Boston College NT&AG seminar

Abstracts for Spring 2015 ( Older abstracts ) p>Apr. 30: Steven J. Miller (Williams College). Finite conductor models for zeros near the central point of elliptic curve L-functions
Abstract:
Random Matrix Theory has successfully modeled the behavior of zeros of elliptic curve L-functions in the limit of large conductors. In this talk we explore the behavior of zeros near the central point for one-parameter families of elliptic curves with rank over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except possibly at the central point for deep arithmetic reasons; these families provide a fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though theory suggests the family zeros (those we believe exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical calculations show this is not the case; however, the discrepency is likely due to small conductors, and unlike excess rank is observed to noticeably decrease as we increase the conductors. We shall mix theory and experiment and see some surprisingly results, which leads us to conjecture that an excised orthogonal ensemble correctly models the small conductor behavior

Apr. 23: Chao Li (Harvard) Level raising mod 2 and arbitrary 2-Selmer ranks
Abstract:
We prove a level raising mod p=2 theorem for elliptic curves over Q, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the p-part of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd p. This is joint work with Bao V. Le Hung.

Apr. 16:Paolo Stellari (Universita degli Studi di Milano) Uniqueness of dg enhancements in geometric contexts
Abstract:
It was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper. In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasi-coherent sheaves on any scheme. This is a joint work in progress with A. Canonaco.

Apr. 9:Yuchen Zhang (Michigan) The volume of isolated singularities
Abstract:
Boucksom, de Fernex and Favre defined a non-log-canonical volume to study isolated singularities and developed several geometric results. Their definition is based on the intersection number of nef envelope of the log discrepancy b-divisor. In this talk, I will give an alternate definition of this volume by log canonical modification.

Mar. 26:Sho Tanimoto (Rice) Height zeta functions of algebraic groups
Abstract:
One of fundamental questions in diophantine geometry is to understand asymptotic formulas for counting functions of rational points of bounded height, and for each counting function, one can associate a Dirichlet series socalled height zeta functions. Height zeta functions have been studied for a variety of equivariant compactifications of homogeneous spaces including toric varieties and equivariant compactifications of vector groups. In this talk, I will review some results in this area, and report a recent progress toward understanding of height zeta functions of one-sided equivariant compactifications of algebraic groups. This is joint work with Ramin Takloo-Bighash

Mar. 12:Michael Magee (IAS) Expanding maps and continued fractions
Abstract:
In this talk I'll discuss a lattice point count for a thin semigroup inside SL_2(Z). It is important for applications that I'll describe that one can perform this count uniformly throughout congruence classes. The approach to counting is dynamical - with input from both the real place and finite primes. At the real place methods of Dolgopyat that were further built upon by Naud are at play. At finite primes the necessary mixing property follows from work of Bourgain and Gamburd that uses tripling estimates in SL_2(F_p) due to Helfgott as key input. This is joint work with Hee Oh and Dale Winter.

Feb. 19: Charlotte Chan (U. of Michigan), ) p-adic Deligne-Lusztig constructions and the local Langlands correspondence
Abstract:
The representation theory of SL2(Fq) can be studied by studying the geometry of the Drinfeld curve. This is a special case of Deligne-Lusztig theory, which gives a beautiful geometric construction of the irreducible representations of finite reductive groups. I will discuss recent progress in studying Lusztig's conjectural construction of a p-adic analogue of this story. It turns out that for division algebras, the cohomology of the p-adic Deligne-Lusztig (ind-)scheme gives rise to supercuspidal representations of arbitrary depth and furthermore gives a geometric realization of the local Langlands and Jacquet-Langlands correspondences. (This talk is based on arXiv:1406.6122 and forthcoming work.)

