Algebra-Number Theory Archives for Academic Year 2016

Organizational meeting

When: Wed, September 9, 2015 - 2:00pm
Where: Math 1311
Speaker: None () -
Abstract: Organizational meeting for Algebra and Number Theory, and Lie Groups and Representation Theory Seminars.

Motivic decomposition of projective pseudo-homogeneous varieties

When: Wed, September 16, 2015 - 2:00pm
Where: Math 1311
Speaker: Srimathy Srinivasan (University of Maryland) -

Cup products and the Heisenberg group

When: Mon, September 21, 2015 - 2:00pm
Where: Math 1311
Speaker: N Ramachandran (UMD) -
Abstract: (joint work with E. Aldrovandi) This talk will discuss a basic question about cup products of cohomology classes of degree one and relate it to the Heisenberg group. This has applications to codimension two algebraic cycles.

Exponentiable motivic measures

When: Mon, October 12, 2015 - 2:00pm
Where: Math 1311
Speaker: N Ramachandran (UMD) -
Abstract: (joint with G. Tabuada). Motivic measures provide refined Euler characteristics of algebraic varieties. The talk will present recent results on motivic measures whose associated Kapranov zeta functions have nice properties with respect to products of varieties.

Mod p isogeny classes on Shimura varieties of Hodge type with parahoric level structure

When: Mon, October 19, 2015 - 2:00pm
Where: Math 1311
Speaker: Rong Zhou, Harvard University
Abstract: We give a description of the structure of a mod-p isogeny class on integral models of Shimura varieties constructed by Kisin and Pappas. This description is as a certain subset of the conjectural description of the isogeny class as predicted by the Langlands-Rapoport conjecture, and we reduce the problem of showing equality to results about connected components of affine Deligne-Lusztig varieties and combinatorics of affine Weyl groups.

The Shareshian-Wachs conjecture on Hessenberg Varieties

When: Mon, October 26, 2015 - 2:00pm
Where: Math 1311
Speaker: Patrick Brosnan (UMCP) -
Abstract: Tim Chow and I recently proved a conjecture of Shareshian and Wachs, which relates a certain action (defined by Tymoczko) of the symmetric group on the cohomology of Hessenberg varieties (in type A) to a certain formal power series called the chromatic symmetric function introduced by Stanley. I'll try to explain what all these words mean and sketch our proof of the conjecture.

Uniformization of the moduli space of abelian 6-folds

When: Fri, October 30, 2015 - 2:00pm
Where: Math 1311
Speaker: Valery Alexeev (University of Georgia) -
Abstract: The moduli spaces Ag of principally polarized abelian varieties of dimension g are of general type for g>=7 and unirational for g<=5; the latter is proved by uniformizing Ag by moduli spaces of curves with extra data (Jacobians, Pryms). In this work (joint work with Donagi, Farkas, Izadi, Ortega) we prove a conjecture of Kanev providing a similar uniformization result for genus g=6. We use it to study the geometry of A_6. The principal technique for the proof is the study of degenerations of abelian varieties.

Hecke algebras and $p$-adic groups

When: Wed, November 11, 2015 - 2:00pm
Where: Math 1311
Speaker: Xuhua He (University of Maryland) -
Abstract: In a 1957 paper, Tits explained the analogy between the symmetric group $S_n$ and the general linear group over a finite field $\mathbb F_q$ and indicated that $S_n$ should be regarded as the general linear group over $\mathbb F_1$, the field of one element.

Following Tits' philosophy, we may informally regard the affine Weyl groups as the reductive group over $\mathbb Q_1$, the $1$-adic field. Although it might be premature to develop the theory of $1$-adic field at the current stage, we do have a fairly good understanding on the conjugacy classes of the affine Weyl groups, together with the length function on it, and such knowledge allows us to reveal a great part of the structure of the conjugacy classes of $p$-adic groups. In the first talk, we will explain how such knowledge is used to understand the Frobenius-twisted conjugacy classes of loop groups and discuss some further applications to Shimura varieties. In the second talk, we will explain how such knowledge is used in the study of cocenters and representations of affine Hecke algebras and discuss some further application to the representations of $p$-adic groups.

