Algebra-Number Theory Archives for Academic Year 2015

Organizational meeting

When: Tue, September 2, 2014 - 1:00pm
Where: MATH 2300
Speaker: Adam Lizzi (UMD) -

Organizational Meeting

When: Wed, September 3, 2014 - 2:00pm
Where: Math 1311
Speaker: () -

The Chow ring of a finite group

When: Wed, September 10, 2014 - 2:00pm
Where: Math 1311
Speaker: Burt Totaro (UCLA) -
Abstract: A natural class of singular varieties consists of the quotients of vector spaces by linear actions of a finite group G. We can ask what the Chow group of algebraic cycles is, for such a quotient. By taking larger and larger representations of G, we can package these Chow groups into a ring, called the Chow ring of the classifying space of G, or (for short) the Chow ring of G. It maps to the cohomology ring of G, usually not by an isomorphism.

We present the latest tools for computing Chow rings of finite groups. These tools give complete calculations for all "small" groups and many other finite groups. A surprising point is that Chow rings become "wild", in a precise sense, for some slightly larger finite groups.

Kottwitz-Rapoport conjecture on crystals with additional structure

When: Mon, September 22, 2014 - 2:00pm
Where: Math 1311
Speaker: Xuhua He (UMD) -
Abstract: In 1973, Mazur showed that the Newton polygon of a crystal lies below the Hodge polygon of the associated isocrystal and the two polygons have the same end points. In 2003, Kottwitz and Rapoport showed that the converse is true, i.e., given two such polygons, there exists a crystal with given polygons as its Hodge polygon and Newton polygon respectively. Kottwitz and Rapoport conjectured a similar statement for crystals with additional structure. This conjecture plays an important role in the study of reduction of Shimura varieties. In this talk, I will explain this conjecture, its relation to the Shimura varieties, and I will discuss some ideas of the proof.

Higher direct images of the structure sheaf in birational geometry

When: Mon, October 6, 2014 - 2:00pm
Where: Math 1311
Speaker: Andre Chatzistamatiou (Universitat Duisburg-Essen) -

Abstract: (Joint work with Kay Rulling.) For a birational projective morphism between regular excellent noetherian schemes, we will show that the higher direct images of the structure sheaf vanish. This generalizes results in positive characteristic to an arithmetic setup.

Arithmetic quotients of the mapping class group

When: Mon, October 6, 2014 - 3:00pm
Where: Math 1313
Speaker: Justin Malestein (University of Bonn) -
Abstract: In this talk, I will discuss a procedure for obtaining infinitely many “virtual” arithmetic quotients of mapping class groups, (surjective maps up to finite index). Specifically, for any irreducible rational representation of a finite group of rank less than g, we produce a corresponding virtual arithmetic quotient of the genus g mapping class group. Particular choices of irreducible representations of finite groups yield arithmetic quotients of type Sp(2m), SO(2m, 2m), and SU(m, m) for arbitrarily large m in every genus. Joint with F. Grunewald, M. Larsen, and A. Lubotzky.​

Where are all the hyperkaehler varieties?

When: Mon, October 13, 2014 - 2:00pm
Where: Math 1311
Speaker: Nicolas Addington (Duke University) -
Abstract: Curves with trivial canonical divisor are elliptic curves,
perhaps the richest and most exciting class of curves. In higher
dimensions, there are three basic classes of varieties with trivial
canonical divisor: abelian varieties, Calabi-Yau varieties, and compact
hyperkaehler varieties. The first are the most direct analogue of
elliptic curves; the second are plentiful and important in physics; the
third are the most mysterious, and very few examples are known. I will
discuss a new example due to Lehn et al. built from twisted cubics
on cubic fourfolds, and my recent proof (joint with Lehn) that the new
example is deformation-equivalent to an old example. The problem is
purely classical, but it is hugely clarified by thinking about derived
categories. I'll conclude by outlining a very speculative approach
to explaining why we can't seem to find new hyperkaehlers.

