Search Site

##### Events
MIT
Mon 12 Jan 2009, 3:00pm SPECIAL
Topology and related seminars
110 WMAX
TBA
110 WMAX
Mon 12 Jan 2009, 3:00pm-4:00pm

#### Abstract

hide
Jose Manuel Gomez
Department of Mathematics, UBC
Wed 15 Apr 2009, 3:00pm
Topology and related seminars
WMAX 110 (PIMS)
Stable decompositions and almost commuting elements in Lie groups
WMAX 110 (PIMS)
Wed 15 Apr 2009, 3:00pm-4:00pm

#### Abstract

In this talk I show that the space of almost commuting elements in a compact Lie group G splits after one suspension.
hide
Jon Carlson
University of Georgia
Wed 22 Apr 2009, 3:00pm
Department Colloquium / Topology and related seminars
WMAX 1110 (PIMS)
Modules of constant Jordan type
WMAX 1110 (PIMS)
Wed 22 Apr 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
Duane Randall
Loyola University, New Orleans
Wed 13 May 2009, 3:00pm
Topology and related seminars
WMAX 110
Span of Orthogonal Sphere Bundles over Spheres
WMAX 110
Wed 13 May 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Lille/PIMS visitor
Wed 22 Jul 2009, 3:00pm
Topology and related seminars
WMAX 216 (PIMS upstairs)
Splitting methods in topology and geometry
WMAX 216 (PIMS upstairs)
Wed 22 Jul 2009, 3:00pm-4:00pm

#### Abstract

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).
hide
University of Western Ontario
Wed 5 Aug 2009, 3:00pm
Topology and related seminars
WMAX 110
Cocycle categories
WMAX 110
Wed 5 Aug 2009, 3:00pm-4:00pm

#### Abstract

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
hide
University of Western Ontario
Fri 7 Aug 2009, 3:00pm SPECIAL
Topology and related seminars
WMAX 110
Pointed torsors and Galois groups
WMAX 110
Fri 7 Aug 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
Indiana University
Wed 19 Aug 2009, 3:00pm
Topology and related seminars
216 WMAX
Mapping Tori of Self-homotopy equivalences of lens spaces
216 WMAX
Wed 19 Aug 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Bonn
Wed 16 Sep 2009, 3:00pm
Topology and related seminars
110 WMAX
Hilbert Uniformization I: moduli spaces of surfaces
110 WMAX
Wed 16 Sep 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Bonn
Wed 23 Sep 2009, 3:00pm
Topology and related seminars
110 WMAX
Hilbert Uniformization II: homology of moduli spaces
110 WMAX
Wed 23 Sep 2009, 3:00pm-4:00pm

#### Abstract

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
hide
Yale University
Fri 25 Sep 2009, 4:00pm SPECIAL
Topology and related seminars
110 WMAX
Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated group
110 WMAX
Fri 25 Sep 2009, 4:00pm-5:00pm

#### Abstract

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
hide
UBC
Wed 7 Oct 2009, 3:00pm
Topology and related seminars
110 WMAX
Survey of profinite homotopy theory
110 WMAX
Wed 7 Oct 2009, 3:00pm-4:00pm

#### Abstract

hide
University of Oregon
Wed 14 Oct 2009, 3:00pm
Topology and related seminars
110 WMAX
The moduli spaces of generalized Morse functions
110 WMAX
Wed 14 Oct 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Victoria
Thu 15 Oct 2009, 3:00pm SPECIAL
Topology and related seminars
216 WMAX
Equivariant Lefschetz invariants via analysis
216 WMAX
Thu 15 Oct 2009, 3:00pm-4:00pm

#### Abstract

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
hide
UBC
Wed 28 Oct 2009, 3:00pm
Topology and related seminars
110 WMAX
Finiteness conditions for diagrams of spaces with applications to algebraic spheres
110 WMAX
Wed 28 Oct 2009, 3:00pm-4:00pm

#### Abstract

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
hide
University of Regina
Wed 4 Nov 2009, 3:00pm
Topology and related seminars
110 WMAX
Nullity classes and t-structures in the derived category of a ring
110 WMAX
Wed 4 Nov 2009, 3:00pm-4:00pm

#### Abstract

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)
hide
Stanford University
Wed 18 Nov 2009, 3:00pm
Topology and related seminars
110 WMAX
Generalized Morse Theory for Maps to Two Dimensional Manifolds: PL-immersions of Polygons and Protein Folding
110 WMAX
Wed 18 Nov 2009, 3:00pm-4:00pm

#### Abstract

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
hide
UBC
Wed 25 Nov 2009, 3:00pm
Topology and related seminars
110 WMAX
Profinite homotopy theory
110 WMAX
Wed 25 Nov 2009, 3:00pm-4:00pm

#### Abstract

hide
U. of Michigan
Tue 8 Dec 2009, 3:00pm SPECIAL
Topology and related seminars
216 WMAX
Fully irreducible outer automorphisms of a free group
216 WMAX
Tue 8 Dec 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
U. of Michigan
Thu 10 Dec 2009, 3:00pm SPECIAL
Topology and related seminars
110 WMAX
Non-actions of the mapping class group.
110 WMAX
Thu 10 Dec 2009, 3:00pm-4:00pm

#### Abstract

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.
hide
Brown University
Sat 12 Dec 2009, 4:00pm SPECIAL
Topology and related seminars
110 WMAX
TBA
110 WMAX
Sat 12 Dec 2009, 4:00pm-5:00pm

#### Abstract

hide
MIT
Tue 12 Jan 2010, 3:00pm SPECIAL
Topology and related seminars
MATX 1102
On the relationship between EO_n and TAF
MATX 1102
Tue 12 Jan 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
Richard Kent
Brown University
Tue 12 Jan 2010, 4:00pm SPECIAL
Topology and related seminars
MATX 1102
Slicing, skinning, and grafting
MATX 1102
Tue 12 Jan 2010, 4:00pm-5:00pm

#### Abstract

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.

hide
MIT
Wed 20 Jan 2010, 3:00pm
Topology and related seminars
110 WMAX
An infinite loop space machine for symmetric monoidal 2-categories
110 WMAX
Wed 20 Jan 2010, 3:00pm-4:00pm

