Employing an affine version of the plactic algebra (which arises in theRobinson-Schensted-Knuth correspondence) one can define non-commutative Schur polynomials. The latter can be employed to construct a combinatorial ring with integer structure constants. This combinatorial ring turns out to be isomorphic to what is called the su(n) WZNW fusion ring in the physics and the su(n) Verlinde algebra (extension over C) in the mathematics literature. There is a simple physical description of this ring in terms of quantum particles hoping on the affine su(n) Dynkin diagram. Many of the known complicated results concerning the fusion ring can be derived in a novel and elementary way. Using the particle picture one also arrives at new recursion formulae for the structure constants which are dimensions of moduli spaces of generalized theta functions. I explain the close connection with the small quantum cohomology ring of the Grassmannian and present a simple reduction formula which allows to relate the structure constants of the su(n) Verlinde algebra with Gromov-Witten invariants.

One particularly easy way to compute generating functions for 3D Young diagrams, and for "pyramid partitions", is to use the commutation properties of vertex operators. In fact, the vertex operator method turns out to apply to a broader class of box-counting / dimer cover problems.

We will describe this more general class of problems, and explicitly give their generating functions. All of these generating functions can be readily turned into Donaldson-Thomas partition functions for the associated quivers (modulo a superpotential) by introducing signs on certain variables.

In this talk I will overview some results about derived categories of toric stacks. In particular the problem of existense of strong exceptional collections of line bundles. Some connections of this problem to Mirror symmetry and combinatorics of polytopes will be mentioned.

If H is a variation of Hodge structure over a variety S, then there is a family of complex tori J(H) over S associated to H. Admissible normal functions are certain sections of J(H) over S. Roughly speaking, they are the ones that have the possibility of coming from algebraic geometry.

I will explain recent work with Gregory Pearlstein proving that the locus where a section of J(H) vanishes is an algebraic subvariety of S. This answers a conjecture of Griffiths and Green.

I will discuss some deformation properties of Fano
varieties. The general methods rely on the investigation of the
variation of the cone of effective curves and, more generally, of the
Mori chamber decomposition, which, according to Mori theory, encode
information on the geometry of the variety. The talk is based on
joint work with C. Hacon.

Kostant's remarkable formula (which I will recall) generalizes to smooth Schubert varieties in the flag variety G/B of an algebraic group G. On the other hand, Sara Billey noticed that rationally smooth Schubert varieties in G/B give an analogous formula, though it often doesn't agree with the remarkable formula in the singular case. This motivates the question of which rationally smooth Schubert varieties are smooth. I will show that there is a neat answer hinted at in the title.

In the first of a two lecture series (to be completed by Dave Anderson immediately following), we present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex $G$-variety $X$ by its associated arc space $J_{\infty} X$, with its induced $G$-action. If $X$ admits an `equivariant affine paving', then we deduce an explicit geometric basis for the equivariant cohomology ring. Moreover, under appropriate hypotheses, we obtain explicit bijections between bases for the equivariant cohomology rings of smooth varieties related by an equivariant, properbirational map. As an initial application, we present a geometric basis for the equivariant cohomology ring of a smooth toric variety.

Let G be an algebraic group acting on a smooth complex variety X. In joint
work with Alan Stapledon, we present a new perspective on the G-equivariant
cohomology of X, which replaces the action of G on X with the induced action
of the respective arc spaces. I will explain how this point of view allows
one to interpret the cup product of classes of subvarieties geometrically
via contact loci in the arc space, at least under suitable hypotheses on the
singularities. As an explicit example, I'll discuss GL_n acting on the space
of matrices.

Goresky and MacPherson observed that certain pairs of
algebraic varieties with torus actions have equivariant cohomology
rings that are "dual" in a sense that I will define. Examples of
such pairs come up naturally in both representation theory and
combinatorics. I will explain how this duality is in fact a shadow
of a much deeper relationship, in which certain categories of sheaves
on the varieties are Koszul dual to each other.

I will report on joint work with E. Macri on the space of stability
conditions for the derived category of the total space of the canonical
bundle on the projective plane. It is a 3–dimensional manifold, with
many chamber decompositions coming from the behaviour of moduli spaces
of stable objects under change of stability conditions.

I will explain how this space is related to classical results by Drezet
and Le Potier on stable vector bundles on the projective plane. Using
the space helps to determine the group of auto-equivalences, which
includes a subgroup isomorphic to \Gamma_1(3). Finally, via mirror symmetry,
it contains a universal cover of the moduli space of elliptic curves
with \Gamma_1(3)–level structure.

I will present work-in-progress on the construction of compactifications of the moduli space of curves with A_k-singularities. These spaces conjecturally give moduli interpretations of certain log canonical models of the moduli space of curves. This is joint work with David Smyth and Fred van der Wyck.

It is expected that a general K3 surface does not admit
self rational maps of degree > 1. I'll give a proof of this conjecture for
K3 surfaces of genus at least 4.

MATH ANNEX 1102 (relocated because of PIMS closure)

Mon 21 Dec 2009, 1:30pm-3:00pm

Abstract

In his influential works, A. Okounkov showed how to associate a convex body to a very ample G-line bundle L on a projective G-variety X such that it projects to the moment polytope of X and the push-forward of the Lebesgue measure on it gives the Duistermaat-Heckamnn measure for the correspoding Hamiltonian action. He used this to prove the log-concavity of multiplicities in this case. Motivated by his work, recently Lazarsfeld-Mustata and Kaveh-Khovanskii developed a general theory of Newton-Okounkov bodies (without presence of a G-action).

In this talk, I will go back to the case where X has a G-action. I discuss how to associate different convex bodies to a graded G-algebra which in particular encode information about the multiplicities of the G-action. Using this I will define the Duistermaat-Heckmann measure for a graded G-algebra and prove a Brunn-Minkowski inequality for it. Also I will prove a Fujita approximation type result (from the theory of line bundles) for this Duistermaat-Heckmann measure. This talk is based on a preprint in preparation joint with A. G. Khovanskii.

In classical Lie theory a homomorphism of Lie groups f : H--> G, with H simply connected, is uniquely given by its effect on
the Lie algebras Lie(f) : Lie(H) --> Lie(G). When f : H --> G is a weak
morphism of Lie 2-groups, with H 2-connected (i.e., \pi_iH vanish
for i=0,1,2), we prove that f is uniquely given by Lie(f), where
Lie(f) : Lie(H) --> Lie(G) is the induced morphism in the derived
category of 2-terms diff. graded Lie algebras. We also exhibit a
functorial construction of the 2-connected cover H<2> of a Lie
2-group H.

Let G be a finite group. A G-gerbe over a space X may be
intuitively thought of as a fiber bundle over X with fibers being the
classifying space (stack) BG. In particular BG itself is the G-gerbe
over a point. A more interesting class of examples consist of G-gerbes
over BQ, which are equivalent to extensions of the finite group Q by G.
Considerations from physics have led to conjectures asserting that
the geometry of a G-gerbe Y over X is equivalent to certain "twisted"
geometry of a "dual" space Y'. A lot of progresses have be made recently
towards proving these conjectures in general. In this talk we'll try to
explain these conjectures in the elementary concrete examples of G-gerbes
over a point or BQ.

The stack of stable maps parameterizes maps from a complete curves having at worst nodal singularities into a smooth scheme. Generally this stack is not smooth, but we will explain how it can be made smooth by relaxing the condition that the source curves be complete. Although the resulting stack is not fibered in groupoids, and therefore may not be easily accessible to geometric intuition, it is a natural setting in which to construct the virtual fundamental class. We will discuss how this generalization can be used to prove a conjecture of Abramovich and Fantechi relating the virtual fundamental classes of two different moduli spaces parameterizing stable maps into mildly singular schemes.

In the first half of the talk, I will explain the notion of PT stability, as defined by Bayer. I will also explain how it is related to classical stability conditions on sheaves, and other Bridgeland-type stability conditions. In the second half of the talk, I will discuss results on the moduli space of PT-stable objects from my thesis. In particular, I will explain how to use semistable reduction to obtain the valuative criterion of completeness for PT-stable objects.

I will present a global construction of the Neron model for degenerating families of intermediate Jacobians; a classical case would be families of abelian varieties. The construction is based on Saito's theory of mixed Hodge modules; a nice feature is that it works in any dimension, and does not require normal crossing or unipotent monodromy assumptions. As a corollary, we obtain a new proof for the theorem of Brosnan-Pearlstein that, on an algebraic variety, the zero locus of an admissible normal function is an algebraic subvariety.

Geometrization is a process of replacing finite sets by algebraic varieties over finite field and functions on such sets by sheaves on the corresponding variety. I will explain the meaning of the above sentence and state some applications.

I will introduce the notion of categorical resolution of singularities which is based on the concept of
a smooth DG algebra. Then I will compare this notion with the traditional resolution in algebraic geometry and
give some examples.

I will discuss joint work in progress with Rajesh Kulkarni
on the moduli of maximal orders on surfaces. In contrast to the
"classical" case of Azumaya algebras, ramified maximal orders have
several potentially interesting moduli spaces. I will discuss three
different scheme structures on the same set of points: a naive
structure, a structure arising from a non-commutative version of
Koll\'ar's condition on moduli of stable surfaces, and a structure
that comes from hidden Azumaya algebras on stacky models of the
underlying surface. Only (?) the third admits a natural
compactification carrying a virtual fundamental class, giving rise to
potentially new numerical invariants of division algebras over
function fields of surfaces.

Abstract: Recently Kaledin proved a non-commutative generalization of
the Deligne-Illusie theorem about the de Rham complex of an algebraic
variety in characteristic p.
I will explain how his approach can be used to prove new results in
commutative algebraic geometry.

Given a system of polynomial equations in some variables
and
depending on one parameter, when can every solution which is a power
series in the parameter be approximated to arbitrary order by
solutions
which are polynomial in the parameter? Hassett observed that a
necessary
condition is that the generic fiber is "rationally connected", i.e.,
for a
general choice of the parameter, every pair of solutions are
interpolated
by a family of solutions which are the output of a polynomial
function in
one variable. Hassett and Tschinkel conjecture the converse holds:
if a
general fiber is rationally connected, then "weak approximation"
holds.
I will review progress by Hassett -- Tschinkel, Colliot-Th\'el\`ene
--
Gille, A. Knecht, Hassett, de Jong -- Starr, and Chenyang Xu. Then I
will
present a new perspective by Mike Roth and myself using "pseudo ideal
sheaves", a higher codimension analogue of Fulton's effective pseudo
divisors. I will also mention a theorem of Zhiyu Tian, who used this
perspective to relate weak approximation to equivariant rational
connectedness, thereby proving many new cases of weak approximation.