Feb. 5: Tony Varilly-Alvarado (Rice) Special cubic fourfolds and K3 surfaces: an arithmetic perspective
Abstract:
Cubic fourfolds containing a surface not homologous to a complete intersection often have nonspecial cohomology isomorphic to the primitive cohomology of a K3 surface "twisted" by an element of the Brauer group. This isomorphism is usually a manifestation of a geometric correspondence, which has consequences for the distribution of rational points on K3 surfaces over number fields. We will discuss this circle of ideas, including some recent developments in joint work with McKinnie, Sawon and Tanimoto on p-torsion Brauer classes of K3 surfaces and with Tanimoto on the Kodaira dimension of the moduli space of special cubic fourfolds of fixed discriminant

Jan. 29: Alex Kontorovich (Rutgers) Equidistribution of Shears and Applications
Abstract:
A "shear" is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete group (possibly of infinite co-volume). In joint work with Dubi Kelmer, we prove the regularized equidistribution of shears under large translates. We give applications including to moments of GL(2) automorphic L-functions, and to counting integer points on affine homogeneous varieties. No prior knowledge of these topics will be assumed.

Jan. 15: Ethan Cotterill (Universidade Federal Fluminense) Dimension counts for singular rational curves
Abstract:
Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. sub-semigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus g \leq 8, the dimension of unibranch singularities is as one would naively expect, but that expectations fail as soon as g=9. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to g when g \leq 5. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins. Abstracts for Fall 2014 ( Older abstracts )

Dec. 04: Igor Minevich (BC), Cohomology of Topological Groups and Grothendieck Topologies
Abstract:
For an abstract group G, there is only one "canonical" theory H^n(G, A) of group cohomology for a G-module A. There are many different ways to generalize this theory when G is a topological group and A is a topological G-module. Some examples are the continuous cochain theory, Moore's measurable cochain theory, and Wigner's semisimplicial theory. Lichtenbaum was able to use the semisimplicial theory to state a conjecture about special values of the zeta function of a number field. The questions then arise: (1) how do we compare the different theories, and (2) can we apply other theories just as Lichtenbaum did? We will talk about how Grothendieck topologies can help answer these questions.

Nov. 13: Letao Zhang (Stony Brook), Character formula on cohomologies of deformations of Hilbert schemes of K3 surfaces
Abstract:
Let X be a hyperkaehler manifold deformation equivalent to Hilbert scheme of n points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X, and derive a generating series for deducing the number of canonical Hodge classes on X. The formula indicates the number of Hodge classes on X that remain Hodge under any deformation.

Nov. 06: Andrew Phillips (BC), A Gross-Zagier formula for Shimura curves
Abstract:
The Gross-Zagier theorem on singular moduli gives a formula for the product of differences of j-invariants over all pairs of imaginary quadratic integers with fixed relatively prime fundamental discriminants. I will explain how this theorem can be interpreted as a result about intersection theory on a modular curve and describe a generalization to certain Shimura curves.

Oct. 30: Michael Woodbury (Columbia), An Adelic Kuznetsov Trace Formula for GL(4)
Abstract:
An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications. In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4). I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem.

Oct. 23: Alexander Polishchuk, (U of Oregon) A modular compactification of M_{1,n} from A-infinity structures
Abstract:
I will report on a joint work with Yanki Lekili. We give an elementary construction of a compactification of the moduli space of genus 1 curves with n marked points and give its interpretation as the moduli space of A-infinity structures on a certain finite-dimensional algebra. We show that our compactification coincides with the moduli space of (n-1)-stable curves considered by David Smyth and derive some geometric consequences (e.g., show normality).

Oct. 16: Jenia Tevelev (UMass Amherst), Algebraic surfaces of general type of geometric genus zero
Abstract:
Algebraic surfaces of general type without non-trivial global holomorphic 2-forms occupy the most mysterious region in geography of surfaces. Standard approaches to their moduli (e.g. by variation of Hodge structure or through the study of canonical models) have been so far largely unsuccessful. By this reason deep questions like the Bloch conjecture are so hard to settle. I will survey what's known about these surfaces with an emphasis on "stable surfaces" on the boundary of the moduli space.

Oct. 09: Aaron Silberstein (UPenn), Geometric Reconstruction in Bogomolov's Program of Anabelian Geometry
Abstract:
This talk will be an introduction to Bogomolov's Program of Anabelian Geometry, as developed by Bogomolov, Tschinkel, Pop, Topaz, and the speaker. We will apply Bogomolov's program - and in particular the technique of geometric reconstruction - to geometric Galois actions and Grothendieck-Teichmüller theory.