Absolute vs. Relative Gromov-Witten Invariants

When: Mon, November 16, 2015 - 2:00pm
Where: Math 1311
Speaker: Mohammad Farajzadeh Tehrani (Simons Center) -
Abstract: We compare absolute and relative Gromov-Witten invariants with the basic contact vector for very ample divisors. Whenever the divisor is sufficiently ample, one might expect that these invariants are the same (up to a natural multiple). We show that this is indeed the case outside of a narrow range of the dimension of the target and the genus of the domain.
We provide explicit examples to show that these invariants are generally different inside of this range. This is joint work with A. Zinger.

Weighted Compactifications of Configuration Spaces

When: Mon, November 23, 2015 - 2:00pm
Where: Math 1311
Speaker: Evangelos Routis (IPMU) -
Abstract: In the early 90's, Fulton and MacPherson provided a natural and beautiful way of compactifying the configuration space F(X,n) of n distinct labeled points on an arbitrary nonsingular variety. In this talk, I will present an alternate compactification of F(X,n), which generalizes the work of Fulton and MacPherson and is parallel to Hassett's weighted generalization of the moduli space of n-pointed stable curves. After discussing its main properties, I will give a presentation of its intersection ring and as an application, I will describe the intersection ring of Hassett's spaces in genus 0. Finally, as time permits, I will discuss some additional moduli problems associated with weighted compactifications.

The local lifting problem, the Oort Conjecture, and its generalizations

When: Mon, November 30, 2015 - 2:00pm
Where: Math 1311
Speaker: Andrew Obus (UVa) -
Abstract: Let k be an algebraically closed field of characteristic p. The local lifting problem asks if the action of a finite group G by k-automorphisms on k[[t]] can be lifted to an action of G by R-automorphisms on R[[t]], where R is some characteristic zero DVR with residue field k. This is motivated by the problem of lifting a Galois branched cover of smooth projective algebraic curves from characteristic p to characteristic zero.

The Oort conjecture (now a theorem of Obus-Wewers and Pop) states that cyclic actions can always be lifted (for some R). We will discuss a generalization of this conjecture to the case of metacyclic actions, as well as recent progress by the speaker on this problem.

Local systems on analytic germ complements

When: Wed, January 27, 2016 - 2:00pm
Where: Math 1311
Speaker: Nero Budur (KU Leuven) -
Abstract: A conjecture of Beauville and Catanese from 1980's stated
that the sets of rank one local systems on a compact Kähler manifold
with prescribed cohomology are special varieties, that is, their
irreducible components are torsion-translated subtori. The conjecture
has finally been fully proved by Botong Wang in 2013. Around the same
time, Wang and I proved that the same holds for all quasi-projective
complex algebraic manifolds. In this talk, we present the recent proof
of a much more subtle case: germ complements of complex analytic sets.
This is a vast generalization of the classical Monodromy Theorem
stating that the eigenvalues of the monodromy on the cohomology of the
Milnor fiber of a germ of a holomorphic function are roots of unity.
The proof uses the Riemann-Hilbert correspondence between D-modules
and perverse sheaves. Joint work with Botong Wang.

Statistics for points on curves over finite fields

When: Thu, February 11, 2016 - 2:00pm
Where: 1311.0
Speaker: Alina Bucur (UCSD) -
Abstract: A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations
should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. Our main focus will be the family of cyclic prime degree covers, which can be approached both via the original combinatorial/analytic approach and via maps from the idele class group. This is joint work with Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, Ekin Ozman and Melanie Matchett Wood.