Rigid cohomology is one flavor of Weil cohomology. This entails for instance that one can asociate to a scheme X over F_p a collection of finite dimensional Q_p-vector spaces H^i(X) (and variants with supports in a closed subscheme or compact support), which enjoy lots and lots of nice properties (e.g. functorality, excision, Gysin, duality, a trace formula -- basically everything one needs to give a proof of the Weil conjectures).

Classically, the construction of rigid cohomology is a bit complicated and requires many choices, so that proving things like functorality (or even that it is well defined) are theorems in their own right. An important recent advance is the construction by le Stum of an `Overconvergent site' which computes the rigid cohomology of X. This site involves no choices and so it trivially well defined, and many things (like functorality) become transparent.

In this talk I'll explain a bit about classical rigid cohomology and the overconvergent site, and explain some new work generalizing rigid cohomology to algebraic stacks (as well as why one would want to do such a thing).

Tropicalization is a technique that transforms algebraic geometric objects to combinatorial objects. Specifically, it associates polyhedral complex to subvarieties of an algebraic torus. One may ask which polyhedral complexes arise in this fashion. We focus on curves which are transformed by tropicalization to immersed graphs. By applying toric geometry and Baker's specializing of linear systems from curves to graphs, we give a new necessary condition for a graph to come from an algebraic curve. In genus 1 and in certain geometric situation, this condition specializes to the well-spacedness condition discovered by Speyer and generalized by Nishinou and Brugalle-Mikhalkin. The techniques in this talk give a combinatorial way of thinking about deformation theory which we hope will have further applications.

I aim to explain a recent paper of my collaborator Bai-Ling Wang in which he proves that there is a generalisation of the Baum-Douglas geometric cycles which realise ordinary K-homology classes to the case of twisted K-homology. We propose that these twisted geometric cycles are D-branes in string theory. There is an analogous picture for manifolds that are not string.

I will present a joint work with Behrend.
For any smooth projective variety, we construct a differential graded scheme (stack) structure on the moduli space of
complexes of coherent sheaves. The construction uses the Hochschild cochain complex of A infinty bi-modules.
As an application, we show that the DT/PT wall crossing can be intepreted as change of stability conditions on
dg schemes.

Let G be a simple Lie group or Kac-Moody group and P a parabolic
subgroup.
One of the goals Schubert calculus is to understand the product
structure
of the cohomology ring H^*(G/P) with respect to its basis of Schubert
classes. If G/P is the Grassmannian, then the structure constants
corresponding to the Schubert basis are the classical
Littlewood-Richardson
coefficients which appear in various topics such as enumerative
geometry,
algebraic combinatorics and representation theory.

In this talk, I will discuss joint work with A. Berenstein in which
we give
a combinatorial formula for these coefficients in terms of the Cartan
matrix corresponding to G. In particular, our formula implies
positivity
of the “generalized” Littlewood-Richardson coefficients in the
case
where the off diagonal Cartan matrix entries are not equal to -1. Moreover, this positivity result does not rely on the geometry
of
the flag variety G/P.

Motivated by applications to Brill-Noether theory and higher-rank Brill-Noether theory, we discuss several variations on Grassmannians. These include "doubly symplectic Grassmannians", which parametrize subspaces which are simultaneously isotropic for a pair of symplectic forms, "linked Grassmannians", which parametrize tuples of subspaces of a chain of vector spaces linked via linear maps, and "symplectic linked Grassmannians", which is an amalgamation of the linked Grassmannian and symplectic Grassmannian.

Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and this construction has recently been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, this Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I'll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. This project was motivated by examples, and as an application, I'll describe some toric degenerations of flag varieties and Schubert varieties. There will be pictures of polytopes.

A 0-dimensional scheme is said to be "smoothable" if it deforms to a disjoint union of points. Determining if a given 0-dimensional scheme is smoothable seems to be quite a difficult problem in general, and I will survey some of the main results in this area of research, including some recent progress that is joint work with David Eisenbud and Mauricio Velasco. In particular, I will explain how Gale duality provides a geometric obstruction to smoothability.

One of the central theorems of classical Lie theory is that all split Cartan subalgebras of a finite dimensional simple Lie algebra over an algebraically closed field are conjugate.This result, due to Chevalley, yields the most elegant proof that the type of the root system of a simple Lie algebra is its invariant. In infinite dimensional Lie theory maximal abelian diagonalizable subalgebras (MADs) play the role of Cartan subalgebras in the classical theory. In the talk we address the problem of conjugacy of MADs in a big class of Lie algebras which are known in the literature as extended affine Lie algebras (EALA). To attack this problem we develop a bridge which connects the world of MADs in infinite dimensional Lie algebras and the world of torsors over Laurent polynomial rings.

To any polynomial over a perfect field of positive characteristic, one may associate an invariant called the F-pure threshold. This invariant, defined using the Frobenius morphism on the ambient space, can be thought of as a positive characteristic analog of the well-known log canonical threshold in characteristic zero. In this talk, we will present some examples of F-pure thresholds, and discuss the relationship between F-pure thresholds and log canonical thresholds. We also point out how these results are related to the longstanding open problem regarding the equivalence of (dense) F-pure type and log canonical singularities for hypersurfaces in complex affine space.

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli by pfaffian method and determine their fundamental topological invariants. The existence of CY 3-folds with the computed invariants was previously conjectured by C. van Enckevort and D. van Straten. We then report mirror symmetry for these non-complete intersection CY 3-folds. We explicitly build their mirror candidates, some of which have 2 LCSLs, and check the mirror phenomenon.

Log canonical thresholds are invariants of singularities that play an important role in birational geometry. After an introduction to these invariants, I will describe recent progress on a conjecture of Shokurov predicting the Ascending Chain Condition for such invariants in any fixed dimension. This is based on joint work with Tommaso de Fernex and Lawrence Ein.

Invariants of singularities are defined in birational geometry via divisorial valuations, and are computed by resolution of singularities. In positive characteristic, one defines similar invariants via the action of the Frobenius morphism. The talk will give an overview of the known results and conjectures relating the two sets of invariants.

In 1960, Feit and Fine were interested in the question posed by the title and to answer it, they found a beautiful formula for the number of pairs of commuting n by n matrices with entries in the field F_q. Their method amounted to finding a stratification of the variety of commuting pairs of matrices into strata each of which is isomorphic to an affine space (of various dimensions). Consequently, their computation can be interpreted as giving a formula for the motivic class of the commuting variety, that is, its class in the Grothendieck group of varieties. We give a simple, new proof of their formula and we generalize it to various other settings. This is joint work with Andrew Morrison.

A quantum cluster algebra is a subalgebra of an ambient skew field of rational functions in finitely many indeterminates. The quantum cluster algebra is generated by a (usually infinite) recursively defined collection called the cluster variables. Explicit expressions for the cluster variables are difficult to compute on their own as the recursion describing them involves division inside this skew field. In this talk I will describe the rank 2 cluster variables explicitly by relating them to varieties associated to valued representations of a quiver with 2 vertices. I will also indicate to what extent the theory. I present is applicable to higher rank quantum cluster algebras.

Informally, the essential dimension of a finite group is the minimal number of parameters required to describe any of its actions. It has connections to Galois cohomology and several open problems in algebra. I will discuss how one can use techniques from birational geometry to compute this invariant and indicate some of its applications to the Noether Problem, inverse Galois theory, and the simplification of polynomials.

Building on I. Dolgachev and V. Iskovskikh's recent work classifying finite subgroups of the plane Cremona group, I will classify all finite groups of essential dimension 2. In addition, I show that the symmetric group of degree 7 has essential dimension 4 using Yu. Prokhorov's classification of all finite simple groups with faithful actions on rationally connected threefolds.

I will explain an interpretation of Illusie's results on the deformation theory of commutative rings in terms of the cohomological classification of torsors and gerbes. Then I'll show how this point of view can be used to solve some other deformation problems. I'll also indicate some deformation problems that I don't know how to approach this way.

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to provide a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a result regarding topological obstructions for this problem. I will also address the question of whether a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.

The Hilbert scheme of projective space is a fundamental moduli space in algebraic geometry. The naive generalization of the Hilbert scheme can fail to exist for some spaces of interest, however. D. Rydh and I have generalized the Hilbert scheme, to the Hilbert stack, and have shown that the Hilbert stack is always algebraic. I will describe the Hilbert stack and some of the ideas behind the proof.

The structure of the zero set of a multivariate polynomial is a topic of
wide interest, in view of its ubiquity in problems of analysis, algebra,
partial differential equations, probability and geometry. The study of
such sets originated in the pioneering work of Jung, Abhyankar and
Hironaka and has seen substantial recent advances in an algebraic setting.

In this talk, I will mention a few situations in analysis where the study of
polynomial zero sets plays a critical role, and discuss prior work in this
analytical framework in two dimensions. Our main result (joint with
Tristan Collins and Allan Greenleaf) is a formulation of an algorithm for
resolving singularities of a real-analytic function in any dimension with
a view to applying it to a class of problems in harmonic analysis.

This is a report on my joint work with Prakash Belkale and Nicolas Ressayre. We prove a generalization of Fulton’s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.

Vanishing theorems and rational singularities are closely related and play important roles in classification theory as well as other areas of algebraic geometry. In this talk I will discuss these roles their interrelations and a new notion that helps understand these singularities and their connections to the singularities of the minimal model program and moduli theory of higher dimensional varieties.

We consider moduli spaces of semistable sheaves on an elliptically fibered K3 surface, so that the first Chern class of the sheaves is a numerical section. For pairs of complementary such moduli spaces subject to numerical restrictions, we establish the strange duality isomorphism on sections of theta line bundles. We will also present applications to Brill-Noether theory for sheaves on a K3.

I will describe an action of a quantized Heisenberg algebra on the (derived) categories of coherent sheaves on Hilbert schemes of ALE spaces (crepant resolutions of C^2/G). This action essentially lifts the actions of Nakajima and Grojnowski on the cohomology of these spaces. (Joint with Tony Licata.)