#### Abstract

Abstract:
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.
hide
Wed 27 Jan 2010, 3:00pm
Topology and related seminars
110 WMAX
Optimal bounds for the colored Tverberg problem
110 WMAX
Wed 27 Jan 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
MIT
Tue 2 Feb 2010, 3:00pm SPECIAL
Topology and related seminars
216 WMAX
An infinite presentation for the Torelli group
216 WMAX
Tue 2 Feb 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Tue 9 Feb 2010, 3:00pm SPECIAL
Topology and related seminars
216 WMAX
Orderings, eigenvalues and surgery
216 WMAX
Tue 9 Feb 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
Tetsuya Ito
University of Tokyo
Wed 3 Mar 2010, 3:00pm
Topology and related seminars
110 WMAX
Braid orderings and knot genus
110 WMAX
Wed 3 Mar 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Victoria
Wed 3 Mar 2010, 4:00pm
Topology and related seminars
110 WMAX
110 WMAX
Wed 3 Mar 2010, 4:00pm-5:00pm

#### Abstract

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.
hide
UBC
Wed 17 Mar 2010, 3:00pm
Topology and related seminars
110 WMAX
Fault-tolerant cluster state quantum computation, and how homology helps describing it
110 WMAX
Wed 17 Mar 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Oregon
Wed 24 Mar 2010, 3:00pm
Topology and related seminars
110 WMAX
Relations amongst motivic Hopf maps
110 WMAX
Wed 24 Mar 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Lethbridge
Wed 31 Mar 2010, 3:00pm
Topology and related seminars
110 WMAX
Introduction to Ratner's Theorems on Unipotent Flows
110 WMAX
Wed 31 Mar 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
U. of Illinois at Urbana-Champaign
Thu 1 Apr 2010, 3:00pm SPECIAL
Topology and related seminars
216 WMAX
In pursuit of DeRham cohomology for E_{\infty}-algebras
216 WMAX
Thu 1 Apr 2010, 3:00pm-4:00pm

#### Abstract

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
hide
Union College
Wed 7 Apr 2010, 3:00pm
Topology and related seminars
110 WMAX
The Calculus of Functors and DeRham cohomology
110 WMAX
Wed 7 Apr 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Calgary
Thu 15 Apr 2010, 3:00pm SPECIAL
Topology and related seminars
110 WMAX
Stable splittings of mapping spaces
110 WMAX
Thu 15 Apr 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 28 Apr 2010, 3:00pm
Topology and related seminars
WMAX 110
Heegaard-Floer Homology and applications
WMAX 110
Wed 28 Apr 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Illinois at Urbana-Champaign
Wed 23 Jun 2010, 3:00pm
Topology and related seminars
WMAX 110
A (mostly) intuitive introduction to Goodwillie's Calculus of Functors
WMAX 110
Wed 23 Jun 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 7 Jul 2010, 4:00pm
Topology and related seminars
WMAX 110
Finiteness Obstructions for G-Spaces up to hG-equivalence
WMAX 110
Wed 7 Jul 2010, 4:00pm-5:00pm

#### Abstract

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.
hide
University of Western Ontario
Wed 21 Jul 2010, 3:00pm
Topology and related seminars
WMAX 216
Galois descent and pro objects
WMAX 216
Wed 21 Jul 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Athens Panepistimioupolis
Fri 20 Aug 2010, 3:00pm
Topology and related seminars
WMAX 110
Gorenstein dimension and proper actions
WMAX 110
Fri 20 Aug 2010, 3:00pm-4:00am

#### Abstract

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
hide
UBC
Wed 15 Sep 2010, 3:00pm
Topology and related seminars
WMAX 110
Proofs of the Strengthened Hanna Neumann Conjecture I
WMAX 110
Wed 15 Sep 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Mon 20 Sep 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 20 Sep 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on the homotopy theory of moduli spaces
hide
UBC
Wed 22 Sep 2010, 3:00pm
Topology and related seminars
WMAX 110
Proofs of the Strengthened Hanna Neumann Conjecture II
WMAX 110
Wed 22 Sep 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Mon 27 Sep 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 27 Sep 2010, 2:00pm-3:00pm

#### Abstract

Pontryagin-Thom Theory II

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.
hide
UBC
Wed 29 Sep 2010, 3:00pm
Topology and related seminars
WMAX 110
Homotopy fixed points of profinite actions
WMAX 110
Wed 29 Sep 2010, 3:00pm-4:00pm

#### Abstract

TBA
hide
UBC
Mon 4 Oct 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 4 Oct 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on the homotopy theory of moduli spaces
hide
UBC
Wed 6 Oct 2010, 3:00pm
Topology and related seminars
WMAX 110
Vector fields and the Kervaire invariant
WMAX 110
Wed 6 Oct 2010, 3:00pm-4:00pm

#### Abstract

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
hide
Stanford University
Wed 13 Oct 2010, 3:00pm
Topology and related seminars
WMAX 110
Homology of moduli spaces of manifolds
WMAX 110
Wed 13 Oct 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Mon 18 Oct 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 18 Oct 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on the homotopy theory of moduli spaces
hide
University of Strasbourg/Pims
Wed 20 Oct 2010, 3:00pm
Topology and related seminars
WMAX 110
Lazy cohomology
WMAX 110
Wed 20 Oct 2010, 3:00pm-4:00pm

#### Abstract

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
hide
UBC
Mon 25 Oct 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 25 Oct 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on the homotopy theory of moduli spaces
hide
University of Strasbourg/PIMS
Wed 27 Oct 2010, 3:00pm
Topology and related seminars
WMAX 110
A link invariant with values in the Witt ring
WMAX 110
Wed 27 Oct 2010, 3:00pm-4:00pm

#### Abstract

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.

hide
UBC
Mon 1 Nov 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 1 Nov 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on homotopy theory of moduli spaces
hide
University of Rome
Wed 3 Nov 2010, 3:00pm
Topology and related seminars
WMAX 110
Infinitesimal index and some cohomology computations
WMAX 110
Wed 3 Nov 2010, 3:00pm-4:00pm

#### Abstract

(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 de fine a map, called in nitesimal 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.
hide
UBC
Mon 8 Nov 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 8 Nov 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on homotopy theory of moduli spaces
hide
UBC
Wed 10 Nov 2010, 3:00pm
Topology and related seminars
WMAX 110
Constructing free actions of p-groups on products of spheres
WMAX 110
Wed 10 Nov 2010, 3:00pm-4:00pm