Oct. 02: Brian Lehmann (BC), The geometric constants in Manin's Conjecture
Abstract:
Manin's Conjecture predicts that the behavior of rational points on a variety X is controlled by certain geometric constants associated to X. I will discuss how the minimal model program can be used to analyze the behavior of these constants. This is joint work with Sho Tanimoto and Yuri Tschinkel.

Sep. 25:Matt Lamoureux (BC), Stirling’s formula in number fields
Abstract:
Stirling's formula says n! is asymptotic to (n^n/e^n)sqrt{2 pi n}, or equivalently log(n!) = nlog(n) - n + (1/2)log(2 pi n) + o(1), as n approaches infinity. I will discuss what an analogue of this formula looks like in a number field K, where there is an unexpected contribution of zeros of the zeta-function of K when K is not Q.

Sep. 18: Ilya Vinogradov(Bristol) Directions in Hyperbolic Lattices
Abstract:
Let Gamma be a lattice in the group of isometries of the hyperbolic n-space. Given two points w and z in H^n, we analyze local statistics of the set of directions of orbit Gamma.w as viewed from z.

Sep. 11: Lior Bary-Soroker (Tel Aviv University) Simultaneous prime polynomial values of linear functions .
Abstract:
The theme of the talk is number theory in function fields. I will mainly discuss the recent resolution of a function field analogue of a folklore conjecture on the number of simultaneous prime values of several linear functions. This conjecture vastly generalizes both the twin prime conjecture and the Goldbach conjecture, hence in the number field setting is considered as hopelessly open.

Abstracts for Spring 2014

May. 01 2pm Angelica Cueto (Columbia U), Faithful tropicalization of the Grassmannian of planes
Abstract:
In this talk, we show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich sense. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian (inside the split torus) as a space of phylogenetic trees by Speyer-Sturmfels. We also show that both sets have piecewiselinear structures that are compatible with our homeomorphism and characterize the fibers of the tropicalization map as affinoid domains with a unique Shilov boundary point. Our homeomorphism identify each point in the tropical Grassmannian with the Shilov boundary point on its fiber. Time permitted, we will discuss the combinatorics of the aforementioned space of trees inside tropical projective space.
This is joint work with M. Haebich and A. Werner (Math. Ann. (2014), in press).

May. 01 3pm Izzet Coskun (UIC), Effective divisors on moduli spaces of sheaves on the plane
Abstract:
In this talk, I will describe joint work with Jack Huizenga and Matthew Woolf on determining when Brill-Noether divisors on moduli spaces of sheaves on the plane are effective. The proof is inspired by Bridgeland stability and relies on remarkable number theoretic properties of exceptional slopes.

Apr. 24: Andrew Sutherland (MIT)), The refined Sato-Tate conjecture
Abstract:
The classical Sato-Tate conjecture (now a theorem) is concerned with the distribution of the number of points on the reductions modulo primes of a fixed elliptic curve E over the rational numbers. This distribution corresponds to that given by a random matrix model, using the Haar measure on the special unitary group SU(2), or a subgroup thereof when E has complex multiplication.
The Sato-Tate conjecture generalizes naturally to abelian varieties over number fields. It associates to an abelian variety A of dimension g a compact subgroup of the unitary symplectic group USp(2g) whose Haar measure governs the distribution of arithmetic data attached to the abelian variety; this subgroup of USp(2g) is called the Sato-Tate group of A. While the Sato-Tate conjecture remains open for g > 1, I will present recent work that has culminated in a complete classification of the 52 Sato-Tate groups that can and do arise when g = 2. I will also describe numerical computations that support the conjecture, along with animated visualizations of this data, and discuss the prospects for obtaining similar results in genus g > 2.

Apr. 10:François Charles (Université Paris-Sud), Zarhin's trick for K3 surfaces and the Tate conjecture
Abstract:
f A is an abelian variety, Zarhin's trick constructs a principal polarization on a power of the product of A and its dual. We prove a version of Zarhin's trick for K3 surfaces over arbitrary fields using results of Mukai. As an application, we give a simple proof of the Tate conjecture for K3 surfaces over finite fields (proved in characteristic at least 3 by Maulik, Pera and the author) without using the Kuga-Satake correspondence.