After reviewing some basic constructions with multiplier ideals on complex algebraic varieties, we recall the definition of multiplier ideals in positive characteristic and highlight the failure of some desirable properties to carry over in this setting. This leads us to a related measure of singularities coming from commutative algebra -- the test ideal -- which seems to exhibit better behavior than the multiplier ideal in positive characteristic. While test ideals were first introduced in the theory of tight closure, our goal in this talk will be to describe a new and algebro-geometric characterization of test ideals using regular alterations. This characterization is also holds for multiplier ideals in characteristic zero (but not in positive characteristic!!!), providing a kind of uniform description with new insight and intuition. Time permitting, we will use this result to give an analogue of Nadel Vanishing in positive characteristic.

In this talk, I will explain SYZ proposal on describing mirror symmetry as a Fourier-Mukia transformation along special Lagrangian torus fibration. By computing certain open Gromov-Witten invariants, we show that the mirror map is the same as the SYZ map for certain toric Calabi-Yau manifolds.

In this talk I will give an introduction to Donaldson-Thomas invariants, and then their motivic incarnation. I'll discuss motivic vanishing cycles and lambda rings, before moving to the main example of the talk - the one loop quiver with potential. It turns out that the motivic DT invariants in this simple example have a neat presentation, and in a break with other worked out examples these invariants really involve the mondromy of the motivic vanishing cycle.

We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_X^{r}(-n)-->F where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique. In a sequel to this project (arXiv:1101.2251), we show how to compute similar invariants associated to frozen triples using Kontsevich Soibelman, Joyce-Song wall-crossing techniques.

Projectivized toric vector bundles are a large class of rational varieties that share some of the pleasant properties of toric varieties and other Mori dream spaces. Hering, Mustata and Payne proved that the Mori cones of these varieties are polyhedral and asked if their Cox rings are indeed finitely generated. We present the complete answer to this question. There are several proofs of a positive answer in the rank two case [Hausen-Suss, Gonzalez]. One of these proofs relies on the simple structure of the Okounkov body of these varieties with respect to a special flag of subvarieties. For higher ranks we study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces [Gonzalez-Hering-Payne-Suss]. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.

I will talk on joint work in progress with Behrang Noohi. We study the GIT problem given by the differential graded Lie algebra of Hochschild cochains of a finite graded algebra. This will lead to a definition of stability for non-commutative polarized projective schemes, and to the construction of quasi-projective moduli spaces for them. These moduli spaces are differential graded schemes. There may be new moduli spaces with symmetric obstruction theories coming out of this.

Let L be a reductive Lie algebra. The strong Macdonald theorems of Fishel, Grojnowski, and Teleman state that the cohomology algebras of L[z]/z^N and L[z,s] (where s is an odd variable) are free skew-commutative algebras with generators in certain degrees. The theorems were originally conjectured by Hanlon and Feigin as Lie algebra cohomology extensions of Macdonald's constant term identity in algebraic combinatorics. The proof uses ideas from the Kahler geometry of the loop Grassmannian.

I will explain how to extend Fishel, Grojnowski, and Teleman's ideas to generalized flag varieties of (twisted) loop groups, and consequently get strong Macdonald theorems for p[s] and p/z^N p when p is a parahoric. When p has a non-trivial parabolic component the cohomology of p/z^N p is no longer free, as it contains a factor which is isomorphic to the cohomology algebra of the flag variety of the corresponding parabolic.

We will introduce and study the notion of an equivariant pretheory. Basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.

We construct an isomorphism from Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) over a perfect field to a corresponding Hom group in Voevodsky's triangulated category of effective motivic complexes.

Let X be a smooth projective scheme over a field F. We say that Rost nilpotence is true for X in the category of Chow motives with integral coefficients if for any field extension E/F the kernel of

CH_2(S x S) --> CH_2(S_E x S_E)

consists of nilpotent correspondences. In my talk I will present a proof of Rost nilpotence for surfaces over fields of characteristic zero which uses Rost's theory of cycle modules.

This talk will explore a technique of using equivariant cohomology to say something about the action of a group on the cohomology of a space. In particular, we will look at examples of cohomology of flag varieties and configuration spaces. Also, we will look at a family of algebras with an algebro-geometric interpretation that admits an S_n action, and use the results we developed to make progress toward a result about these algebras.

Topological essential dimension of a covering is the minimal dimension of a base-space such that the original covering can be induced from some covering over this base-space.

We will see how to compute the topological essential dimension for coverings over tori.

Surprisingly this question turns out to be useful in obtaining estimates in Klein's resolvent problem: what is the minimal number k such that the equation z^n+a_1z^n+...+a_n=0 with complex coefficients a_1,...,a_n can be reduced by means of a rational substitution y=R(z,a_1,...,a_n) to an equation on y depending on k algebraically independent parameters.

We will also obtain some bounds in the analogue of this question for other algebraic functions and get a sharp result for functions on C^n unramified outside of coordinate hyperplanes.

The physicists Bershadsky, Cecotti, Ooguri and Vafa argued that the mirror to the theory of Gromov-Witten invariants is provided by a certain quantum field theory on Calabi-Yau varieties. I'll describe joint work in progress with Si Li, which gives a rigorous construction of the BCOV quantum field theory. In the case of the elliptic curve, Li has shown that our theory recovers the Gromov-Witten invariants of the mirror curve, and so proving mirror symmetry in this example.

Using techniques from the theory of Banach algebras, Kuranishi constructed an analytic germ of the moduli space of deformations of a holomorphic vector bundle on a compact complex manifold. In order to extend his results to deformation theory of perfect complexes, we introduce higher analytic stacks, which are simplicial Banach analytic varieties satisfying a horn filler condition modeled on that satisfied by Kan complexes. We show that there is a natural way to attach a higher analytic stack to a Banach algebra, and apply this to the deformation theory of perfect complexes. This is a joint work with Kai Behrend.

The coherent constructible correspondence matches coherent sheaves on a toric variety to constructible sheaves on a compact torus T^n. Microlocal sheaf theory allows one to view the latter sort of object as a Lagrangian submanifold in the symplectic manifold T^n x R^n, making this a form of mirror symmetry. I will discuss this correspondence, and an extension of of it to hypersurfaces in toric varieties, which in some sense matches coherent sheaves to Legendrian submanifolds of the contact manifold T^n x S^{n-1}.

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

I will survey the constructions of holomorphic Chern-Simons functionals for the moduli spaces of sheaves on CY 3-folds and several applications to the theoretical and computational aspects of Donaldson-Thomas theory.

We characterize Schubert varieties (for GLn) which are local complete intersections (lci) by the combinatorial notion of pattern avoidance. For the Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement only requires ordinary pattern avoidance, showing the other Schubert varieties are not lci appears to require more complicated combinatorial ideas which have their own geometric underpinnings. The Schubert varieties defined by inclusions, originally introduced by Reiner and Gasharov, turn out to be an important subclass of lci Schubert varieties. Using the explicit equations at the identity for the lci Schubert varieties, we can recover formulas for some of their local singularity invariants at the identity as well as explicit presentations for their cohomology rings.

This is joint work with Henning Ulfarsson (Reykjavik U.).

If X is a finite CW complex of small dimension, information about the homotopy groups of unitary groups can be translated into cohomological classification results for complex vector bundles on X. I will explain how A^1-homotopy theory can be used in an analogous fashion in the classification of vector bundles of on smooth affine varieties of small dimension. In particular, I will explain some joint work (in progress) with J. Fasel which shows how to give a complete classification of vector bundles on smooth affine 3-folds over certain fields. No knowledge of A^1-homotopy theory will be assumed.

We explain how the computation of motivic Donaldson-Thomas invariants associated to a quiver with potential reduces to the computation of the motivic classes of simpler quiver varieties. This has led to the calculation of these invariants for some interesting Calabi-Yau geometries derived equivalent to a quiver with potential. Here we observe q-deformations of the classical generating series.

In this talk we will present a formula to count the number of hyperelliptic curves on a polarized Abelian surface, up to translation. This formula is obtained using orbifold Gromov-Witten theory, the crepant resolution conjection and the Yau-Zaslow formula to related hyperelliptic curves to rational curves on the Kummer surface Km(A). We will show how this formula can be described in terms of certain generating functions studied by P. A. MacMahon, which turn out to be quasimodular forms.

The Mori cone of curves of the Grothendieck-Knudsen moduli space of stable rational curves with n markings, is conjecturally generated by the one-dimensional strata (the so-called F-curves). A result of Keel and McKernan states that a hypothetical counterexample must come from rigid curves that intersect the interior. In this talk I will show several ways of constructing rigid curves. In all the examples a reduction mod p argument shows that the classes of the rigid curves that we construct can be decomposed as sums of F-curves. This is joint work with Jenia Tevelev.

Let X be a variety over a field k, with a fixed rational point x_0 in X(k). Nori defined a profinite group scheme N(X,x_0), usually called Nori's fundamental group, with the property that homomorphisms N(X,x_0) to a fixed finite group scheme G correspond to G-torsors P --> X, with a fixed rational point in the inverse image of x_0 in P. If k is algebraically closed this coincides with Grothendieck's fundamental group, but is in general very different. Nori's main theorem is that if X is complete, the category of finite-dimensional representations of N(X,x_0) is equivalent to an abelian subcategory of the category of vector bundles on X, the category of essentially finite bundles.
After describing Nori's results, I will explain my work in collaboration with Niels Borne, from the University of Lille, in which we extend them by removing the dependence on the base point, substituting Nori's fundamental group with a gerbe (in characteristic 0 this had already been done by Deligne), and give a simpler definition of essentially finite bundle, and a more direct and general proof of the correspondence between representations and essentially finite bundles.

The Kodaira vanishing theorem and its generalizations are extremely important tools in higher dimensional geometry and the failure of these theorems in positive characteristic causes great difficulties in extending the existing theories to that realm. In this talk I will discuss new results about cases where an appropriate vanishing theorem holds and cases where the expected one fails even in characteristic zero. These results are joint works (separately) with Christopher Hacon and with János Kollár.

In this talk we compare several different stratifications of parameter spaces of sheaves. The starting point is the infinite Yang-Mills stratification of the space of vector bundles on a compact Riemann surface, which is equal to the stratification by Harder-Narasimhan types. We then go on to look at finite stratifications of some quot schemes associated to a certain group action (the geometric invariant theory quotient for this action is a moduli space for sheaves) and relate this to a stratification of the quot scheme by Harder-Narasimhan types. Finally we discuss the limitations of the finite stratifications and how we could instead modify the set up to get infinite stratifications.