#### Abstract

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).
hide
UBC
Mon 15 Nov 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 15 Nov 2010, 2:00pm-3:00pm

#### Abstract

This is a working seminar on homotopy theory of moduli spaces
hide
Bowdoin College
Wed 17 Nov 2010, 4:00pm
Topology and related seminars
WMAX 110
Multiplicative structures on Lie groupoids
WMAX 110
Wed 17 Nov 2010, 4:00pm-5:00pm

#### Abstract

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.
hide
UBC
Mon 22 Nov 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 22 Nov 2010, 2:00pm-4:00pm

#### Abstract

Sheaves and their Realization:
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.

hide
Queen's University Belfast
Wed 24 Nov 2010, 3:00pm
Topology and related seminars
WMAX 110
The derived category of regular toric schemes: a combinatorial approach
WMAX 110
Wed 24 Nov 2010, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Mon 29 Nov 2010, 2:00pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 29 Nov 2010, 2:00pm-3:00pm

#### Abstract

Sheaves of Categories and the Main Theorem
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.
hide
UBC
Mon 6 Dec 2010, 1:30pm
Topology and related seminars
WMAX 110
Working seminar on homotopy theory of moduli spaces
WMAX 110
Mon 6 Dec 2010, 1:30pm-3:00pm

#### Abstract

This is the last talk of this series.

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.

hide
University of Strasbourg/PIMS
Mon 10 Jan 2011, 2:00pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 10 Jan 2011, 2:00pm-3:00pm

#### Abstract

This is the first talk of a series in Persistent homology
hide
University of Toronto
Wed 12 Jan 2011, 4:00pm
Topology and related seminars
WMAX 110
A homomorphic universal finite type invariant of knotted trivalent graphs
WMAX 110
Wed 12 Jan 2011, 4:00pm-5:00pm

#### Abstract

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.
hide
University of Strasbourg/PIMS
Mon 17 Jan 2011, 2:00pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 17 Jan 2011, 2:00pm-3:00pm

#### Abstract

This is the second talk of a series in Persistent homology
hide
University of Regina
Wed 19 Jan 2011, 4:00pm
Topology and related seminars
WMAX 110
LS category
WMAX 110
Wed 19 Jan 2011, 4:00pm-5:00pm

#### Abstract

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.
hide
UBC
Mon 24 Jan 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 24 Jan 2011, 4:30pm-5:30pm

#### Abstract

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
hide
UBC
Wed 26 Jan 2011, 4:00pm
Topology and related seminars
WMAX 110
Bundles, Cohomology and Truncated Symmetric Polynomials
WMAX 110
Wed 26 Jan 2011, 4:00pm-5:00pm

#### Abstract

We discuss generators for the ideal of truncated
symmetric polynomials and some applications to
topology. This is joint work with Z. Reichstein.
hide
UBC
Mon 31 Jan 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 31 Jan 2011, 4:30pm-5:30pm

#### Abstract

This talk is the second talk in a series of talks about Hochschild (co)homology and
cyclic (co)homology
hide
University of Victoria
Wed 2 Feb 2011, 4:00pm
Topology and related seminars
WMAX 110
A homology theory for hyperbolic dynamical systems
WMAX 110
Wed 2 Feb 2011, 4:00pm-5:00pm

#### Abstract

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.)
hide
University of Strasbourg/PIMS
Mon 21 Feb 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 21 Feb 2011, 4:30pm-5:30pm

#### Abstract

This is a continuation on the topic of cyclic homology
hide
UBC
Wed 23 Feb 2011, 4:00pm
Topology and related seminars
WMAX 110
Equivariant K-theory and maximal rank subgroups
WMAX 110
Wed 23 Feb 2011, 4:00pm-5:00pm

#### Abstract

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).
hide
UBC
Mon 28 Feb 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 28 Feb 2011, 4:30pm-5:30pm

#### Abstract

This talk is a continuation on cyclic homology and Hochschild homology.
hide
UBC
Mon 7 Mar 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 7 Mar 2011, 4:30pm-5:30pm

#### Abstract

This is a working seminar in Topology. We will start a series of talk
hide
UBC
Mon 14 Mar 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 14 Mar 2011, 4:30pm-5:30pm

#### Abstract

This is a continuation on the mini cycle about
Simplicial Presheaves in Topology and Algebraic Geometry
hide
UBC
Wed 16 Mar 2011, 4:00pm
Topology and related seminars
WMAX 110
Homotopy theory and spaces of representations
WMAX 110
Wed 16 Mar 2011, 4:00pm-5:00pm

#### Abstract

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.
hide
UBC
Mon 21 Mar 2011, 4:30pm
Topology and related seminars
WMAX 110
Working seminar in Topology
WMAX 110
Mon 21 Mar 2011, 4:30pm-5:30pm

#### Abstract

This is a continuation on the mini cycle about
Simplicial Presheaves in Topology and Algebraic Geometry
hide
UCL Louvain
Wed 23 Mar 2011, 4:00pm
Topology and related seminars
WMAX 110
On the homology of the space of long knots.
WMAX 110
Wed 23 Mar 2011, 4:00pm-5:00pm

#### Abstract

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
hide
McMaster University
Wed 6 Apr 2011, 3:00pm
Topology and related seminars
WMAX 110
Group actions on spheres and non-compact space forms
WMAX 110
Wed 6 Apr 2011, 3:00pm-4:00pm

#### Abstract

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.
hide
Stanford University
Wed 6 Apr 2011, 4:00pm
Topology and related seminars
WMAX 110
Combinatorial models for p-completions of compact Lie groups
WMAX 110
Wed 6 Apr 2011, 4:00pm-5:00pm

#### Abstract

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.
hide
CNRS and University of Strasbourg
Tue 10 May 2011, 4:00pm
Topology and related seminars
WMAX 110
A continuous non-abelian one-cocycle with palindromic values
WMAX 110
Tue 10 May 2011, 4:00pm-5:00pm

#### Abstract

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.)
hide
UQAM
Mon 25 Jul 2011, 11:00am SPECIAL
Topology and related seminars
WMAX 110 (PIMS)
Left-orderability and Dehn surgery
WMAX 110 (PIMS)
Mon 25 Jul 2011, 11:00am-12:30pm