Apr. 03: Kamal Khuri-Makdisi(AUB), Eisenstein series of weight 1
Abstract:
Let N >= 3. In this talk, I will sketch a proof that the ring generated by Eisenstein series of weight 1 on the principal congruence subgroup Gamma(N) contains all modular forms in weights 2 and above. This means that the only forms that are not seen by polynomials in these Eisenstein series are cusp forms of weight 1. This result gives rise to a systematic way to produce equations for the modular curve X(N).

Mar. 27: Dubi Kelmer (BC) ), Equidistribution of translated geodesics and counting integer points
Abstract:
I will describe new results on the equidistribution of sheared unbounded geodesics on a hyperbolic surface and show how it can be used to get precise asymptotics for the number of integer solutions to equation such as x^2+y^2-z^2=1 that lie in growing balls. This is based on joint work with A. Kontorovich.

Mar. 20: Alexei Oblomkov (UMass), Topology of the Affine Springer Fibers: examples and applications to the representation theory
Abstract:
The talk is based on joint work with Zhiwei Yun. Given an element $a\in \mathfrak{g}[[t]]$, the affine Springer is the locus inside affine Grassmanian of type $\mathfrak{g}$ fixed by $a$. In my talk I explain the construction of the action of the rational Cherednik algebra on the cohomology of the affine Springer fibers that admit $\mathbb{C}^*$-action. To demonstrate the richness of the subject I will discuss in details the Affine Springer Fibers for the algebras $\mathfrak{g}$ of rank $2$: many familiar objects like elliptic curves and $K3$-surfaces will appear as examples of $\CC^*$-fixed locus of such Springer fibers. If time permits I few consequences of the construction, in particular, a combinatorial formula for the dimensions of the finite dimensional spherical representation of the rational Cherednik algebras.

Mar. 13: Nicola Tarasca(Utah), Double total ramifications for curves of genus 2
Abstract:
Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to P1 of degree d totally ramified at the two marked points have codimension two. In this talk I will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. I will also discuss the relation with the related work of Hain, Grushevsky-Zakharov, and Chen-Coskun.

Feb. 27: Tomer Schlank (MIT), Etale Homotopy, Sections and Diophantine Equations
Abstract:
From the view point of algebraic geometry solutions to a Diophantine equation are just sections of a corresponding map of schemes X- > S. When schemes are usually considered as a certain type of "Spaces" . When considering sections of maps of spaces f:X->S in the realm of algebraic topology Bousfield and Kan developed an Obstruction-Classification Theory using the cohomology of the S with coefficients in the homotopy groups of the fiber of f. In this talk we will describe a way to transfer Bousfield - Kan theory to the realm of algebraic geometry. Thus yielding a theory of homotopical obstructions for solutions for Diophantine equations . This would be achieved by a generalizing the étale homotopy type defined by Artin and Mazur to a relative setting X ? S . In the case of Diophantine equation over a number filed i.e. when S is the spectrum of a number field, this theory can be used to obtain a unified view of classical arithmetic obstructions such as the Brauer-Manin obstruction and descent obstructions. If time permits I will present also applications to Glaios theory.

Feb. 20: Melody Chan (Harvard), Fano schemes of determinants and permanents
Abstract:
I will discuss joint work with Nathan Ilten in which, motivated loosely by questions from geometric complexity theory, we study the geometry of the Fano schemes parametrizing the k-dimensional linear subspaces of matrices all of whose r x r subdeterminants or subpermanents vanish. For example, we will completely answer the following question: when is the Fano scheme parametrizing k-planes of singular n x n matrices connected?

Feb. 13: Brendan Hassett (Rice), K3 surfaces over local fields and derived equivalence
Abstract:
Isogenies of elliptic curves are a fundamental tool in Diophantine geometry. For K3 surfaces, the notion of derived equivalence plays an analogous role. By a result of Lieblich-Olsson, two derived equivalent K3 surfaces over a finite field have the same number of points. We discuss analogous results over local fields. (joint with Tschinkel)

Feb. 06: David Treumann (BC), Functoriality, Smith theory, and the Brauer homomorphism.
Abstract:
Smith theory is a technique for relating the mod p homologies of X and of the fixed points of X by an automorphism of order p. I will discuss how, in the setting of locally symmetric spaces, it provides an easy method (no trace formula) for lifting mod p automorphic forms from G^{sigma} to G, where G is an arithmetic group and sigma is an automorphism of G of order p. This lift is compatible with Hecke actions via an analog of the Brauer homomorphism from modular representation theory, and is often compatible with a homomorphism of L-groups on the Galois side. The talk is based on joint work with Akshay Venkatesh. I hope understanding the talk will require less number theory background than understanding the abstract.