-2 curves are a favorite toy of birational geometers working in dimension 3 - they are slightly more complicated cousins of the resolved conifold. In this talk I'll try to give a reasonably self contained introduction to the theory of motivic DT invariants, and integrality, by explaining how this theory plays out in the case of "noncommutative" -2 curves. It turns out that, in common with the noncommutative conifold, the motivic DT partition function for -2 curves have a strikingly nice form, confirming the integrality conjecture in this case.

Let X be a Calabi-Yau threefold. We study the symmetric trilinear form on the integral second cohomology group of X defined by the cup product. Our study is motivated by C.P.C. Wall's classification theorem, which roughly says that the diffeomorphism class of a spin sixfold is determined by the trilinear form. We investigate the interplay between the Chern classes and the trilinear form of X, and demonstrate some numerical relations between them. If time permits, we also discuss some properties of the associated cubic form. This talk is based on a joint work with P.H.M. Wilson.

The Gopakumar-Marino-Vafa formula, proven almost ten years ago, evaluates certain triple Hodge integrals on moduli spaces of curves in terms of Schur functions. It has since been realized that the GMV formula is a special case of the Gromov-Witten/Donaldson-Thomas correspondence for Calabi-Yau threefolds.

In this talk, I will introduce an orbifold generalization of the GMV formula which evaluates certain abelian Hodge integrals in terms of loop Schur functions. I will introduce local Z_n gerbes over the projective line and show how the gerby GMV formula can be used to prove the GW/DT correspondence for this class of orbifolds. With the remaining time, I will sketch the main ideas in the proof of the formula and discuss generalizations to other geometries.

Let p be a prime, r >= 3, and n_i = p^{a_i} for positive integers a_1,...,a_r. Set G = GL_{n_1} x ... x GL_{n_r}, and let \mu be a central subgroup of G. The Galois cohomology set H^1(K, G/\mu) classifies r-tuples of central simple algebras satisfying linear equations in the Brauer group Br(K). We study the essential dimension of G/\mu by constructing the 'code' associated to /mu.

The equivariant embeddings of a split torus have been well-known since the 70s. The isomorphism classes of such embeddings are classified by combinatorial objects called fans (after Demazure). In this talk, we address the classification of the embeddings of a non-necessary split torus and ask: Are the isomorphisms classes of such embeddings classified by Galois-stable fans? If time permits, we will discuss the analogous results in the setting of spherical homogeneous spaces.

The Schubert subvarieties of a rational homogeneous variety X are distinguished by the fact that their homology classes form an additive basis of the integer homology of X. In general, the Schubert varieties are singular.

The cominuscule rational homogeneous varieties are those admitting the structure of a compact Hermitian symmetric space (eg. complex Grassmannians). In this case, type-dependent characterizations of the singular loci are known.

I will discuss a type-independent description, by representation theoretic data, of the singular loci. The result is based on a characterization (joint with D. The) of the Schubert varieties by an non-negative integer and a marked Dynkin diagram.

(If there is time left, I will discuss the project in which the integer-diagram characterization arose as a technical lemma. This work aims to determine whether or not the Schubert classes admit any algebraic representatives (other than the Schubert varieties). It is a remarkable consequence of Kostant's work that these algebraic representatives are solutions of a system of PDE; as a consequence, differential geometric techniques may be applied to this algebro-topological question.)

One remarkable application of classical Schubert calculus on the cohomology of the Grassmannian is its close connection to the eigenvalue problem on sums of hermitian matrices. The eigenvalue problem asks: Given three sequences of real numbers, do there exist hermitian matrices A+B=C with eigenvalues given by the three sequences? This problem has a generalization to eigenvalues of majorized sums of hermitian matrices where we replace "A+B=C" with "A+B>C".

In this talk, I discuss joint work with D. Anderson and A. Yong where we show that the eigenvalue problem on majorized sums is related to the Schubert calculus on the torus-equivariant cohomology of the Grassmannian in the same way that classical Schubert calculus is related to eigenvalue problem on usual sums of Hermitian matrices. One consequence of this connection is a generalization of the celebrated saturation theorem to T-equivariant Schubert calculus.

Higher stacks arise in many contexts in algebraic geometry and differential topology. The simplest type are higher principal bundles, special cases of which include principal bundles and n-gerbes. Locally, these objects are presentable by higher cocycles on a hypercover. With ordinary principal bundles, we obtain a bundle from a cocycle by using the cocycle to construct a Lie groupoid over the trivial bundle on the cover, and then passing to its orbit space. We establish the existence of an analogous construction for arbitrary higher principal bundles. Unpacking this construction in examples, we recover the familiar definitions of principal bundles, bundle gerbes, multiplicative gerbes and their equivariant versions, now seen as instances of a single construction. Applications beyond this include establishing a representability criterion for connected simplicial presheaves, and a Lie's 3rd theorem for finite dimensional L_oo-algebras.

Let K be a regular noetherian commutative ring. I consider finite type commutative K-algebras and K-schemes. I will begin by explaining the theory of rigid residue complexes on K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes on K-schemes and their functorial properties. For any map between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map of schemes is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.

Then I will move to finite type Deligne-Mumford K-stacks. Any such stack has a rigid residue complex on it, and for any map between stacks there is a trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish by presenting two conjectures, that refer to Grothendieck Duality for proper maps between DM stacks. A key condition here is that of tame map of stacks.

Codimension 2 complete intersections in P^N have a natural parameter space that is a projective bundle over a projective space given by the data of the two equations.

In this talk, we will be interested in the birational geometry of this parameter space. In particular, is it a Mori dream space? If this is the case, is it possible to describe its MMP explicitly? We will give motivations for these questions and answers in particular cases.

Let X and X' be two smooth Deligne-Mumford stacks. We call dash arrow X-->X' a Crepant Transformation if there exists a third smooth Deligne-Mumford stack Y and two morphisms \phi:Y-> X, \phi': Y-> X' such that the pullbacks of canonical divisors are equivalent, i.e. \phi^*K_{X}\cong \phi'^*K_{X'}. The crepant transformation conjecture says that the Gromov-Witten theory of X and X' is equivalent if X-->X' is a crepant transformation. This conjecture was well studied in two cases: the first one is the case when X and X' are both smooth varieties; the other is the case that there is a real morphism X-> |X'| to the coarse moduli space of X', resolving the singularities of X'. In this talk I will present some recent progress for this conjecture, especially in the case when both X and X' are smooth Deligne-Mumford stacks.

I will explain how to associate a Satake-type isomorphism to certain characters of the compact torus of a split reductive group over a local field. I will then discuss the geometric analogue of this isomorphism. (Joint work with Travis Schedler).

I'll discuss two results regarding how DT invariants (of smooth and projective Calabi-Yau threefolds) change under birational modifications. The first deals with flops and the second is related to the McKay correspondence and work of Jim Bryan and David Steinberg.

For a smooth projective variety over a finite field, Poonen’s Bertini Theorem computes the probability that a high degree hypersurface section of that variety will be smooth. We prove a semiample generalization of Poonen's result, where the probability of smoothness is computed as a product of local probabilities taken over the ﬁbers of a corresponding morphism. This is joint with Melanie Matchett Wood.

Note for Attendees

ESB 4133 is the library room attached to the PIMS lounge.

A divisor on a curve is called "special'' if its linear equivalence class is larger than expected. On a hyperelliptic curve, all such come from pullbacks of points from the line. But one can ask subtler questions. Fix a degree zero divisor Z; consider the space parameterizing divisors D where D and D+Z are both special. In other words, we wish to study the intersection of the theta divisor with a translate; the main goal is to understand its singularities and its cohomology.

The real motivation comes from number theory. Consider, in products of the moduli space of elliptic curves, points whose coordinates all correspond to curves with complex multiplication. The Andre-Oort conjecture controls the Zariski closure of sequences of such points (and in this case is a theorem of Pila) and a rather stronger equidistribution statement was conjectured by Zhang. The locus introduced above arises naturally in the consideration of a function field analogue of this conjecture. This talk presents joint work with Jacob Tsimerman.

By a theorem of Behrend Donaldson-Thomas invariants can be defined interns of a certain constructible function. We will compute this function at all points in the Hilbert scheme of points in three dimensions and see that it is constant. As a corollary we see that this Hilbert scheme of points is generically reduced and its components have the same dimension mod 2. This gives an application of the techniques of BPS state counting to a problem in Algebraic Geometry.

We define an action of a Heisenberg algebra on categories of coherent sheaves on Hilbert schemes of points of C^2. This lifts the constructions of Nakajima and Grojnowski from cohomology to derived categories. Vertex operator techniques are then used to extend this to an action of sl_infty. We end with applications to knot homology and a discussion of future research directions.

AKSZ Theory is a topological version of the Sigma Model in quantum field theory, and includes many of the most important topological field theories. I will present two generalizations of the usual AKSZ construction. The first is closely related to the generalization from symplectic to Poisson geometry. (AKSZ theory has already incorporated an analogous step from the geometry of cotangent bundles to the geometry of symplectic manifolds.) The second generalization is to phrase the construction in an algebrotopological language (rather than the usual language of infinite-dimensional smooth manifolds), which allows in particular for lattice versions of the theory to be proposed. From this new point of view, renormalization theory is easily recognized as the way one constructs strongly homotopy algebraic objects when their strict versions are unavailable. Time permitting, I will end by discussing an application of lattice Poisson AKSZ theory to the deformation quantization problem for Poisson manifolds: a _one_-dimensional version of the theory leads to a universal star-product in which all coefficients are rational numbers.

Suppose that a polarized self-morphism \phi of X dominates a polarized self-morphism \psi of Y. Szpiro and Tucker asked if, if \phi is isotrivial, then \psi also descends to an isotrivial morphism. We give an affirmative answer in a large set of cases, including the case Y = P^1. At heart is a result of Petsche, Szpiro, and Tepper on isotriviality and potential good reduction for self-maps of P^n, which we extend to more general polarized self-morphisms of projective varieties.

I will discuss a combinatorial problem which comes from algebraic geometry. The problem, loosely, is to show that two theories for "counting" "curves" (Pandharipande-Thomas theory and reduced Donaldson-Thomas theory) give the same answer. I will prove a combinatorial version of this correspondence in a special case (X is toric Calabi-Yau), where the difficult geometry reduces to a study of the "topological vertex'' (a certain generating function) in these two theories. The combinatorial objects in question are plane partitions, perfect matchings on the honeycomb lattice and the double dimer model.