#### Abstract

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.

hide
University of Chicago
Wed 14 Sep 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Isometries of aspherical manifolds
WMAX 216 (PIMS)
Wed 14 Sep 2011, 3:00pm-4:30pm

#### Abstract

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.
hide
UBC
Wed 21 Sep 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Expander graphs, metrics and knots.
WMAX 216 (PIMS)
Wed 21 Sep 2011, 3:00pm-4:00pm

#### Abstract

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.
hide
Prof. Masao Hara
Tokai University, Japan
Wed 5 Oct 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Minimal crossing diagram and Jones polynomial
WMAX 216 (PIMS)
Wed 5 Oct 2011, 3:00pm-4:30pm

#### Abstract

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.
hide
University of Michigan
Wed 12 Oct 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Bounds on eigenvalues of the Laplacian for certain hyperbolic 3-manifolds
WMAX 216 (PIMS)
Wed 12 Oct 2011, 3:00pm-4:00pm

#### Abstract

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}.
hide
UBC
Wed 19 Oct 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Abstract Commensurators of the Johnson Filtration
WMAX 216 (PIMS)
Wed 19 Oct 2011, 3:00pm-4:00pm

#### Abstract

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.
hide
The University of Texas at Austin
Wed 26 Oct 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
The density conjecture and a short survey on Kleinian groups
WMAX 216 (PIMS)
Wed 26 Oct 2011, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 2 Nov 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Equivariant K-theory for Actions with Maximal Rank Isotropy
WMAX 216 (PIMS)
Wed 2 Nov 2011, 3:00pm-4:00pm

#### Abstract

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.
hide
Barnard College and Stony Brook
Mon 7 Nov 2011, 1:00pm SPECIAL
Topology and related seminars
WMAX 216 (PIMS)
Displaceability in symplectic toric manifolds
WMAX 216 (PIMS)
Mon 7 Nov 2011, 1:00pm-2:00pm

#### Abstract

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.
hide
University of Utah
Wed 9 Nov 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Minimal Algebraic Laminations and Arational Trees
WMAX 216 (PIMS)
Wed 9 Nov 2011, 3:00pm-4:00pm

#### Abstract

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.)
hide
Alberto Verjovsky
UNAM Cuernavaca
Wed 23 Nov 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Hedlund's theorem for compact, minimal laminations by surfaces with negative curvature
WMAX 216 (PIMS)
Wed 23 Nov 2011, 3:00pm-4:00pm

#### Abstract

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.
hide
Princeton University
Wed 30 Nov 2011, 4:15pm SPECIAL
Topology and related seminars
WMAX 216 (PIMS)
Finite-sheeted covers of 3-manifolds and the Cohomology of Solenoids
WMAX 216 (PIMS)
Wed 30 Nov 2011, 4:15pm-5:15pm

#### Abstract

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.
hide
UBC
Wed 11 Jan 2012, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Ordering knot groups
WMAX 216 (PIMS)
Wed 11 Jan 2012, 3:00pm-4:00pm

#### Abstract

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.

hide
Columbia University
Wed 18 Jan 2012, 3:00pm SPECIAL
Algebraic Geometry Seminar / Topology and related seminars
WMAX 110 (PIMS)
WMAX 110 (PIMS)
Wed 18 Jan 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
Jeff Smith
UBC
Wed 1 Feb 2012, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
The Adem HW problem and the moduli space of commuting endomorphisms.
WMAX 216 (PIMS)
Wed 1 Feb 2012, 3:00pm-4:00pm

#### Abstract

hide
Harvard University
Wed 8 Feb 2012, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Mapping class groups and finite covers
WMAX 216 (PIMS)
Wed 8 Feb 2012, 3:00pm-4:00pm

#### Abstract

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.

hide
UCLA
Fri 10 Feb 2012, 2:00pm SPECIAL
Topology and related seminars
WMAX 216 (PIMS)
A conjecture relating Heegaard Floer homology and the fundamental group
WMAX 216 (PIMS)
Fri 10 Feb 2012, 2:00pm-3:00pm

#### Abstract

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.

hide
Technion
Wed 15 Feb 2012, 3:00pm
Probability Seminar / Topology and related seminars
WMAX 110
DRAT: Doing random algebraic topology
WMAX 110
Wed 15 Feb 2012, 3:00pm-4:00pm

#### Abstract

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.


hide
Galo Higuera Rojo
UBC
Wed 7 Mar 2012, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Spaces of homomorphisms and commuting matrices
WMAX 216 (PIMS)
Wed 7 Mar 2012, 3:00pm-4:00pm

#### Abstract

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)).

hide
Universitat Autònoma de Barcelona
Wed 14 Mar 2012, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Cohomology of Kac-Moody groups over finite fields
WMAX 216 (PIMS)
Wed 14 Mar 2012, 3:00pm-4:00pm

#### Abstract

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.

hide
Stanford University
Tue 20 Mar 2012, 3:00pm
Topology and related seminars
ESB 4127
TBA
ESB 4127
Tue 20 Mar 2012, 3:00pm-4:00pm

#### Abstract

hide
UBC
Wed 21 Mar 2012, 3:00pm
Probability Seminar / Topology and related seminars
WMAX 110 (PIMS)
Distributional limits of Riemannian manifolds and graphs with sublinear genus growth
WMAX 110 (PIMS)
Wed 21 Mar 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Michigan
Wed 28 Mar 2012, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)
Strong accessibility of torsion free (relatively) hyperbolic groups
WMAX 216 (PIMS)
Wed 28 Mar 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Chicago
Mon 2 Apr 2012, 2:00pm SPECIAL
Topology and related seminars
WMAX 110 (PIMS)
The cohomology groups of the pure string motion group are uniformly representation stable
WMAX 110 (PIMS)
Mon 2 Apr 2012, 2:00pm-3:00pm

#### Abstract

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.
hide
University of Tokyo
Wed 4 Apr 2012, 4:00pm SPECIAL
Topology and related seminars
WMAX 216
Self-linking number of transverse knots in general open books
WMAX 216
Wed 4 Apr 2012, 4:00pm-5:00pm

#### Abstract

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).

