn complex matrices that generate the matrix algebra Mat_n(C) as an algebra over C. The projective general linear group acts on U_n^r by simultaneous conjugation, and one can form the quotient: B_n^r. These two families of spaces have many interesting homotopical properties. Notably, the spaces B_n^r approximate the classifying space BPGL_n(C); I will show that there is a map B_n^r->BPGL_n(C) that induces an isomorphism on homotopy groups in degrees <2(r-1)(n-1)-1. Lastly, I will sketch the computation of the singular cohomology groups of U_2^r and B_2^r in degrees <4r-6.

Many recent concordance invariants of knots come from perturbing the differential on a knot homology theory to get a complex with trivial homology but an interesting filtration. I will describe the invariant coming from Rasmussen's E(-1) spectral sequence from Khovanov homology in this way, and show that it gives a bound on the nonorientable slice genus.

Online (Ask Liam Watson or Ben Williams for the Zoom link)

Wed 29 Jul 2020, 10:00am-11:00am

Abstract

Joint work with Ajneet Dhillon. The classical Skolem--Noether Theorem by Giraud shows us:
(1) how we can assign to an Azumaya algebra A on a scheme X a class in H^2(X,\mathbf G_m) and
(2) how Azumaya algebras correspond to twisted vector bundles.
The Derived Skolem--Noether Theorem by Lieblich generalizes this to étale forms of derived endomorphism algebras of perfect coherent sheaves in the derived 1-category. We show that in general for a co-family of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving the assignments (1) and (2). In the case of complexes over a qcqs scheme X, the long exact sequence on homotopy sheaves splits giving in particular the aforementioned versions. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and IndCoh and crystals in Derived Algebraic Geometry in characteristic 0.

## Note for Attendees

There will be tea and cookies in the PIMS 1st floor lounge at approximately 2:45pm.