There will be many pictures. This is a combinatorics talk, so no algebraic geometry will be used, except as an oracle for predicting the answer.

Given a planar curve singularity, Oblomkov and Shende conjectured a precise relationship between the geometry of its Hilbert scheme of points and the HOMFLY polynomial of the associated link. I will explain a proof of this conjecture, as well as a generalization to colored invariants proposed by Diaconescu, Hua, and Soibelman.

Hessenberg varieties are closed subvarieties of the full flag variety. Examples of Hessenberg varieties include both Springer fibers and the flag variety. In this talk we will show that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor of the Lie algebra are paved by affines. We then provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements, generalizing results of Tymoczko.

I will apply Martha Precup's theorem on affine pavings to describe the equivariant cohomology algebras of (regular) Springer fibres in terms of certain Weyl group orbits. This will also yield a simple description of Springer's representation of W on the cohomology of the above Springer fibres.

This talk addresses the problem of how to analyze and discuss singularities of a variety X that "naturally'' sits inside a flag manifold. Our three main examples are Schubert varieties, Richardson varieties and Peterson varieties. The overarching theme is to use combinatorics and commutative algebra to study the "patch ideals", which encode local coordinates and equations of X. Thereby, we obtain formulas and conjectures about X's invariants. We will report on projects with (subsets of) Erik Insko (Florida Gulf Coast U.), Allen Knutson (Cornell), Li Li (Oakland University) and Alexander Woo (U. Idaho).

The existence of a height pairing on the equivalence relation defining Bloch's higher cycle groups is a surprising consequence of some recent joint work by myself and Xi Chen on a nontrivial K_1-class on a self-product of a general K3 surface. I will explain how this pairing comes about.

We discuss possible generalizations of the concept of Schubert and Grothendieck polynomials to the context of an arbitrary algebraic oriented cohomology theory. We apply these techniques to a rational formal group law and
obtain formulas for the respective polynomials in the A_n-cases. This is a joint project with C. Zhong.

The classical example of a log scheme is a variety X with a normal crossing divisor D. One can study differential forms on X with logarithmic (that is, order one) poles along D. Dual to these are log tangent vectors on (X, D), which have "zeroes along D." As ordinary jet schemes generalise tangent spaces, log jet schemes generalise log tangent spaces. We'll introduce the construction of log jet schemes for log schemes in the sense of K. Kato, which replace the divisor D with some combinatorial data, and some of their properties. This talk won't assume familiarity with jet schemes or log geometry.

I will introduce a class of Calabi-Yau manifolds associated to the polytope tilings. Their mirrors provide new insights in the toric mirror symmetry, and are also closely related to certain modular forms. This is a joint work with Siu-Cheong Lau.

The multiplicative Horn problem asks what constraints the eigenvalues of two n x n unitary matrices place on the eigenvalues of their product. The solution of this problem, due to Belkale, Kumar, Woodward, and others, expresses these constraints as a convex polyhedron in 3n dimensions and describes the facets of this polyhedron more or less explicitly. I will explain how the vertices of the polyhedron may instead be described in terms of fixed points of a torus action on a symplectic stratified space, constructed as a quotient of the so-called universal group-valued implosion.

An Abelian variety (of complex dimension g) is an algebraic geometer's version of a torus — it is a variety which is topologically equivalent to a (real) 2g-dimensional torus. Geometers consider the problem of counting the number of curves on an Abelian variety subject to some set of constraints. In dimensions g=1,2, and 3, these geometric numbers have a surprising connections to number theory and combinatorics. They occur as the coefficients of Fourier expansions of various modular forms and they can also be determined in terms of combinatorics of 2D and 3D partitions (a.k.a. box counting). We illustrate this using only elementary ideas from topology and combinatorics in the case of g=1. For g=2 and g=3, we describe recent theorems and conjectures which complete determine the enumerative geometry of Abelian surfaces and threefolds in terms of Jacobi forms and in the process we indicate how Jacobi forms arise from the combinatorics of box counting.

It has been recently shown by Chen-Donaldson-Sun that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the property of K-stability. In general, however, this does not lead to an effective criterion for deciding whether such a metric exists, since verifying the property of K-stability requires one to consider infinitely many special degenerations called test configurations. I will discuss recent joint work with H. Süß in which we show that for Fano manifolds with complexity-one torus actions, there are only finitely many test configurations one needs to consider. This leads to an effective method for verifying K-stability, and hence the existence of a Kähler-Einstein metric. As an application, we provide new examples of Kähler-Einstein Fano threefolds.

This is a report on joint work with Zsolt Patakfalvi. We prove a strengthening of Kollár's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also confirm the Iitaka-Viehweg conjecture on the subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.

After introducing the class of the classifying stack of a (finite) group, BG, in the Grothendieck ring of algebraic stacks, I will present certain cohomological invariants for a group - the Ekedahl invariants.

I am going to show that the class of BG is trivial if G is a finite subgroup of GL_3(k) or if G is a finite linear (or projective) reflection group. (k is a algebraically closed field of characteristic zero.) I will also show that the Ekedahl invariants of the discrete 5-Heisenberg group are trivial.

These results relate naturally to Noether's Problem and to its obstruction, the Bogomolov multiplier.

At the end of the talk, I will link these results to the study of the motivic classes of the quotient varieties V/G by showing that such classes and the classes of BG exhibit the same combinatorial structure. Therefore, despite the title and technical terminology I will aim at making the talk enjoyable also by the combinatorial community.

The Shimura varieties X attached to orthogonal and unitary groups come equipped with a large collection of so-called special cycles. Examples include Heegner divisors on modular curves and Hirzebruch-Zagier cycles on Hilbert modular surfaces. We will review work of Borcherds and Bruinier using regularised theta lifts for the pair (SL_2,O(V)) to construct Green currents for special divisors. Then we will explain how to construct other interesting currents on X using the dual pair (Sp_4,O(V)). We will show that one obtains currents in the image of the regulator map of a certain motivic complex of X. Finally, we will describe how an argument using the Siegel-Weil formula allows to relate the values of these currents to the product of a special value of an L-function and a period on a certain subgroup of Sp_4.

Cox rings of algebraic varieties were defined by Hu and Keel in relation to the minimal model program. The main question in the theory is to determine if the Cox ring of a variety is finitely generated. Such varieties are called Mori Dream Spaces. In this talk I will discuss examples of varieties that are not Mori Dream Spaces. These include toric surfaces blown up at a point and the moduli spaces of rational curves with n points. This is a joint work with Jose Gonzalez.

Given a scheme X and a normal crossings divisor D in X, the Olsson fan of X and D is an algebraic stack that encodes the combinatorics of the components of D and their intersections. I will describe Olsson fans and show how they are constructed. Then I will discuss the moduli space of stable maps from curves into an Olsson fan, and highlighting a number of applications to Gromov-Witten theory.

We explore a relationship between combinatorics and certain moduli spaces appearing in symplectic geometry. The combinatorics comes from the theory of cluster algebras, a kind of unified theory of canonical bases in representation theory and algebraic geometry. Some basic features of cluster algebras are that they are defined from purely combinatorial data (for example, a quiver) and they are coordinate rings of varieties covered by algebraic tori with transition functions of a special, universal form. Despite the originally representation-theoretic motivation for the subject, connections between cluster theory and symplectic geometry emerged later through the appearance of similar formulae in wall-crossing and mirror symmetry.

We will discuss recent work expanding on this connection, in particular providing a universal framework for interpreting cluster varieties as moduli spaces of objects in Fukaya categories of Weinstein 4-manifolds. In simple examples these moduli spaces reduce to well-known ones, such as character varieties of surfaces and positroid cells in the Grassmannian. An accompanying theme, which plays a key role both technically and in relating the symplectic perspective to more standard representation-theoretic ones, is the role of categories of microlocal sheaves as topological counterparts of Fukaya categories. This is joint work with Vivek Shende, David Treumann, and Eric Zaslow.

The concept of basis of a vector space over a field generalizes to the concept of generators of a module over a ring. However, generators carry very little information about the structure of the module, in contrast to bases, which are very useful in the study of vector spaces. Hilbert introduced the approach to describe the structure of modules by free resolutions. Hilbert's Syzygy Theorem shows that minimal free resolutions over a polynomial ring are finite. By a result of Serre, it follows that most minimal free resolutions over quotient rings are infinite. We will discuss the structure of such resolutions. The concept of matrix factorization was introduced by Eisenbud 35 year ago, and it describes completely the asymptotic structure of minimal free resolutions over a hypersurface. Matrix factorizations have applications in many fields of mathematics: for the study of cluster algebras, Cohen-Macaulay modules, knot theory, moduli of curves, quiver and group representations, and singularity theory. Starting with Kapustin and Li, physicists discovered amazing connections with string theory. In a current joint work with Eisenbud, we introduce the concept of matrix factorization for complete intersection rings and show that it suffices to describe the asymptotic structure of minimal free resolutions over complete intersections.

In this talk I will introduce the generalization of relative Donaldson-Thomas theory to 3-dimensional smooth Deligne-Mumford stacks. We adopt Jun Li’s construction of expanded pairs and degenerations and prove an orbifold DT degeneration formula. I’ll also talk about the application in the case of local gerby curves, and its relationship to the work of Okounkov-Pandharipande and Maulik-Oblomkov.

The ADE braid group acts faithfully on the derived category of coherent sheaves on the resolution of the associated Kleinian singularity. In other words, the ADE braid groups are "2-linear" groups. In a similar spirit, the free group is a 2-linear group. In this talk we'll describe a few proofs of these results and explain how spherical twists and triangulated categories are related to some open problems in group theory.

The geometric Satake equivalence relates the category of perverse sheaves on the affine Grassmannian and the representation category of a semisimple group G. We will discuss a quantum K-theoretic version of this equivalence. In this setup the representation category of G is replaced with (a quantum version) of coherent sheaves on G/G. This is joint work Joel Kamnitzer.

The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit.

This talk is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space, which the relevant category is the homotopy category of L_\infty-algebras. Time permitting, I will explain a example of homotopy probability space which law corresponds to variations Hodge structures on a toric hypersurface.

I am going to discuss Alexeev's and Brion's moduli space parametrizing maps
from broken toric varieties into a fixed toric variety V. Following ideas
of Olsson, I will explain how one can obtain a modular description of the
main irreducible component of Alexeev's and Brion's space, using an
analogous moduli space K(V) parametrizing logarithmic maps from broken
toric varieties into V. The resulting space K(V) is in fact a toric stack
-- it is a stacky enrichment of an appropriate Chow quotient of V. I will
conclude by explaining why K(V) and the Chow quotient stack coincide, and
describe explicitly the combinatorial data that determine the latter.