hide
Université de Louvain
Thu 12 Apr 2012, 2:00pm SPECIAL
Topology and related seminars
WMAX 216 (PIMS)
From the eversion of the sphere to spaces of knots
WMAX 216 (PIMS)
Thu 12 Apr 2012, 2:00pm-3:00pm

#### Abstract

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.
hide
Jon Carlson
University of Georgia
Mon 30 Jul 2012, 3:00pm SPECIAL
Topology and related seminars
WMAX 110 (PIMS)
Classifying thick subcategories of the stable category
WMAX 110 (PIMS)
Mon 30 Jul 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
Man Chuen Cheng
UBC
Wed 12 Sep 2012, 3:15pm
Topology and related seminars
PIMS Lounge, ESB 4133
A Duality Theorem for Quotient Stacks with respect to Morava K-theory
PIMS Lounge, ESB 4133
Wed 12 Sep 2012, 3:15pm-4:15pm

#### Abstract

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.
hide
University of Iowa
Wed 19 Sep 2012, 3:15pm
Topology and related seminars
PIMS Lounge, ESB 4133
Open book foliation and fractional Dehn twist coefficient
PIMS Lounge, ESB 4133
Wed 19 Sep 2012, 3:15pm-4:15pm

#### Abstract

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.
hide
Man Chuen Cheng
UBC
Tue 25 Sep 2012, 3:00pm SPECIAL
Topology and related seminars
ESB 4127
A Duality Theorem for Quotient Stacks with respect to Morava K-theory, Part II
ESB 4127
Tue 25 Sep 2012, 3:00pm-4:00pm

#### Abstract

This is the continuation of my previous talk.

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.
hide
UBC
Wed 3 Oct 2012, 3:00pm
Topology and related seminars
ESB 4127
Knotted geodesics on hyperbolic surfaces
ESB 4127
Wed 3 Oct 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 10 Oct 2012, 3:00pm
Topology and related seminars
ESB 4127
Topology meets essential dimension
ESB 4127
Wed 10 Oct 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 17 Oct 2012, 3:00pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Homotopy colimits of classifying spaces of finite groups and K-theory
ESB 4133 (PIMS Lounge)
Wed 17 Oct 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Southern California
Wed 31 Oct 2012, 3:00pm
Topology and related seminars
ESB 4127
The Brauer Group and Obstruction Theory
ESB 4127
Wed 31 Oct 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 14 Nov 2012, 3:00pm
Topology and related seminars
ESB 4127
Cohomology and K-theory of Crystallographic Groups I
ESB 4127
Wed 14 Nov 2012, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 21 Nov 2012, 3:00pm
Topology and related seminars
ESB 4127
Cohomology and K-theory of Crystallographic Groups II
ESB 4127
Wed 21 Nov 2012, 3:00pm-4:00pm

#### Abstract

This is continuation from the previous week:

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.
hide
UBC
Wed 9 Jan 2013, 3:10pm
Topology and related seminars
ESB 4127
Holomorphic maps between moduli spaces
ESB 4127
Wed 9 Jan 2013, 3:10pm-4:10pm

#### Abstract

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.
hide
Enrique Torres
Wed 16 Jan 2013, 3:00pm
Topology and related seminars
ESB 4127
Topological Complexity
ESB 4127
Wed 16 Jan 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 30 Jan 2013, 3:00pm
Topology and related seminars
ESB 4127
Retractions of representation varieties of nilpotent groups
ESB 4127
Wed 30 Jan 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
Stanford University
Wed 13 Feb 2013, 3:00pm
Topology and related seminars
ESB 4127
Homological stability for moduli spaces of high dimensional manifolds
ESB 4127
Wed 13 Feb 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
Korea Advanced Institute of Science and Technology
Thu 21 Feb 2013, 3:00pm SPECIAL
Topology and related seminars
ESB 2012
Tabulation of prime knots by arc index
ESB 2012
Thu 21 Feb 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 6 Mar 2013, 3:00pm
Topology and related seminars
ESB 4127
Relative twisting in Outer space
ESB 4127
Wed 6 Mar 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 13 Mar 2013, 3:00pm
Topology and related seminars
ESB 4127
A topological approach to orderable groups
ESB 4127
Wed 13 Mar 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
Stanford University
Wed 20 Mar 2013, 3:00pm
Topology and related seminars
ESB 4127
Stability in the unstable cohomology of mapping class groups, SL_n(Z), and Aut(F_n)
ESB 4127
Wed 20 Mar 2013, 3:00pm-4:00pm

#### Abstract

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.

hide
Centre de recherches mathématiques - Université de Montréal
Thu 4 Apr 2013, 3:00pm SPECIAL
Topology and related seminars
ESB 2012
A Short Elementary Survey of Symplectic Topology
ESB 2012
Thu 4 Apr 2013, 3:00pm-4:00pm

#### Abstract

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.

#### Note for Attendees

Note special day for this seminar.
hide
University of Ottawa
Wed 7 Aug 2013, 3:00pm SPECIAL
Topology and related seminars
ESB 4127
Equivariant oriented cohomology of a projective homogeneous variety: Algebraic model.
ESB 4127
Wed 7 Aug 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Augsburg
Thu 5 Sep 2013, 4:00pm
Topology and related seminars
ESB 2012
Positive scalar curvature: Existence and Classification
ESB 2012
Thu 5 Sep 2013, 4:00pm-5:00pm

#### Abstract

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.
hide
University of Augsburg
Wed 11 Sep 2013, 3:00pm SPECIAL
Topology and related seminars
ESB 4133
The stable free rank of symmetry of products of spheres
ESB 4133
Wed 11 Sep 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
Univeristy of Oxford
Wed 18 Sep 2013, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Configurations spaces revisited and homology stability for some diffeomorphism groups
ESB 4133
Wed 18 Sep 2013, 3:15pm-4:15pm

#### Abstract

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.
hide
Carlos Dominguez
UBC
Thu 3 Oct 2013, 3:00pm
Topology and related seminars
ESB 4133
The integral cohomology of configuration spaces of pairs of points in real projective spaces
ESB 4133
Thu 3 Oct 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 9 Oct 2013, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Groups of PL homeomorphisms
ESB 4133
Wed 9 Oct 2013, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Thu 17 Oct 2013, 3:00pm
Topology and related seminars
ESB 4133
The topology of nilpotent representations in reductive groups and their maximal compact subgroups
ESB 4133
Thu 17 Oct 2013, 3:00pm-4:00pm