Whittaker functions and geometric crystals

When: Mon, October 20, 2014 - 2:00pm
Where: Math 1311
Speaker: Thomas Lam (University of Michigan) -
Abstract: I will talk about a formula for Archimedean Whittaker
functions as integrals over Berenstein and Kazhdan's geometric
crystals. This formula is a geometric analogue of the expression for
an irreducible character of a complex semisimple Lie algebra as a sum
over Kashiwara's crystals. The formula is closely related to mirror
symmetry phenomenona for flag varieties, and to the study of directed
polymers in probability.

Toric geometry and moduli spaces of sheaves I

When: Mon, October 27, 2014 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: I am giving a number of mainly expository talks on the moduli spaces of coherent sheaves and their invariants. Torus action can be used to explore some aspects of the geometry of these moduli spaces such as Euler characteristic, cell decomposition, Poincare polynomial,... Using this, we will give combinatorial descriptions of these invariants and prove that some of the invariants have modular properties.
Our main examples are the moduli spaces of rank 1 and rank 2 torsion free sheaves on smooth toric varieties and smooth toric DM stacks. The talks will include a gentle introduction to Donaldson-Thomas invariants in the toric setting. In the way, I will mention some of my contributions.

Noncommutative motives and its applications

When: Fri, October 31, 2014 - 2:00pm
Where: Math 1311
Speaker: Goncalo Tabuada (MIT) -
Abstract: I will explain how the recent theory of noncommutative motives
allows us to address the following "commutative" problems and conjectures:
(i) Computation of motivic decompositions; (ii) Voevodsky's nilpotence
conjecture; (iii) Paranjape-Srinivas' conjecture on complete intersections.
(Special Time and Day)

Toric geometry and moduli spaces of sheaves II

When: Mon, November 3, 2014 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -

Abstract: In the first talk, we reviewed some of the properties of varieties with torus action. We later used these to find Euler charactersitics and Betti numbers of the Hilbert scheme of points on a toric surface. The generating function of Euler characteristics can be expressed in terms of modular eta function. The Hilbert scheme of points on a toric surface can be realized as a moduli space of rank 1 torsion free sheaves. In the second talk, we will use toric techniques to find the Euler characteristics of the moduli spaces of rank 2 torsion free sheaves on toric surfaces. These moduli spaces arise when one tries to compactify the moduli spaces of rank 2 vector bundles. As in rank 1, physicists expect (by S-duality) the generating functions of the Euler characteristics of rank 2 moduli spaces to be modular. We will have a discussion about modularity. Finally, if time allows we will turn into the moduli spaces of rank 1 and 2 torsion free sheaves on toric Deligne-Mumford stacks of dimension 2.

Derivative of the symmetric square p-adic L-function via pull-back formula

When: Wed, November 5, 2014 - 2:00pm
Where: Math 1311
Speaker: Giovanni Rosso (Columbia University) -
Abstract: Let f be a weight k modular form Steinberg at p. Under certain
hypotheses on the conductor of f, we give a formula for the derivative
at s=k-1 of the symmetric square p-adic L-function of f, thus proving a
conjecture of Benois on trivial zeros. Crucial is the construction of the p-adic L-function by Boecherer and Schmidt which we will recall.

On the remodeling conjecture for toric Calabi-Yau 3-orbifolds

When: Mon, November 10, 2014 - 2:00pm
Where: Math 1311
Speaker: Zhengyu Zong (Columbia University)
Abstract: Based on the work of Eynard-Orantin and Marino, the
remodeling conjecture was proposed in the papers of
Bouchard-Klemm-Marino-Pasquetti in 2007 and 2008. The remodeling
conjecture can be viewed as an all genus mirror symmetry for toric
Calabi-Yau 3-orbifolds. It relates the higher genus open Gromov-Witten
potential of a toric Calabi-Yau 3-orbifold to the higher genus B-model
potential which is obtained by applying the topological recursion on
the mirror curve. In this talk, I will explain the proof of the
remodeling conjecture for general toric Calabi-Yau 3-orbifolds. This
work is joint with Bohan Fang and Chiu-Chu Melissa Liu.