Joint work with Michael Temkin, we extend the old results with Karu, Matsuki and Wlodarczyk from varieties to qe schemes and use this to prove factorization for various other categories.

Let S be a K3 surface. Generating series of Gromov-Witten invariants of the product geometry SxP1 are conjectured to be quasi-Jacobi forms. We sketch a proof of this conjecture for classes of degree 1 or 2 over P1 using genus bounds on hyperelliptic curves in K3 surfaces by Ciliberto and Knutsen. This has applications to a GW/Hilb correspondence for K3 surfaces, and curve counting on SxE, where E is an elliptic curve.

In this talk I will survey some recent and ongoing work of myself and collaborators (David Ben-Zvi, David Nadler, Hendrik Orem), and others, concerning certain topological field theories associated to a complex reductive group G. The basic example of such a theory, assigns the cohomology of the character variety (i.e. moduli of representations of the fundamental group) to a topological surface. To a point, it assigns the categorical group algebra of D-modules on G. I will discuss various approaches to studying this theory, including work from my thesis on parabolic induction and restriction functors, work in progress with Ben-Zvi and Nadler on a monoidal quantization of the the group scheme of regular centralizers using translation functors on Whittaker modules, and a categorical highest weight theorem with Ben-Zvi, Nadler and Orem. Our work is partly motivated by the "Arithmetic Harmonic Analysis" developed by Hausel, Rodriguez-Villegas, and Lettalier, to study the cohomology of character and quiver varieties.

Hall algebras play a prominent role in the interactions between algebraic geometry and representation theory. Recently, "refined" versions of them, called K-theoretic Hall algebras, were introduced by Schiffmann and Vasserot. They have notable connections with the geometric Langlands correspondence, the theory of quantum groups and gauge theories.
In the first part of the talk, I will give an overview of the theory of Hall algebras. In the second part, I will describe some (new) examples of K-theoretic Hall algebras. These algebras are related to some stacks of a certain kind of sheaves on noncompact surfaces. (Work in progress with Olivier Schiffmann.)

An Azumaya algebra of degree n is an algebra locally isomorphic to an nxn matrix algebra, a concept that generalizes that of central simple algebras over fields. The Brauer group consists of equivalence classes of Azumaya algebras and the index of a class in the Brauer group is defined to be the greatest common divisor of the degrees of all Azumaya algebras in that class.

Suppose p and q are relatively prime positive integers. Whereas an Azumaya algebra of degree pq need not, in general, decompose as a tensor product of algebras of degrees p and q, we show that a Brauer class of index pq does decompose as a sum of Brauer classes of indices p and q. The argument requires only the representation theory of GLn, and therefore establishes the result in contexts where one does not have recourse to the theory of fields, for instance in the theory of Azumaya algebras on topological spaces.

Given a smooth surface, the generating series of Euler characteristics of its Hilbert schemes of points can be given in closed form by (a specialisation of) Goettsche's formula. I will discuss a generalisation of this formula to surfaces with rational double points. A certain representation of the affine Lie algebra corresponding to the surface singularity (via the McKay correspondence), and its crystal basis theory, play an important role in our approach. Joint work with András Némethi and Balázs Szendrői.

Let g>2 be a positive integer, and let M_g be the moduli space of smooth curves of genus g over \mathbb{C}. The classical Franchetta conjecture asserts that the Picard group of the generic curve C_{\mu} over Mg is freely generated by its cotangent bundle. It was proved by Arbarello and Cornalba in 1980, Then Mestrano ('87) and Kouvidakis ('91) deducted the Strong Franchetta conjecture, which asserts that the rational points of the relative picard scheme Pic_{C_{\mu} / \mu} are precisely the multiples of the cotangent bundle.

We will show that a suitably modified version of the Franchetta conjecture holds for a different moduli problem over \mathbb{C}, that of principally polarised abelian varieties (p.p.a.v.) of genus g\geq 3 with n-level structure. The abelian Franchetta conjecture states that the generic p.p.a.v. of genus g with n-level structure X_{g,n} has Picard group isomorphic to \mathbb{Z} \oplus (\mathbb{Z}/n\mathbb{Z})^{2g}, where the free part is generated by the bundle inducing the polarization, and the torsion part comes from the level structure.

In the abelian case, the ``weak" statement immediately implies the corresponding ``strong" statement regarding the rational points of the relative Picard scheme. Using duality, we will use this to compute the Picard group of the universal abelian variety \mathscr{X}_{g,n} over the moduli stack \mathscr{C}_{g,n} of genus g p.p.a.v. with n-level structure.

I will report on joint work with D. Carchedi, S. Scherotzke and N. Sibilla, about a comparison between two objects obtained from a fs log scheme over the complex numbers: the "infinite root stack" and the "Kato-Nakayama space". I will also hint at more recent work that explains how parabolic sheaves (with real or rational weights) interact with the picture.

I will be as little technical as possible and focus on examples rather than on the general theory.

Motivated by algebraic geometry, one studies non-commutative analogs of resolutions of singularities. In short, a non-commutative resolution (=NCR) of a commutative ring R is an endomorphism ring of a certain R-module of finite global dimension. However, it is not clear how to construct non-commutative resolutions in general and which structure they have. The most prominent example of NCRs comes from the classical McKay correspondence that relates the geometry of so-called Kleinian surface singularities with the representation theory of finite subgroups of SL(2,\mathbb{C}).

In this talk we will first review this fascinating result, exhibiting the connection to the ubiquitious Coxeter-Dynkin diagrams. Moreover, we will comment on an algebraic version of the correspondence, due to Maurice Auslander.

This leads to joint work in progress with Ragnar Buchweitz and Colin Ingalls about a version of the McKay correspondence when G in GL(n,\mathbb{C}) is a finite group generated by reflections: The group G acts linearly on the polynomial ring S in n variables over \mathbb{C}. When G is generated by reflections, then the discriminant D of the group action of G on S is a hypersurface with a singular locus of codimension 1. We give a natural construction of a NCR of the coordinate ring of D as a quotient of the skew group ring A=S*G. We will explain this construction, which gives a new view on Knörrer's periodicity theorem for matrix factorizations and allows to extend Auslander's theorem to reflection groups.

I will review the notion of a t-structure and discuss some recent uses of t-structures on categories of coherent sheaves in (geometric) representation theory. After reviewing some traditional methods to obtain t-structures I will present a new construction that uses categorical Lie actions. As an application one recovers the category of "exotic sheaves", used in a recent proof of Lusztig's conjectures on a canonical bases for the Grothendieck group of Springer fibers by Bezrukavnikov and Mirković. The new construction is purely geometric, instead of using deep results from modular representation theory. This is joint work with Sabin Cautis.

The minimal multiple of the diagonal to admit a decomposition in the sense of Bloch and Srinivas is called the torsion order of a smooth projective variety. It is bounded above by the greatest common divisor of the degrees of all unirational parameterizations, and is a stable birational invariant. Recently, a degeneration method initiated by Voisin, and developed by Colliot-Thélène and Pirutka, has led to a breakthrough in establishing lower bounds for the torsion order, hence obstructions to stable rationality. The power of this method lies in its mix of inputs from algebraic cycles, Hodge theory, algebraic K-theory, birational geometry, and singularity theory. I will survey the state of the art of this theory, which includes recent work of Chatzistamatiou and Levine, as well as provide some new examples.

We give a new definition of the derived Maurer-Cartan locus MC^*(L), as a functor from differential graded Lie algebras to cosimplicial schemes, whose definition is sufficiently straightforward that it generalizes well to other settings such as analytic geometry. If L is differential graded Lie algebra, let L_+ be the truncation of L in positive degrees i>0. We prove that the differential graded algebra of functions on the cosimplicial scheme MC^*(L) is quasi-isomorphic to the Chevalley-Eilenberg complex of L_+, which is the usual definition of the derived Maurer-Cartan locus in characteristic zero.

We will not assume any prior knowledge of stacks for this talk. Toric stacks, like toric varieties, form a concrete class of examples which are particularly amenable to computation. We give an introduction to the subject and explain how we have used toric stacks to obtain an unexpected result in cycle theory. We end the talk by discussing some conjectures recently introduced by myself and Dan Edidin.

By physical considerations Huang, Katz and Klemm conjecture that the generating series of Donaldson-Thomas invariants of an elliptic Calabi-Yau threefold is a Jacobi form. In this talk I will explain a mathematical approach to proving part of their conjecture. The method uses an autoequivalence of the derived category, and wallcrossing techniques developed by Toda. This leads to strong structure results for curve counting invariants. As a leading example we will discuss the elliptic fibration over P2 in degree 1.

I will give an introduction to gauged Gromov-Witten theory. The theory naturally leads to studying compactifications of the moduli space of G bundles on nodal curves, which I'll discuss briefly. Then I'll focus on a version of gauged Gromov-Witten theory developed by Woodward and Gonzalez and I'll present a theorem which is joint work with Woodward and Gonzalez on the properness of the moduli of scaled gauged maps satisfying a stability condition introduced by Mundet and Schmitt.

In this talk, I will discuss work giving in many cases a complete description of the group of rational sections of the relative Jacobian of a linear system of curves on a surface. By specializing to the case of spectral curves, we are able to determine very explicitly the group of sections of the Hitchin fibration. We will also discuss work in progress to extend this work to principal G-Higgs bundles for more general groups G.

Tautological relations are certain equations in the Chow ring of the moduli space of curves. I will discuss a family of such relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line. These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations. This is joint work with various subsets of S. Grushevsky, F. Janda, X. Wang, and D. Zakharov.

First suggested by Witten in the early 1990's, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves along with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.

The study of Grothendieck rings of varieties in the context of real algebraic geometry has begun since the apparition of motivic integration. Several such rings are of interest, depending notably on the class of functions we are interested in. We will discuss recent progress in the cases of real algebraic varieties and of arc-symmetric sets.