#### Abstract

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).
hide
Indiana University
Thu 24 Oct 2013, 3:00pm
Topology and related seminars
ESB 4133
Any finite group acts freely and homologically trivially on a product of spheres
ESB 4133
Thu 24 Oct 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Thu 14 Nov 2013, 3:00pm
Topology and related seminars
ESB 4133
Homotopy Colimits of Classifying Spaces of Abelian Groups
ESB 4133
Thu 14 Nov 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
University of Victoria
Thu 21 Nov 2013, 3:00pm
Topology and related seminars
ESB 4133
ESB 4133
Thu 21 Nov 2013, 3:00pm-4:00pm

#### Abstract

hide
University of Michigan
Thu 28 Nov 2013, 3:00pm
Topology and related seminars
ESB 4133
Mirror symmetry and modular forms
ESB 4133
Thu 28 Nov 2013, 3:00pm-4:00pm

#### Abstract

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.
hide
UBC
Wed 22 Jan 2014, 4:00pm SPECIAL
Topology and related seminars
ESB 2012
Period & Index in Locally Ringed Topoi
ESB 2012
Wed 22 Jan 2014, 4:00pm-5:00pm

#### Abstract

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.
hide
UBC
Wed 29 Jan 2014, 3:15pm
Topology and related seminars
ESB 4133
Period & Index in Locally Ringed Topoi II
ESB 4133
Wed 29 Jan 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
Technion - Israel Institute of Technology
Wed 5 Feb 2014, 4:00pm SPECIAL
Topology and related seminars
ESB 4133
The curve complex and 3-manifolds
ESB 4133
Wed 5 Feb 2014, 4:00pm-5:00pm

#### Abstract

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.
hide
University of Texas at Austin.
Wed 12 Feb 2014, 3:15pm
Topology and related seminars
ESB 4133
Masur criterion analog for OUT(F) and applications
ESB 4133
Wed 12 Feb 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
Georgia Institute of Technology
Wed 26 Feb 2014, 3:15pm
Topology and related seminars
ESB 4133
Towards a motivic simplicial EHP spectral sequence
ESB 4133
Wed 26 Feb 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
University of Texas at Austin
Wed 5 Mar 2014, 3:15pm
Topology and related seminars
ESB 4133
Right-angled Artin subgroups of mapping class groups and Out(F)
ESB 4133
Wed 5 Mar 2014, 3:15pm-4:15pm

#### Abstract

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).
hide
University of Washington
Wed 19 Mar 2014, 3:15pm
Topology and related seminars
ESB 4133
Algebraic topology and algebraic torsors
ESB 4133
Wed 19 Mar 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 26 Mar 2014, 3:15pm
Topology and related seminars
ESB 4133
A Classifying Space for Commutativity for Lie Groups
ESB 4133
Wed 26 Mar 2014, 3:15pm-4:15pm

#### Abstract

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.

hide
McMaster University
Wed 2 Apr 2014, 3:15pm
Topology and related seminars
ESB 4133
Finite group actions and chain complexes over the orbit category
ESB 4133
Wed 2 Apr 2014, 3:15pm-4:15pm

#### Abstract

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).
hide
University of Victoria
Wed 9 Apr 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 2012
Triangulations of 4-manifolds and a table of knots in homotopy 4-spheres
ESB 2012
Wed 9 Apr 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
Yale University
Tue 15 Apr 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Anti-trees and right-angled Artin subgroups of planar braid groups
ESB 4133
Tue 15 Apr 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
CIMAT, Mexico
Fri 6 Jun 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Compact Lie groups, p-compact groups and p-local H-spaces
ESB 4133
Fri 6 Jun 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
University of Utah
Mon 9 Jun 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Laminations, trees, and Out(F)
ESB 4133
Mon 9 Jun 2014, 3:15pm-4:15pm

#### Abstract

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).

hide
Toulouse, France
Thu 12 Jun 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Discontinuous action of Out(Fn) and discrete representations
ESB 4133
Thu 12 Jun 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
Université de Caen
Thu 3 Jul 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
Laver tables
ESB 4133
Thu 3 Jul 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
KTH Royal Institute of Technology, Sweden
Wed 16 Jul 2014, 3:15pm SPECIAL
Topology and related seminars
ESB 4133
String topology of classifying spaces
ESB 4133
Wed 16 Jul 2014, 3:15pm-4:15pm

#### Abstract

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
hide
National University of Mexico
Thu 18 Sep 2014, 2:15pm SPECIAL
Topology and related seminars
ESB 4133
On the Chern classes of singular varieties
ESB 4133
Thu 18 Sep 2014, 2:15pm-3:15pm

#### Abstract

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).
hide
UBC
Wed 24 Sep 2014, 3:15pm
Topology and related seminars
ESB 4133
Algebraic K-Theory of Group rings and its applications
ESB 4133
Wed 24 Sep 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 1 Oct 2014, 3:15pm
Topology and related seminars
ESB 4133 (may move to Thursday)
The Status of the Farrell-Jones conjecture
ESB 4133 (may move to Thursday)
Wed 1 Oct 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
Man Chuen Cheng
UBC
Wed 15 Oct 2014, 3:15pm
Topology and related seminars
ESB 4133
The space of almost commuting tuples of unitary matrices
ESB 4133
Wed 15 Oct 2014, 3:15pm-4:15pm

#### Abstract

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.