Abstracts for Fall 2013

Nov. 21: Alina Marian (Northeastern) Strange duality on abelian surfaces
Abstract:
Strange duality emerges as a symmetry in the geometry of determinant line bundles over spaces of sheaves on a variety. This symmetry has several distinct flavors when the underlying variety is an abelian surface. I will discuss recent results in the abelian case, obtained in joint work with Barbara Bolognese and Dragos Oprea.

Nov. 14: Anand Deopurkar (Columbia University) GIT Stability of First Syzygies of Canonical Curves
Abstract:
The GIT quotients arising from the generators of the ideals of (pluri)canonically embedded curves are expected to yield certain log canonical models of M_g . As a way to reach the canonical model, we can also consider the quotients arising from the syzygies. I will introduce these GIT quotients, explain their significance, and outline a proof of the first step towards their description, namely that if g is odd, then the first of these quotients is non-empty. This is joint work with Fedorchuk and Swinarski.

Nov. 7: Keith Merrill (Tufts) Approximations on Quadratic Hypersurfaces
Abstract:
The field of Diophantine approximation seeks to quantify the density of a subset Q in a metric space X. We will motivate this notion by recalling four hallmark theorems of the field which answer this question for $\mathbb{Q} \subset \mathbb{R}$. We will then discuss analogues of these theorems in the case of quadratic hypersurfaces with a dense set of rational points. Our proofs rely on dynamical reformulations of these questions. Based on joint work with L. Fishman, D. Kleinbock, and D. Simmons.

Oct. 31: Wei Ho (Columbia) Some applications of explicit moduli spaces for genus one curves
Abstract:
We will first discuss how to describe certain moduli spaces related to genus one curves in an explicit manner. Using ideas from these descriptions, one may construct genus one curves mapping to Brauer-Severi varieties (joint work with A.J. de Jong). Another arithmetic application of these moduli spaces gives upper bounds for average ranks of elliptic curves in various families (joint work with M. Bhargava)

Oct. 24: Sebastian Casalaina-Martin(University of Colorado), Extending the Prym map to toroidal compactifications of the moduli of abelian varieties.
Abstract:
Extending period maps to toroidal compactifications of the moduli of abelian varieties has been of interest since their construction. For curves, it was shown by Mumford and Namikawa that the Torelli map extends to a morphism from the moduli of stable curves to the second Voronoi compactification. Recently, Alexeev and Brunyate showed that the Torelli map also extends to the perfect cone compactification. The case of admissible covers and Prym varieties has been investigated in the literature as well. Friedman and Smith showed that the Prym map fails to extend to a morphism from Beauville's moduli of admissible covers to either of these toroidal compactifications. Recently Alexeev, Birkenhake and Hulek gave a characterization of the indeterminacy locus of the map to the second Voronoi compactification. In this talk I will discuss joint work with Grushevsky, Hulek and Laza where we investigate the indeterminacy locus of the Prym map to the perfect cone compactification, as well as a resolution of the rational map.

Oct. 24: Radu Laza (Stony Brook) The KSBA compactication for the moduli space of degree two K3 pairs.
Abstract:
A classical (and still open) problem in algebraic geometry is the search for a geometric compactication for the moduli of polarized K3 surfaces (X,L). If one considers instead K3 pairs (X,H) with H a divisor in the linear system |L|, the resulting moduli space has a natural geometric compactication given by the general MMP framework (pioneered by Kollar, Shepherd-Barron, and Alexeev). In this talk, I will discuss the existence of a good compactication for the moduli of K3 pairs in all degrees, and then discuss in detail the degree 2 case.