A rim hook rule for the equivariant quantum cohomology of the Grassmannian

When: Mon, November 17, 2014 - 2:00pm
Where: Math 1311
Speaker: Elizabeth Beazley (Haverford College) -
Abstract: The theory of quantum cohomology was initially developed in the early 1990s by physicists working in the field of superstring theory. Mathematicians then discovered applications to enumerative algebraic geometry, counting the number of rational curves of a given degree satisfying certain incidence conditions. The driving question in modern quantum cohomology is to find non-recursive, positive combinatorial formulas for expressing the quantum product of elements in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of k planes in complex n-space by doing classical multiplication and then applying a combinatorial "rim hook rule" which yields the quantum parameter. In this talk, we will describe joint work with Anna Bertiger and Kaisa Taipale which provides a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Time permitting, we will mention potential applications to finding families of polynomial representatives for the equivariant homology of the affine Grassmannian via the Peterson isomorphism.

Toric geometry and moduli spaces of sheaves III

When: Mon, November 24, 2014 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: The main part of this talk will be devoted to study the moduli spaces of coherent sheaves on smooth toric Deligne-Mumford stacks in dimension 2. In some special cases as in the case of varieties, the Euler characteristics of the moduli spaces of torsion free sheaves in rank 1 and 2 are expressed in terms of modular forms.

If time allows, we move on to the moduli spaces of rank 1 torsion free sheaves on nonsingular toric 3-folds. This will lead to the famous Donaldson-Thomas invariants which were later linked to Gromov-Witten invariants by Maulik-Nekrasov-Okounkov-Pandharipande.

Toric geometry and moduli spaces of sheaves IV

When: Mon, December 1, 2014 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: We study the moduli spaces of rank 1 torsion free sheaves on nonsingular toric 3-folds. This will lead to the famous Donaldson-Thomas invariants which were later linked to Gromov-Witten invariants by Maulik-Nekrasov-Okounkov-Pandharipande.

Stability conditions on abelian threefolds

When: Mon, February 9, 2015 - 2:00pm
Where: Math 1311
Speaker: Emanuele Macri (Northeastern University)-
Abstract: I will present a new proof and a generalization a result by Maciocia
and Piyaratne on the existence of Bridgeland stability conditions on
any abelian threefold. As an application, we deduce the existence of
Bridgeland stability conditions on a number of Calabi-Yau threefolds,
namely Calabi-Yau threefolds of abelian type and Kummer threefolds.

As in the work of Maciocia and Piyaratne, the idea is to show a
Bogomolov-Gieseker type inequality involving Chern classes of certain
stable objects in the derived category; this was conjectured by Bayer,
Toda, and myself. Our approach uses the multiplication maps on abelian
threefolds instead of Fourier-Mukai transforms.

This is joint work with Arend Bayer and Paolo Stellari.

Equations of Tropical Varieties

When: Mon, February 16, 2015 - 2:00pm
Where: Math 1311
Speaker: Jeff Giansiracusa (Swansea University) -
Abstract: Tropical geometry is a combinatorial shadow of algebraic geometry over a nonarchimedean field that encodes information about things like intersections and enumerative invariants. Usually one defines tropical varieties as certain polyhedral subsets of R^n satisfying a balancing condition. I'll show how these arise as the solution sets to certain systems of polynomial equations over the tropical semiring T = (R union -infinity, max, +) related to matroids. This yields a notion of tropical Hilbert polynomials, and in this framework there is a universal tropicalization that is closely related to the Berkovich analytification and the moduli space of valuations.

Supersingular locus of Hilbert modular variety and Tate conjecture

When: Wed, February 18, 2015 - 2:00pm
Where: Math1311
Speaker: Liang Xiao (University of Connecticut) -
Abstract: We give a description of the supersingular locus of a Hilbert modular variety with hyperspecial level structure. In the special case when the prime number p is inert of even degree in the totally real field F, we show that the irreducible components of the supersingular locus generate Tate classes in the cohomology of the special fiber of the Hilbert modular variety, under some genericity condition. This is a joint work with Yichao Tian.