This is joint work with Adem.

hide
UBC
Wed 22 Oct 2014, 3:15pm
Topology and related seminars
ESB 4133
Generalized torsion in knot groups
ESB 4133
Wed 22 Oct 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
University of Utah
Wed 29 Oct 2014, 3:15pm
Topology and related seminars
ESB 4133
Constructing aspherical manifolds with a given fundamental group
ESB 4133
Wed 29 Oct 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 12 Nov 2014, 3:15pm
Topology and related seminars
ESB 4133
Differentiable Stacks and Foliation Theory, Part I
ESB 4133
Wed 12 Nov 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 19 Nov 2014, 3:15pm
Topology and related seminars
ESB 4133
Differentiable Stacks and Foliation Theory, Part II
ESB 4133
Wed 19 Nov 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
MIT
Wed 26 Nov 2014, 3:15pm
Topology and related seminars
ESB 4133
The six operations of Grothendieck in equivariant motivic homotopy theory
ESB 4133
Wed 26 Nov 2014, 3:15pm-4:15pm

#### Abstract

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.
hide
Bilkent University and McMaster University
Wed 4 Mar 2015, 3:15pm
Topology and related seminars
ESB 4133
Finite group actions on homotopy spheres
ESB 4133
Wed 4 Mar 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
Osaka University
Wed 11 Mar 2015, 3:15pm
Topology and related seminars
ESB 4133
Pseudo-Anosovs with small dilatations in the hyperelliptic handlebody groups and spherical Hilden groups
ESB 4133
Wed 11 Mar 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Tue 17 Mar 2015, 2:00pm SPECIAL
Topology and related seminars
ESB 4133
On the volumes of complements of geodesics on surfaces
ESB 4133
Tue 17 Mar 2015, 2:00pm-3:00pm

#### Abstract

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.
hide
UBC
Wed 25 Mar 2015, 3:15pm
Topology and related seminars
ESB 4133
Categorification of orbifold products and derived loop stacks
ESB 4133
Wed 25 Mar 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
Université Paris 13
Wed 13 May 2015, 3:15pm
Topology and related seminars
ESB 4133
Automorphisms of p-completed classifying spaces of groups of Lie type
ESB 4133
Wed 13 May 2015, 3:15pm-4:15pm

#### Abstract

hide
U.Western Ontario
Tue 19 May 2015, 3:15pm SPECIAL
Topology and related seminars
ESB 4127
Path categories and algorithms
ESB 4127
Tue 19 May 2015, 3:15pm-4:15pm

#### Abstract

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.

hide
UBC Math
Wed 23 Sep 2015, 3:15pm
Topology and related seminars
ESB 4133
Subspace arrangements and BNS invariants.
ESB 4133
Wed 23 Sep 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC Math
Wed 30 Sep 2015, 3:15pm
Topology and related seminars
ESB 4113
Subspace arrangements and BNS invariants, continued.
ESB 4113
Wed 30 Sep 2015, 3:15pm-4:15pm

#### Abstract

(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.
hide
Camille Horbez
University of Utah
Wed 7 Oct 2015, 3:15pm
Topology and related seminars
ESB 4133
Subgroups of the automorphism group of a free product
ESB 4133
Wed 7 Oct 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
Bert Guillou
University of Kentucky
Tue 13 Oct 2015, 4:00pm
Topology and related seminars
ESB 4133
The eta-local motivic sphere
ESB 4133
Tue 13 Oct 2015, 4:00pm-5:00pm

#### Abstract

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.

#### Note for Attendees

Note different day and time.
hide
Kyle Ormsby
Reed College
Wed 21 Oct 2015, 3:15pm
Topology and related seminars
ESB 4133
Tensor triangular geometry of the stable motivic homotopy category
ESB 4133
Wed 21 Oct 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
Omar Antolin Camarena
UBC Math
Wed 28 Oct 2015, 3:15pm
Topology and related seminars
ESB 4133
A simple universal property of Thom ring spectra
ESB 4133
Wed 28 Oct 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
Eiko Kin
Osaka University
Wed 4 Nov 2015, 3:15pm
Topology and related seminars
ESB 4133
Braids, automorphisms and orderings
ESB 4133
Wed 4 Nov 2015, 3:15pm-4:15pm

#### Abstract

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.
hide
University of British Columbia
Wed 13 Jan 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Homotopy Group Actions I
ESB 4133 (PIMS Lounge)
Wed 13 Jan 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
University of British Columbia
Wed 20 Jan 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Homotopy Group Actions II
ESB 4133 (PIMS Lounge)
Wed 20 Jan 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
Henrik Rüping
University of British Columbia
Wed 27 Jan 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Controlled Algebra
ESB 4133 (PIMS Lounge)
Wed 27 Jan 2016, 3:15pm-4:15pm

#### Abstract

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.

hide
Henrik Rüping
University of British Columbia
Wed 3 Feb 2016, 4:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Controlled Algebra, Part II
ESB 4133 (PIMS Lounge)
Wed 3 Feb 2016, 4:15pm-5:15pm

#### Abstract

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.
hide
Bernardo Villarreal Herrera
University of British Columbia
Wed 10 Feb 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Cosimplicial groups and spaces of homomorphisms
ESB 4133 (PIMS Lounge)
Wed 10 Feb 2016, 3:15pm-4:15pm

#### Abstract

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.

#### Note for Attendees

The seminar is back to the regular time!
hide
Daniel Sheinbaum
University of British Columbia
Wed 24 Feb 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Topology of Fermi surfaces and Anomalies
ESB 4133 (PIMS Lounge)
Wed 24 Feb 2016, 3:15pm-4:15pm

#### Abstract

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 = S1, known as the spectral flow and its relation to quantum anomalies. This is work in progress joint with Alejandro Adem and Gordon W. Semenoff.
hide
University of British Columbia
Wed 2 Mar 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
How to get a Calculus named after you
ESB 4133 (PIMS Lounge)
Wed 2 Mar 2016, 3:15pm-4:15pm

#### Abstract

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".
hide
Harvard University
Thu 17 Mar 2016, 3:30pm
Topology and related seminars
ESB 2012
Rotation-Invariance in Algebraic K-Theory
ESB 2012
Thu 17 Mar 2016, 3:30pm-4:30pm

#### Abstract

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.


#### Note for Attendees

Note the special day, time and location!
hide
University of British Columbia
Wed 23 Mar 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
The EHP sequence in A¹ algebraic topology
ESB 4133 (PIMS Lounge)
Wed 23 Mar 2016, 3:15pm-4:15pm

#### Abstract

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 A1 algebraic topology, and some implications this has for the unstable A1 homotopy groups of spheres.

#### Note for Attendees

Back to the regular time and place!
hide
University of British Columbia
Wed 13 Apr 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Minimal volume cusped hyperbolic 3-manifolds and their groups.
ESB 4133 (PIMS Lounge)
Wed 13 Apr 2016, 3:15pm-4:15pm

#### Abstract

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).

hide
Université de Strasbourg
Fri 3 Jun 2016, 11:00am
Topology and related seminars
ESB 4133 (PIMS Lounge)
Some group cohomology calculations in stable homotopy theory
ESB 4133 (PIMS Lounge)
Fri 3 Jun 2016, 11:00am-12:00pm

#### Abstract

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.