Oct. 17: Ellen Goldstein (BC) Nilpotent Orbit Closures in the Symplectic and Orthogonal Groups
Abstract:
Consider the action of a linear algebraic group G on its Lie algebra, Lie G, given by the adjoint representation. For a nilpotent element X of Lie G, the orbit of X is its conjugacy class, and the closure of the orbit is not, in general, a smooth variety. But what about normal? I'll discuss some known results when working over an algebraically closed field of characteristic 0, and how far these extend when we allow positive characteristics, focusing on the particular cases when G is a symplectic or orthogonal group.

Oct. 10: Alex Wright (University of Chicago) Affine submanifolds of the moduli space of abelian differentials
Abstract:
Moduli spaces (strata) of abelian differentials (holomorphic one forms) on Riemann surfaces have a natural affine structure, in that they each have an atlas of charts to C^n with transition functions in GL(n,Z). I will discuss submanifolds of these moduli spaces that are locally defined by linear equations with real coefficients. The two dimensional case corresponds to Teichmuller curves. The main goal will be to explain how algebraic geometry may provide insight into recent conjectures coming from Teichmuller dynamics.

Oct. 3: Laura Rider (MIT) Parity sheaves on the affine Grassmannian and the Mirkovic--Vilonen conjecture
Abstract:
Let G be a connected complex reductive group, and let Gr denote its affine Grassmannian. The topology of Gr encodes the representation theory of the split Langlands dual group G^\vee over any field k via the geometric Satake equivalence due to Mirkovic--Vilonen. This result raises the possibility of using the universal coefficient theorem of topology to compare representations over different fields. With that in mind, Mirkovic and Vilonen conjectured that the local intersection cohomology of the affine Grassmannian with integer coefficients is torsion-free. I will discuss the proof of (a slight modification of) the Mirkovic--Vilonen conjecture. This is joint work with Pramod Achar.

Sep. 26: Li-Mei Lim (BC) Counting Square Discriminants
Abstract:
I will describe a recent result in which we count the number of binary quadratic forms of fixed discriminant and with bounded coefficients. Our methods involve shifted convolution sums of Fourier coefficients of an automorphic form.

Sep. 19: Anand Patel (BC) Picard groups of some moduli spaces of curves
Abstract:
In this talk I will discuss some conjectures about Picard groups of Severi varieties and Hurwitz spaces. Some recent results, joint with A. Deopurkar, will be presented.

Sep. 12: Dan Ciubotaru (Utah) Spin representations of Weyl groups and elliptic theory for affine Hecke algebras
Abstract:
The classification of irreducible characters of pin double covers of Weyl groups began with Schur (1911) for the symmetric group, and was completed by A. Morris, Read and others in 1970's. Recently, motivated by the study of Dirac cohomology for modules over the graded affine Hecke algebras that appear in the representation theory of reductive p-adic groups, a new realization of these irreducible characters emerged. As I will explain in my talk, this realization is intimately related to Springer theory for ordinary representations of Weyl groups and with elliptic representation theory for Weyl groups and affine Hecke algebras. The talk is based on joint work with D. Barbasch, X. He, E. Opdam, and P. Trapa.

Abstracts for Sprimg 2013

Apr. 25: Charlie Siegel (KAVLI IPMU) The Schottky Problem in genus 5
Abstract:
I will talk about the Schottky problem, it's history, and particularly in the resolution of the Schottky-Jung conjecture in genus five via the study of the Prym map on genus six curves and finite incidence geometry.

Apr. 18: Thanos D. Papaïoannou (Harvard), The non-abelian p-curvature conjecture
Abstract:
We will start on the shores of algebraic differential equations and the classical p-curvature conjecture. After frolicking for a bit and working on a few examples, we will walk from obvious idea to obvious idea and onto a little boat of analogies, and we'll sail to the shores of Carlos Simpson's non-abelian world. There we will explore, first, then formulate the correct analogue of the p-curvature conjecture, and sketch its proof in the case of non-abelian de Rham cohomology. This will be a special case of Bost's celebrated conjecture on the algebraicity of the leaves of foliations on varieties defined over number fields. The talk will require no previous exposure to the p-curvature conjecture or non-abelian cohomology of the audience.

Apr. 11: Dustin Clausen (MIT), Arithmetic duality in algebraic K-theory
Abstract:
We will talk about a duality in the algebraic K-theory of number rings which is akin to the Poitou-Tate duality in Galois cohomology. A key feature is that the fundamental class'' which controls this duality can be explicitly constructed by a homotopy-theoretic procedure. As a consequence, we will give a homotopy-theoretic description of the Artin map, one which makes the reciprocity law visible.