Special values of Goss L-series for Drinfeld modules and log-algebraicity

When: Fri, March 6, 2015 - 2:00pm
Where: Math 1311
Speaker: Matt Papanikolas (Texas A & M) -

Abstract: We will explore the theory of Goss L-functions, which are defined by Dirichlet series over function fields in positive characteristic and which take values in the function field. Much like one finds over number fields, Goss L-functions arise naturally from Galois representations associated to Drinfeld modules and more generally to higher dimensional Drinfeld modules of Anderson. For Goss L-series for Dirichlet characters, Anderson introduced the idea of realizing special L-values via specializations of certain 'log-algebraic' power series identities on rank one Drinfeld modules. His identities lead to formulas for evaluations L(1,chi) in terms of torsion points on Drinfeld modules. Subsequently we have generalized Anderson's identities to tensor powers of the Carlitz module, and multivariable versions have been obtained by Angl ès, Pellarin, and Tavares-Ribeiro. In the present talk, using work of Taelman as a starting point, we will investigate log-algebraicity identities for Drinfeld modules of rank greater than one, leading to identities between special values of Goss L-functions, Drinfeld logarithms, and special points. Joint work with C.-Y. Chang and A. El-Guindy.

Gauss-Manin representations of conformal block local systems

When: Mon, March 9, 2015 - 2:00pm
Where: MATH 1311
Speaker: Prakash Belkale (University of North Carolina at Chapel Hill) -
Abstract: Conformal blocks, or spaces of generalized theta functions (attached to a group G and representations at a level k), give projective local systems on moduli spaces of curves with marked points. One can ask if they are realizable in geometry, i.e., as local subsystems of suitable Gauss-Manin local systems of cohomology of families of smooth projective varieties.

I will discuss (in genus 0) the proof of Gawedzki et al's conjecture that Schechtman-Varchenko forms are square integrable (this was proved first for sl(2) by Ramadas). Together with the flatness results of Schechtman-Varchenko,
and the work of Ramadas, one obtains the desired realization and a unitary metric on conformal blocks.

I will also discuss recent results on the image of conformal block local systems in cohomology (joint with S. Mukhopadhyay), and the known results regarding cohomological realizations of the entire space of invariants.

Galois groups of Mori polynomials, semistable curves and monodromy

When: Mon, March 9, 2015 - 3:30pm
Where: MATH 3206
Speaker: Yuri Zarhin (Penn State) -
Abstract: We study the monodromy of a certain class of semistable hyperelliptic curves over the rationals that was introduced by Shigefumi Mori forty years ago. Using ideas of Chris Hall, we prove that the corresponding $\ell$-adic monodromy groups are (almost) as large as possible. We also discuss an explicit construction of two-dimensional families of hyperelliptic curves over an arbitrary global field with big monodromy.

Algebraic cycles and the fundamental group

When: Mon, March 23, 2015 - 2:00pm
Where: Math 1311
Speaker: Ramesh Sreekantan (Indian Statistical Institute, Bangalore) -
Abstract: Hain and Pulte showed that a natural extension of mixed Hodge structures arising from the mixed Hodge structure on the fundamental group of an algebraic curve is related to a natural cycle - the modified diagonal cycle - in the triple product of the curve. In this talk we will provide another example of this phenomena - whereby a natural extension of Hodge structures is related to a natural higher algebraic cycle on the self product of the curve. This is joint work with Subham Sarkar.

On accessible and weakly accessible p-adic period domains

When: Wed, March 25, 2015 - 2:00pm
Where: Math 1311
Speaker: Michael Rapoport (University of Bonn) -
Abstract: After defining (p-adic) period domains, I will determine all period domains for which the period map is surjective. The final answer to this question is due to Scholze.