#### Note for Attendees

Unusual day and time!
hide
University of Aberdeen
Mon 4 Jul 2016, 3:15pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Signature mod eight of surface bundles over surfaces, and cohomology of finite groups
ESB 4133 (PIMS Lounge)
Mon 4 Jul 2016, 3:15pm-4:15pm

#### Note for Attendees

Please note: The topology seminar will take place on Monday instead of the usual Wednesday.
hide
Reed College
Wed 14 Sep 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
ESB 4133 (PIMS Lounge)
Wed 14 Sep 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
Wed 21 Sep 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Multiplicative structures and the twisted Baum-Connes assembly map
ESB 4133 (PIMS Lounge)
Wed 21 Sep 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
University of Illinois, Urbana–Champaign
Wed 28 Sep 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
An equivariant motivic slice filtration
ESB 4133 (PIMS Lounge)
Wed 28 Sep 2016, 3:15pm-4:15pm

#### Abstract

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.

hide
University of Southern California
Wed 5 Oct 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
A new proof of the decomposition theorem
ESB 4133 (PIMS Lounge)
Wed 5 Oct 2016, 3:15pm-4:15pm

#### Abstract

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.

hide
UBC
Wed 12 Oct 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Free actions by elementary abelian 2-groups
ESB 4133 (PIMS Lounge)
Wed 12 Oct 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 19 Oct 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Free actions by elementary abelian p-groups
ESB 4133 (PIMS Lounge)
Wed 19 Oct 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
NTNU Norwegian University of Science and Technology
Wed 26 Oct 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Freaks of algebra -- a topologist's exhibition
ESB 4133 (PIMS Lounge)
Wed 26 Oct 2016, 3:15pm-4:15pm

#### Abstract

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.

hide
McGill
Fri 28 Oct 2016, 11:10am SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Nonpositive Immersions and Counting Cycles
ESB 4133 (PIMS Lounge)
Fri 28 Oct 2016, 11:10am-12:10pm

#### Abstract

The "nonpositive immersion" property is a condition on a 2-complex X that generalizes being a surface. When X has this property, its fundamental 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.
hide
University of Washington
Wed 9 Nov 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Iterated Thom Spectra with Examples
ESB 4133 (PIMS Lounge)
Wed 9 Nov 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
Reed College
Thu 10 Nov 2016, 3:15pm SPECIAL
Topology and related seminars
ESB 4127
The transfer map of free loop spaces
ESB 4127
Thu 10 Nov 2016, 3:15pm-4:15pm

#### Abstract

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)
hide
UBC
Wed 23 Nov 2016, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
A functorial version of Vinogradov’s theorem on free products of orderable groups.
ESB 4133 (PIMS Lounge)
Wed 23 Nov 2016, 3:15pm-4:15pm

#### Abstract

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.
hide
University of Chicago
Wed 25 Jan 2017, 4:15pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Quantitative Nullcobordism and the (in)effectiveness of algebraic topology.
ESB 4133 (PIMS Lounge)
Wed 25 Jan 2017, 4:15pm-5:15pm

#### Abstract

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.
hide
University of Oregon
Wed 1 Feb 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
The mod-two cohomology of symmetric and alternating groups.
ESB 4133 (PIMS Lounge)
Wed 1 Feb 2017, 3:15pm-4:15pm

#### Abstract

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.
hide
North Carolina State University
Wed 15 Feb 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Representation stability in configuration spaces via Whitney homology of the partition lattice
ESB 4133 (PIMS Lounge)
Wed 15 Feb 2017, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 8 Mar 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Nilpotent n-tuples in SU(2)
ESB 4133 (PIMS Lounge)
Wed 8 Mar 2017, 3:15pm-4:15pm

#### Abstract

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.

hide
University of Manitoba
Wed 29 Mar 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
The spaces of left and circular orderings of a group
ESB 4133 (PIMS Lounge)
Wed 29 Mar 2017, 3:15pm-4:15pm

#### Abstract

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.
hide
McMaster University
Wed 5 Apr 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Group actions on homology 3-spheres
ESB 4133 (PIMS Lounge)
Wed 5 Apr 2017, 3:15pm-4:15pm

#### Abstract

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.
hide
UBC
Wed 12 Apr 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Proving Yuzvinsky's conjecture: an instance of applying topology to combinatorics
ESB 4133 (PIMS Lounge)
Wed 12 Apr 2017, 3:15pm-4:15pm

#### Abstract

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).
hide
Université catholique de Louvain
Wed 19 Apr 2017, 3:15pm
Topology and related seminars
ESB 4133
Cosimplicial models for embedding spaces of manifolds and manifold calculus.
ESB 4133
Wed 19 Apr 2017, 3:15pm-4:15pm

#### Abstract

TBA
hide
UBC
Wed 26 Apr 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Mod 2 homology operations for spectral Lie algebras
ESB 4133 (PIMS Lounge)
Wed 26 Apr 2017, 3:15pm-4:15pm

#### Abstract

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.)
hide
MIT
Thu 4 May 2017, 2:00pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Character maps, free loops, and fusion systems
ESB 4133 (PIMS Lounge)
Thu 4 May 2017, 2:00pm-3:00pm

#### Abstract

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.
hide
Harvard University
Thu 4 May 2017, 3:15pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Nonconnective simplicial commutative rings
ESB 4133 (PIMS Lounge)
Thu 4 May 2017, 3:15pm-4:15pm

#### Abstract

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 of simplicial commutative rings; however, from a purely homotopy-theoretic or categorical picture they are slightly mysterious. I will explain ongoing work with Bhargav Bhatt on an extended theory of "generalized rings" which extends this category to allow nonconnective objects. Many "equational" constructions which cannot work with E_infty-rings extend well to generalized rings.
hide
University of Oxford
Thu 15 Jun 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Commutative complex K-theory
ESB 4133 (PIMS Lounge)
Thu 15 Jun 2017, 3:15pm-4:45pm

#### Abstract

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.
hide