Geometry of Nilpotent Cones in Hodge Theory

When: Mon, February 22, 2016 - 2:00pm
Where: Math
Speaker: Greg Pearlstein (Texas A&M) -
Abstract: The local monodromy of a degeneration of smooth complex
projective varieties gives rise to a monodromy cone which plays a
central role in constructing analogs of Mumford's
toroidal compactifications for Hodge structures of arbitrary weight.
In this talk, I will describe several methods for describing the
possible monodromy cones which can arise in a given period domain using
topological boundary components and signed Young diagrams.

Algebraic cobordism group of divisors and counting singular curves with tangency conditions

When: Thu, February 25, 2016 - 1:00pm
Where: MATH 1308
Speaker: Yu-jong Tzeng (University of Minnesota) -

Abstract: The algebraic cobordism theory constructed Morel and Levine
is a universal oriented Borel-Moore homology theory for schemes.
Levine and Pandaripande developed an equivalent algebraic cobordism
theory and many results using degeneration methods in algebraic
geometry can be understood as invariants of this theory. In this talk
I will talk about the generalization of the algebraic cobordism theory
to bundles and divisors on varieties and discuss its application to
count singular curves with tangency conditions. This enumeration
includes but not limited to the number of nodal curves with fixed
tangency multiplicities with a line on the complex projective plane by
Caporaso and Harris.

Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries

When: Mon, February 29, 2016 - 2:00pm
Where: Math 1311
Speaker: June Huh (Institute of Advanced Study) -
Abstract: A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This implies the above mentioned conjectures and their generalization to arbitrary matroids. Joint work with Karim Adiprasito and Eric Katz.

Flat line bundles and arithmetic intersection theory

When: Wed, March 2, 2016 - 2:00pm
Where: MATH 1311
Speaker: Gerard Freixas i Montplet (CNRS-Jussieu) -
Abstract: Arithmetic intersection theory, also known as Arakelov geometry, is an enhancement of intersection theory a la Fulton, suited to the study of cycles on arithmetic varieties. A fundamental result in the formalism is the arithmetic Grothendieck-Riemann-Roch theorem of Gillet-Soulé, that relates the determinant of cohomology of a hermitian vector bundle to some enhanced characteristic classes of such. In some cases, instead of a hermitian metric on vector bundles one rather has a flat connection. It would then be of interest to extend the theory of Gillet and Soulé, plus contributions of Bismut, to this setting. Together with Richard Wentworth, we initiated this program on arithmetic surfaces. Our approach is inspired by Deligne's functorial approach to the Riemann-Roch formula. In this talk, I will report on our construction of an arithmetic intersection theory for flat line bundles on arithmetic surfaces, and explain the Riemann-Roch formula.

Secants of Veronese embeddings and Hopf rings

When: Mon, March 7, 2016 - 2:00pm
Where: Math 1311
Speaker: Steven Sam (University of Wisconsin) -
Abstract: I'll explain a connection between Hopf rings and secant schemes of Veronese embeddings of arbitrary projective schemes and how this can be used to prove the existence of a uniform bound on the degrees of the minimal generators of the ideal of the rth secant scheme independent of the Veronese embedding. This is based on

The moduli space of stable n-pointed genus zero curves is not a Mori Dream Space when n is at least 13

When: Mon, March 21, 2016 - 2:00pm
Where: Math 1311
Speaker: Jose Gonzalez (Yale University) -

Abstract: We prove that the moduli space of stable n-pointed genus zero curves is not a Mori dream space when n is at least 13. We build on the work of Ana-Maria Castravet and Jenia Tevelev who recently obtained the same conclusion when n is at least 134. This talk is a report on joint work with Kalle Karu.

Counting Local Systems on Supersingular Abelian Varieties.