Quasimap theory

When: Fri, April 3, 2015 - 2:00pm
Where: MATH 1311
Speaker: Ionut Ciocan-Fontanine (University of Minnesota) -
Abstract: I will survey the theory of quasimap invariants of (a class of) GIT quotient targets and their relation via wall crossing to the Gromov-Witten invariants of these targets. This theory was developed in joint work with Bumsig Kim, and in part also with Daewoong Cheng and Davesh Maulik.

Moduli spaces of sheaves on Enriques/Bielliptic surfaces

When: Mon, April 13, 2015 - 2:00pm
Where: Math 1311
Speaker: Giulia Sacca (IAS and Stony Brook)
Abstract: From Mukai's fundamental work, it is known that moduli spaces of sheaves on K3 or abelian surfaces are holomorphic symplectic manifolds. The aim of this talk is to describe the geometry of the relative compactified Jacobian of a linear system on an Enriques surface. Among other geometric properties, I will show that these moduli spaces are Calabi-Yau manifolds. Time permetting, I will also describe the corresponding results for bielliptic surfaces.

Donaldson-Thomas Theory and Crepant Resolutions

When: Mon, April 20, 2015 - 2:00pm
Where: Math 1311
Speaker: Dustin Ross (University of Michigan) -
Abstract: For a fixed Calabi-Yau threefold X, Donaldson-Thomas (DT) theory, roughly, is the study of certain Euler characteristics of Hilbert schemes of curves in X. If X is an orbifold with crepant resolution Y, Bryan, Cadman, and Young conjectured that the DT theory of X and Y should be related in a simple way. We prove this conjecture in the toric setting. In this talk, I'll begin by describing the basic notions of DT theory and motivate them through the concrete example of toric varieties. I'll explain how these notions generalize to orbifolds and describe some of the techniques used in the proof of the correspondence.

Nonabelian biextensions and bimonoidal functors

When: Mon, May 11, 2015 - 2:00pm
Where: Math 1311
Speaker: Ettore Aldrovandi (Florida State University) -

Abstract: A biextension is a sort of bilinear analog of the concept of extension. But in a much more precise way, a biextension of abelian groups A and B by C corresponds to a bilinear morphism of the product of A and B into the Picard groupoid of C-torsors.

Our goal is to propose a nonabelian generalization of the notion of biextension, so that biextensions defined in this way give still give rise to bilinear, or more precisely bi-multiplicative, morphisms.

The main application is to classify bimonoidal functors, that is, functors that are monoidal in each variable, among monoidal categories (and stacks) with only mild commutativity assumptions.

Applications and motivations stem from the theory of categorical rings and mod 2 phenomena that distinguish various cohomology theories of rings. If time permits we will illustrate some of these applications.

On Deligne periods of tensor product motives

When: Mon, June 8, 2015 - 11:00am
Where: Math 1313
Speaker: Chandrasheel Bhagawat (IISER (Pune), India) -
Abstract: A classical theorem of Shimura describes the ratio of two successive critical values of the L-function of a holomorphic cusp form f as a ratio of complex numbers which are known as the periods of f. In this talk, I will describe some analogous results on period relations for the ratios of Deligne periods for certain tensor product motives which were proved in a joint work with A. Raghuram. These results provide a motivic interpretation for certain algebraicity results for ratios of successive critical values for Rankin-Selberg L-functions for GL(n) x GL(n') proved by G. Harder & A. Raghuram.

Spectra and arithmetic

When: Mon, June 8, 2015 - 2:00pm
Where: Math 1313
Speaker: Supriya Pisolkar (IISER (Pune), India) -
Abstract: The inverse spectral theory problem in Riemannian geometry,
is to recover properties of a Riemannian manifold from the knowledge
of the spectra of natural differential operators associated to the manifold.
This talk will be a survey of results by Gopal Prasad and A. S. Rapinchuk
which revolves around this question in the more general context of arithmetically
defined locally symmetric spaces. For many of their geometric and arithmetic results
G. Prasad and Rapinchuk have assumed the validity of Schanuel's conjecture.
I will mention towards the end of talk about a joint work with C. S. Rajan
where we have been able to obtain similar but unconditional results.