Apr. 04: Dmitry Zakharov (Stony Brook), The zero section of the universal semi-abelic variety and the double ramification cycle
Abstract:
The moduli space of principally polarized abelian varieties is one of the central objects of study in algebraic geometry. The moduli space is not compact, and admits several natural compactifications. All of these compactifications are extensions of Mumford's partial compactification by semi-abelic varieties of torus rank one. The partial compactification is the base for a universal family that has a natural geometric class, namely the zero section. In our joint work with Samuel Grushevsky, we calculate the Chow class of this zero section. By pulling back this class to the moduli space of curves via the Abel--Jacobi map, we obtain a refinement of Hain's result on the double ramification cycle. I will also describe how the Abel--Jacobi map can be used to obtain interesting relations in the tautological ring of the moduli space of curves.

Mar. 14: Scott Nollet (TCU), Class groups in geometry and algebra
Abstract:
When a local ring is completed, the associated map on class groups is injective. V. Srinivas has asked what are the possible images of this map for a fixed complete local ring (many rings can have isomorphic completions). We use global geometric methods (Noether-Lefschetz theory) to answer this question completely for rational surface singularities. Our method can be adapted to show that arbitrary completions of hypersurface singularities can be written as the completion of geometric unique factorization domains.

Feb. 28: Ben Howard (BC), Gross-Zagier theorems for higher weight modular forms
Abstract:
The main result is an equality of two numbers. The first number is the central derivative of the Rankin-Selberg convolution L-function of a modular form of weight n with the weight theta series attached to a Hermitian lattice of rank n-1. The second number is the Arakelov height pairing of two cycles on a unitary Shimura variety of dimension n-1. When n=2, the result is essentially the Gross-Zagier theorem on heights of Heegner points. This is joint work with Jan Bruinier and Tonghai Yang.

Feb. 21: Yifeng Liu (MIT), Refined Gan-Gross-Prasad conjecture for Bessel periods and some examples
Abstract:
In this talk, we will propose a conjectural formula of the Bessel periods for pairs of orthogonal groups or unitary groups in general cases, extending the work off Ichino-Ikeda and N. Harris where the restricted group is reductive. It also specializes to a conjectural formula about the Fourier-Whittaker coe cients where the restricted group is unipotent. We will look at some new examples where the restricted group is neither reductive nor unipotent using theta lifting.

Jan. 31: Sungmun Cho (Northeastern), Group schemes and local densities.
Abstract:
The celebrated Smith-Minkowski-Siegel mass formula expresses the mass (obstruction to Hasse principle) of a hermitian (or a quadratic) lattice as a product of local factors, called the local densities. This mass formula is an essential tool for the classification of integral hermitian (or quadratic) lattices. In this talk, I will explain the local density formula for hermitian (or quadratic) lattices, with p=2, by constructing a smooth integral model of an appropriate unitary (or orthogonal) group. As a special case, I will give a long awaited proof for the local density formula of Conway and Sloane.

Jan. 24: Kwangho Choiy (Oklahoma State University), Plancherel Measures and Invariants of L-parameters.
Abstract:
The Plancherel measure is well understood for many cases of quasi-split groups over a p-adic feld, but little is known for non-quasi-split p-adic inner forms. In this talk, we shall discuss the behavior of the Plancherel measure between p-adic inner forms. This work supports that the Plancherel measure is invariant on L-packets and preserved by inner forms. As an application, all the known facts involving the Plancherel measure for a quasi-split group can be transferred to its inner forms.

Abstracts for Fall 2012

Dec. 06: Zachary Maddock(Columbia), Regular del Pezzo surfaces with irregularity
Abstract:
Over perfect fields, the geometry of regular del Pezzo surfaces has been classified, but over imperfect fields, the problem remains largely open. We construct the first examples over imperfect fields of regular del Pezzo surfaces X that have positive irregularity h^1(X, O_X ) > 0. Our construction is by quotienting a regular, quasi-linear surface (i.e. a regular variety that is geometrically a non-reduced first-order neighborhood of a plane) by explicit rank 1 foliations. We also find a restriction on the integer pairs that are possible as the anti-canonical degree and irregularity of such surfaces.