When: Mon, March 28, 2016 - 2:00pm
Where: Math 1311
Speaker: Brett Frankel (UPenn) -

Abstract: In a 2008 paper, Hausel and Rodriguez-Villegas studied the moduli space of (twisted) local systems on a Riemann surfaces by computing the number of representations of the fundamental group in GL_n(q). We will discuss some situations where instead of a Riemann surface, one considers an abelian variety defined over an algebraically closed field of characteristic p. For a supersingular abelian variety A, we count the number of representations of the etale fundamental group of A to GL_n(q), where q is a power of p. This count (for fixed n) turns out to be a polynomial in q. The space of such representations is not a scheme, but does have the structure of a constructible set. We give an explicit formula for this polynomial, then state a few theorems which elucidate its features. In particular, we state a new result which generalizes to cosets a theorem of Frobenius about the number of solutions to x^n=1 in a finite group.

Vertical DT invariants and modular forms

When: Thu, April 7, 2016 - 1:00pm
Where: Math 1313
Speaker: Artan Sheshmani (Massachusetts Institute of Technology) -
Abstract: I will talk about derivation of an explicit formula for the generating function of all DT invariants counting "vertical" two dimensional sheaves on K3 fibrations. The final expressions will be shown to satisfy strong modularity properties. In particular I will talk about a new construction of vector valued modular forms which emerges from the geometric framework, exhibiting some of the features of a Hecke transform. This is joint work with Vincent Bouchard, Thomas Creutzig, Emanuel Diaconescu, Charles Doran and Callum Quigley.

Rank 2 stable sheaves on toric threefolds: classical and virtual counts

When: Mon, April 11, 2016 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: We study the torus fixed set of the moduli space of stable rank 2 torsion free sheaves on toric 3-folds. This leads to finding the Euler characteristic of the moduli space and developing a vertex theory for the rank 2 Donaldson-Thomas invariants. These invariants are expressed combinatorially in terms of a new labelled box configuration. We compare this to the vertex theories for the rank 1 Donaldson-Thomas theory of Maulik-Nekrasov-Okounkov-Pandharipande and the stable pair theory of Pandharipande-Thomas.

The decomposition formula for log Gromov-Witten invariants

When: Mon, April 18, 2016 - 2:00pm
Where: Math 1311
Speaker: Qile Chen (Boston College) -
Abstract: One major goal of log Gromov-Witten theory is to reconstruct usual Gromov-Witten invariants of a smooth projective variety X using appropriate invariants of a degeneration of X. The decomposition formula is the first step toward this. It breaks the invariants of a degeneration of X into terms which can be classified by the dual complexes/tropical curves of stable log maps. This leads to a decomposition formula on the level of virtual cycles.

This is a joint work with Dan Abramovich, Mark Gross, and Bernd Siebert.

Equivariant Algebraic K-theory and Derived completion

When: Mon, April 25, 2016 - 2:00pm
Where: Math 1311
Speaker: Roy Joshua (Ohio State University ) -
Abstract: Derived completion is a technique that originated slightly over 20 years ago. However, Gunnar Carlsson may have been the first to realize the true potential of this technique, especially in the context of Algebraic K-theory. The goal of this talk is to discuss some applications of this technique, especially in the context of our joint work with Carlsson, to equivariant algebraic K-theory.

Rank 2 wall-crossing and the Serre correspondence

When: Mon, May 2, 2016 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: We study Quot schemes of 0-dimensional quotients of sheaves R on 3-folds X. When R is rank 2 and reflexive, we prove that the generating function of Euler characteristics of these Quot schemes is a power of the MacMahon function times a polynomial. This polynomial is itself the generating function of a certain 0-dimensional sheaf supported on the locus where R is not locally free.
In the case X = C^3 and R is equivariant, we use our result to prove an explicit product formula for the generating function. This formula was first found using localization techniques and the double dimers. Our results follow from Hartshorne's Serre correspondence and a rank 2 version of a Hall algebra calculation by J. Stoppa and R. P. Thomas.

Motivic Decomposition of Projective Pseudo-Homogeneous Varieties (thesis defense)

When: Mon, May 9, 2016 - 2:00pm
Where: Math 1310
Speaker: Srimathy Srinivasan (University of Maryland) -