In 1997 Caporaso, Harris and Mazur, motivated by uniformity results in Diophantine Geometry, proposed a conjecture about fibered powers of families of varieties of general type. In particular they conjecture that, if X -> B is a family whose general fiber is a variety of general type, then for large n, the n-th fiber power of X over B dominates a variety of general type. The conjecture has been proved by Abramovich and used to deduce interesting results on the distribution of rational points on projective varieties. I will discuss recent work, joint with Kenny Ascher (Brown), generalizing this to pairs (X,D) of a projective scheme and a divisor, and the new challenges that arise when one tries to obtain analogous results for the distribution of integral points in quasi projective varieties.

The Hodge theory of surfaces provides a link between enumerative geometry and modular forms, via the cohomological theta correspondence. I will present an approach to studying the Gromov-Witten invariants of Weierstrass fibrations over P^2, proving part of a conjectural formula coming from topological string theory.

Classically, there are two objects that are particularly interesting to algebraic geometers: hyperelliptic curves and quadrics. The connection between these two seemingly unrelated objects was first revealed by M. Reid, which roughly says that there’s a correspondence between hyperelliptic curves and pencil of quadrics. I’ll give a brief review of Reid’s work and then describe a higher degree generalization of the correspondence.

Let G be a connected algebraic k-group acting on a normal k-variety, where k is a field. We will show that X is covered by open G-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a G-linearized vector bundle on an abelian variety A, where A is a quotient of G. This generalizes a classical result of Sumihiro for actions of smooth connected algebraic groups.

Given a variety X defined over Z, there are two problems which a priori seem to not have a lot to do with each other:

1. Describe the singularities of the complex variety X(C)

2. Fix a prime p and describe how the cardinality of X(Z/p^rZ) depends on r.

A surprising result from the 80s concerning the second problem is that the Poincaré series of X - a formal power series having the above cardinalities as coefficients - is a rational function.

In this talk, I will explain this in more detail and I will present a new notion of stratifications which contributes to both problems: On the one hand, such a stratification specializes to a stratification of X(C) (which has stronger regularity properties than classical Whitney stratifications); on the other hand, using those stratifications, one obtains a (new) geometric proof of the rationality of Poincare series.

Several patterns emerge in collections of Betti tables associated to the powers of a fixed ideal. For example, Wheildon and others demonstrated that the shapes of the nonzero entires of these tables eventually stabilize when the fixed ideal has generators of the same degree. In this talk, I will discuss patterns in the graded Betti numbers of these and other graded systems of ideals. In particular, I will describe ways in which the Betti tables may stabilize, and how different types of stabilization are reflected in the corresponding Boij-Soederberg decompositions.

Recently Bridgeland has introduced the notion of a BPS structure, which is meant to encode the output of unrefined Donaldson-Thomas theory. He studied an associated Riemann-Hilbert problem and found a relation with Gromov-Witten invariants in the case of the conifold. In this talk I will try to give an overview of this work, ending with some potential new directions to explore.

Given a compact Kähler manifold X and a biholomorphic self-map g of X, the topological entropy of g plays an important role in the study of dynamical system (X, g). In this talk, I first talk about a generalization of a surface result, that is, a parabolic automorphism of a compact Kähler surface preserves an elliptic fibration, to hyperkähler manifolds. We give a criterion for the existence of equivariant fibrations on ‘certain’ hyperkähler manifolds from a dynamical viewpoint. Next, I will generalize a finiteness result for the null-entropy subset of a commutative automorphism group due to Dinh–Sibony (2004), to arbitrary virtually solvable groups G of maximal dynamical rank. This is based on joint work with T.-C. Dinh, J. Keum, and D.-Q. Zhang.

I will discuss some recent results and conjectures relating knot invariants (such as HOMFLY-PT polynomial and Khovanov-Rozansky homology) to algebraic geometry of Hilbert schemes of points on the plane. All notions will be introduced in the talk, no preliminary knowledge is assumed. This is a joint work with Andrei Negut and Jacob Rasmussen.

The notion of a power structure is closely related to that of a lambda ring. It is a powerful way to encode operations on certain generating functions. Gusein-Zade, Luengo, and Melle-Hernandez have defined a power structure over the Grothendieck ring of varieties. I will discuss an analog of this on a version of the Grothendieck ring of pretriangulated categories, whose elements represent enhancements of derived categories of coherent sheaves on varieties.

L-theory is often dubbed as "the K-theory" of quadratic forms. It has been used in a crucial way in surgery theory, to determine if two manifolds are cobordant. I will explain how it is easily defined in the derived setting by considering "derived" quadratic forms, and how I have used derived algebraic geometry to prove a rigidity result for L-theory. This will give an application of derived methods to a non-derived problem.

The subject of the talk is two-dimensional cyclic quotients, i.e. two-dimensional toric singularities. We introduce the classical work of Koll'ar/Shephard-Barron relating the components of their deformations and the so-called P-resolutions, we give several combinatorial descriptions of both gadgets, and we will focus on two special components among them - the Artin component allowing a simultaneous resolution and the qG-deformations preserving the Q-Gorenstein property. That is, it becomes important that several (or all) reflexive powers of the dualizing sheaf fit into the deformation as well. We will study this property in dependence on the exponent r. While the answers are already known for deformations over reduced base spaces (char = 0), we will now focus on the infinitesimal theory. (joint work with János Kollár)

There is by now an extensive theory of rational Chow rings of stacks of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known, including some recent developments.

The Banana manifold (or bananafold for short), is a compact Calabi-Yau threefold X which fibers over P^1 with Abelian surface fibers. It has 12 singular fibers which are non-normal toric surfaces whose torus invariant curves are a banana configuration: three P^1’s joined at two points, each of which locally look like the coordinate axes in C^3. We show that the Donaldson-Thomas partition function of X (for curve classes in the fibers) has an explicit product formula which, after a change of variables is the same as the generating function for the equivariant elliptic genera of Hilb(C^2), the Hilbert scheme of points in the plane.

Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover P^n/S_{n-1} -> P^n/S_n, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians.

This talk is concerned with the question of the minimal number of equations necessary to define a given projective variety scheme-theoretically. Every hypersurface is cut out by a single polynomial scheme-theoretically (also set-theoretically and ideal theoretically). Therefore the question is more interesting if a variety has a higher codimension. In this talk, we focus on the case when the codimension is two. If a variety in projective n-space has codimension two, then the minimal number of polynomials necessary to cut out the variety scheme-theoretically is between 2 and n+1. However the varieties cut out by fewer than n+1, but more than 2 polynomials seem very rare. The main goal of this talk is to discuss conditions for a non-singular surface in projective four-space to be cut out by three polynomials.

We study the moduli space of smooth complete intersections of two quadrics by relating it to the geometry of the singular members of the corresponding pencil. We give a new description for this parameter space by using the fact that two quadrics can be simultaneously diagonalized. Using this description we can compute the Picard group, which always happens to be cyclic. For example, we show that the Picard group of the moduli stack of smooth degree 4 Del Pezzo surfaces is Z/4Z.

Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in R^n. A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. When A is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points A may be arbitrary points in R^n. For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set A.

Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n. These are R^n_>-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A.

Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X, and prove several fundamental structural results about this action.

As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) such that any non-trivial p-adic representation of the fundamental group of X, which arises from geometry, is non-trivial mod p^N.

Floer homology has been a central tool to study global aspects of symplectic topology, which is based on pseudoholomorphic curve techniques proposed by Gromov. In this talk, we introduce a so-called wrapped Floer homology. Roughly speaking, this is a certain homology generated by intersection points of two Lagrangians and its differential is given by counting solutions to perturbed Cauchy-Riemann equation. We investigate an entropy-type invariant, called the slow volume growth, of certain symplectomorphisms and give a uniform lower bound of the growth using wrapped Floer homology. We apply our results to examples from real symplectic manifolds, including A_k-singularities and complements of a complex hypersurface. This is joint work with Myeonggi Kwon and Junyoung Lee.

Consider a contraction pi: X -> Y from a smooth Calabi-Yau 3-fold to a singular one. (This is half of an "extremal transition;" the other half would be a smoothing of Y.) In many examples there is an intermediate object called an "exoflop" -- a category of matrix factorizations, derived-equivalent to X, where the critical locus of the superpotential looks like Y with a P^1 sticking out of it, and objects of D(X) that will be killed by pi_* correspond to objects supported at the far end of the P^1. I will discuss one or two interesting examples. This is joint work with Paul Aspinwall.

Nakajima quiver varieties lie at the interface of geometry and representation theory. I will discuss a particular example, hyperpolygon space, which arises from star-shaped quivers. The simplest of these varieties is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we speculate on how this classification might be extended by studying the asymptotic geometry of the variety. In moduli-theoretic terms, this involves driving the stability parameter for the quotient to an irregular value. This is joint work in progress with Harmut Weiss, building on previous work with Jonathan Fisher.

The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. I will explain how the representation type of a finitely-generated algebra (finite, tame, wild) is determined by the essential dimension of the functors of its n-dimensional representations and I will introduce new numerical invariants for algebras. I will then illustrate the theorem and explicitely determine the invariants in the case of quiver algebras.

A famous theorem by Gabriel asserts that two Noetherian schemes X, Y are isomorphic if and only if the categories Coh(X), Coh(Y) are isomorphic. This theorem has been extended in many directions, including algebraic spaces and stacks (if we consider the monoid structure given by tensor product). One more idea to extend the theorem is the following: let X be a scheme of finite type over a field k, and consider the subcategory of Coh(X) given by sheaves supported in dimension at most d-1. We can form the quotient of Coh(X) by this subcategory, which we will call C_d(X). This category should contain enough information to describe the geometry of X "up to subsets of dimension d-1". In a joint work in progress with John Calabrese, we show that this is indeed true, i.e. to any isomorphism f: C_d(Y) ---> C_d(X) we can associate an isomorphism f': U---> V, where U and V are open subset respectively of X and Y whose complement have dimension at most d-1. Additionally, this isomorphism is unique up to subsets of dimension at most d-1. As a corollary of this result, we show that the automorphisms of C_d(X) are in bijection with the set {"automorphisms of X up to subsets of dimension d-1"} x {"line bundles on X up to subsets dimension d-1"}.

Fujita famously conjectured that for a d-dimensional smooth projective variety X with ample divisor H, mH+K_X is basepoint free whenever m\geq d+1. I will discuss recent joint work with Klaus Altmann in which we show this conjecture is true whenever X admits an effective action by a torus of dimension d-1.

In this talk, we will address the following question: given an algebraic group G acting on a variety X, when does thequotientX/G exist? We will provide an answer to this question in the case that G is reductive by giving necessary and sufficient conditions for thequotientto exist. We will discuss various applications to equivariant geometry, moduli problems and Bridgeland stability.

Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring CH(X) onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove new cases G=Spin(11) and G=Spin(12) of this conjecture. On an equivalent note, we compute the Chow ring CH(Y) of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group CH(Y) and determine its order which is equal to 16 777 216. On the other hand, we show that the Chow group of 0-cycles on Y is torsion-free.

The Hikita conjecture relates the cohomology ring of a symplectic resolution to the coordinate ring of another such resolution. I will explain this conjecture, and present a new version of the conjecture involving the quantum cohomology ring. There will be an emphasis on explicit examples.

Consider a smooth projective variety over a number field. The image of the associated (complex) Abel--Jacobi map inside the (transcendental) intermediate Jacobian is an abelian variety. We show that this abelian variety admits a distinguished model over the number field. Among other applications, this tool allows us to answer a recent question of Mazur; recover an old result of Deligne; and give new constructions of period maps over arithmetic bases.

The nilpotent cone has very special geometry which encodes interesting representation theoretic information. It is expected that many of its special properties have analogues for general “symplectic singularities.” This talk will discuss one such analogy for a class of symplectic singularities called hypertoric varieties. The main result, joint with T. Braden, can be described as a duality between nearby and vanishing cycle sheaves on Gale dual hypertoric varieties.

Abstract: The hyperkahler quotient construction (introduced by Hitchin et al in the 1980s) allows us to construct new hyperkahler spaces from suitable group actions on hyperkahler manifolds. This construction is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Hyperkahler implosion is in turn an analogue of symplectic implosion (introduced in a 2002 paper of Guillemin, Jeffrey and Sjamaar) which is related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry.

We prove that the right equivalence class of a super potential in complete free algebra is determined by its Jacobi algebra and the canonical class in its 0-th Hochschild homology represented by the super potential, assuming the Jacobi algebra is finite dimensional. This is a noncommutative version of the famous Mather-Yau theorem in isolated hyper surface singularities. As a consequence, we prove a rigidity theorem for Ginzburg dg-algebra. I will discuss some applications of these results in three dimensional birational geometry. This is a joint work with Guisong Zhou (1803.06128).

The classification of Fano varieties is unknown beyond dimension 3; however, many Fano fourfolds are expected to be GIT theoretic subvarieties of either toric varieties or quiver flag varieties. Quiver flag varieties are a generalization of type A flag varieties and are GIT quotients of vector spaces. In this talk, I will discuss my recent work on quiver flag varieties, including a proof of the Abelian/non-Abelian correspondence for quiver flag varieties, and its application in the large scale computer search for Fano fourfolds that I have carried out in joint work with T. Coates and A. Kasprzyk. We find 139 new Fano fourfolds. I will also discuss the importance of these subvarieties as a testing ground for the conjectures of Coates, Corti, Galkin, Golyshev, Kasprzyk and Tveiten on mirror symmetry for Fano varieties.

The goal of this talk is to create a correspondence between the representation theory of algebraic groups and the topology of Lie groups. The idea is to study the Hodge theory of the classifying stack of a reductive group over a field of characteristic p, the case of characteristic 0 having been studied by Behrend, Bott, Simpson and Teleman. The approach yields new calculations in representation theory, motivated by topology.

The Batyrev--Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown.

In my talk I will explain (partially conjectural) relation between

1) Homology of Hilbert scheme of points on singular curves

2) Knot homology of the links of curve singularities

3) Space functions on the moduli space of maps from the formal disc to the curve singularities.

I will center my talk around discussion of the case of cuspidal curve x^m=y^n and its singularity. In this case it is now known that 1) 2) and 3) are essentially equal. Talk is based on the joint projects with Gorsky, Rozansky, Rasmussen, Shende and Yun.

Let N be the conormal variety of a Schubert variety X. In this talk, we discuss the geometry of N in two cases, when X is cominuscule, and when X is a divisor. In particular, we present a resolution of singularities and a system of defining equations for N, and also describe certain cases when N is normal, Cohen-Macaulay, and Frobenius split. Time permitting, we will also illustrate the close relationship between N and orbital varieties, and discuss the consequences of the above constructions for orbital varieties.

The derived category of a hypersurface is equivalent to the category of matrix factorizations of a certain function on the total space of a line bundle over the ambient space. The hypersurface is smooth if and only if the critical locus of the function is compact. I will present a construction through which a resolution of singularities of the hypersurface corresponds to a compactification of the critical locus of the function, which can be very interesting in examples. This is joint work with Paul Aspinwall and Ed Segal.

I will outline the construction of a natural bivariant theory extending algebraic bordism, which will yield an extension of algebraic cobordism to singular varieties. I will also discuss the connections of this theory to algebraic K-theory and to intersection theory.

Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement's Milnor fiber. This talk is based on joint work with Max Kutler.

Given a smooth projective variety X and a smooth divisor D \subset X, one can study the enumerative geometry of counting curves in X with tangency conditions along D. There are two theories associated to it: relative Gromov-Witten invariants of (X,D) and orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, Abramovich-Cadman-Wise proved that genus zero relative invariants are equal to the genus zero orbifold invariants of root stacks (with a counterexample in genus 1). We show that higher genus orbifold Gromov-Witten invariants of X_{D,r} are polynomials in r and the constant terms are exactly higher genus relative Gromov-Witten invariants of (X,D). If time permits, I will also talk about further results in genus zero which allows us to study structures of genus zero relative Gromov-Witten theory, e.g. Givental formalism for genus zero relative invariants. This is based on joint work with Hisan-Hua Tseng, Honglu Fan and Longting Wu.

The Grothendieck ring of varieties is the target of a rich invariant associated to any algebraic variety which witnesses the interplay between geometric, topological and arithmetic properties of the variety. The motivic Hilbert zeta function is the generating series for classes in this ring associated to a certain compactification of the unordered configuration space, the Hilbert scheme of points, of a variety. In this talk I will discuss the behavior of the motivic Hilbert zeta function of a reduced curve with arbitrary singularities. For planar singularities, there is a large body of work detailing beautiful connections with enumerative geometry, representation theory and topology. I will discuss some conjectural extensions of this picture to non-planar curves.

We introduce a construction call the {\em saturated blowup} of an Artin stack with good moduli space. The saturated blowup is a birational map of stacks which induces a proper birational map on good moduli spaces. Given an Artin stack {\mathcal X} with good moduli space X, there is a canonical sequence of saturated blowups which produces a stack whose rigidification is a DM stack. When the stack is smooth, all of the stacks in the sequence of saturated blowups are also smooth. This construction generalizes earlier work of Kirwan and Reichstein in geometric invariant theory and the talk is based on joint work with David Rydh.

Exceptional groups (over arbitrary rings) are related to octonion algebras, triality and exceptional Jordan algebras. I will talk about recent results of an approach to these objects using certain torsors (principal homogeneous spaces) under smaller exceptional groups, and explain how an explicit understanding of these torsors provides insight into the objects and their interrelations.

Given a principally polarized abelian variety A over a number field (or a function field), one can naturally extract two real numbers that capture the ``complexity'' of A: one is the Faltings height and the other is the N\'eron-Tate height (of a symmetric effective divisor defining the polarization). I will discuss a precise relationship between these two numbers, relating them to some subtle invariants arising from tropical geometry (more precisely, from Berkovich analytic spaces). (Joint work with Robin de Jong.)

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

This is a joint work with B. Calmes, V. Petrov, N. Semenov and K. Zainoulline. In the talk I will discuss a connection between direct sum decompositions of the Chow motive with Z-coefficients of a homogeneous space of a group G, and representations of affine nil Hecke algebras defined in terms of root system of G. This connnection can be used in two directions: prove indecomposability of certain motives as well as get some structural results about Hecke algebras.

Let G/B be a flag variety over C, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. The Bruhat decomposition of G defines subvarieties of G/B called Schubert subvarieties. The codimension 1 Schubert subvarieties are called Schubert divisors. The Chow ring of G/B is generated as an abelian group by the classes of all Schubert varieties, and is "almost" generated as a ring by the classes of Schubert divisors. More precisely, an integer multiple of each element of G/B can be written as a polynomial in Schubert divisors with integer coefficients. In particular, each product of Schubert divisors is a linear combination of Schubert varieties with integer coefficients.

In the first part of my talk I am going to speak about the coefficients of these linear combinations. In particular, I am going to explain how to check if a coefficient of such a linear combination is nonzero and if such a coefficient equals 1. In the second part of my talk, I will say something about an application of my result, namely, how it makes it possible estimate so-called canonical dimension of flag varieties and groups over non-algebraically-closed fields.

The Grothendieck ring of algebraic stacks was introduced by Ekedahl in 2009. It may be viewed as a localization of the more common Grothendieck ring of varieties. If G is a finite group, then the class {BG} of its classifying stack BG is equal to 1 in many cases, but there are examples for which {BG}\neq 1. When G is connected, {BG} has been computed in many cases in a long series of papers, and it always turned out that {BG}*{G}=1. We exhibit the first example of a connected group G for which {BG}*{G}\neq 1. As a consequence, we produce an infinite family of non-constant finite étale group schemes A such that {BA}\neq 1.

In 2013, Esnault and Srinivas proved that as in the de Rham cohomology over the field of complex numbers, the algebraic entropy of an automorphism of a smooth projective surface over a finite field $\mathbb{F}_q$ is taken on the span of the Néron–Severi group inside of $\ell$-adic cohomology. Later, motivated by this and Weil's Riemann Hypothesis, Truong asked whether the spectral radius $\chi_{2k}(f)$ of the pullback $f^*: H^{2k}(X, \mathbb{Q}_\ell) \to H^{2k}(X, \mathbb{Q}_\ell)$ is the same as the spectral radius $\lambda_k(f)$ of the pullback $f^*: N^k(X)_\mathbb{R} \to N^k(X)_\mathbb{R}$, where $f: X \to X$ is a surjective self-morphism of a smooth projective variety $X$ of dimension $n$ defined over an algebraically closed field $\mathbb{k}$ and $N^k(X)$ denotes the finitely generated abelian group of algebraic $(n-k)$-cycles modulo the numerical equivalence. He has shown that $\displaystyle \max_{0\le i\le 2n} \chi_{i}(f) = \max_{0\le k\le n} \lambda_{k}(f)$. We give an affirmative answer to his question in the case of abelian varieties and $k=1$.

## Note for Attendees

ESB 4133 is the library room attached to the PIMS lounge.