Nov. 29: Rajesh Kulkarni (Michigan State), Relative Brauer groups of curves
Abstract:
The relative Brauer group of function fields of curves has recently generated much interest. We will discuss historical interest in this circle of ideas followed by recent developemts. We will then discuss our recent work (together with Aaron Levin). One consequence of our work is that over a global field $K$, given a square-free integer $\ell$ and a finitely generated $\ell$-torsion subgroup G of the Brauer group of $K$, we construct a smooth projective curve with minimal gonality whose relative Brauer group contains G. We will also discuss the relationship with representations of weighted Clifford algebras.

Nov. 15: Anand Patel (Harvard), A Decomposition of Hurwitz Space
Abstract:
In this talk we will describe a natural decomposition of the Hurwitz space $\mathcal{H}_{d,g}$ parametrizing simply-branched covers of $\mathbb{P}^1$. By studying the geometry of this decomposition, we will deduce the irreducibility of the Gieseker-Petri locus $\mathcal{GP}^1_{d} \subset \mathcal{M}_{g}$ where $d = \frac{g+2}{2}$. Furthermore, we will explain the role this decomposition plays in establishing upper bounds for slopes of sweeping curves in the $d$-gonal locus.

Nov. 15: Yaim Cooper (Princeton), Stable quotients and stable maps: A comparison.
Abstract:
In this talk we will discuss recent results comparing stable quotients and stable maps, both comparing the geometry of the moduli spaces themselves, as well as the invariants they can be used to compute. Specifically, in genus 1 we compare the geometries of M_1(P^r,d) and Q_1(P^r,d) and in genus 0 we compare the stable quotient invariants of Fano and Calabi-Yau complete intersections in P^r to the corresponding Gromov-Witten invariants. The latter represents joint work with A. Zinger.

Nov. 08: Ozlem Imamoglu (ETH), On weakly harmonic Maass forms and their Fourier coefficients.
Abstract:
Harmonic Maass forms have been studied extensively in last several years due to their connection to several arithmetic problems. Most notably in the half integral weight case, it can be shown that the Ramanujan's mock theta functions and generating functions of traces of singular moduli can be understood in terms of harmonic Maass forms. A simpler example is provided by the non-holomorphic Eisenstein series of weight 2. In this talk, after giving s short overview of the subject, I will show how to construct a distinguished basis of such forms in the case of weight 2 and study their relation of the regularized inner products of modular functions.

Nov 01: Jack Thorne (Harvard), Automorphy lifting and the Fontaine-Mazur conjecture for U(3).
Abstract:
Recent years have seen great progress in the understanding of the Galois representations associated to unitary groups over CM fields. I will discuss an automorphy lifting theorem in this setting for l-adic Galois representations which are residually reducible, and give an application to the Fontaine-Mazur conjecture for U(3).

Oct. 25: Keerthi Madapusi (Harvard) The Tate conjecture for K3 surfaces over fields of odd characteristic.
Abstract:
The classical Kuga-Satake construction, over the complex numbers, uses Hodge theory to attach to each polarized K3 surface an abelian variety in a natural way. Deligne and Andre extended this to fields of characteristic zero, and their results can be combined with Faltings's isogeny theorem to prove the Tate conjecture for K3 surfaces in characteristic zero. Using the theory of integral canonical models of Shimura varieties of orthogonal type, we extend the Kuga-Satake construction to odd characteristic. We can then deduce the Tate conjecture for K3s in this situation as well (with some exceptions in characteristic 3).

Sep. 27: Yangbo Ye (University of Iowa), Asymptotic Voronoi's summation formula.
Abstract:
Let f be a Maass cusp form for SL(3,Z) with Fourier coefficients A(m,n). The asymptotic expansion of Voronoi's summation formula for A(m,n) is a powerful analytic technique in number theory. In this talk, we will give a proof of this asymptotic formula.

Sep. 20: Yusuf Mustopa (BC), Stability of Syzygy Bundles on Surfaces .
Abstract:
The syzygy bundle associated to a subvariety X of Pn governs the fine structure of the equations which cut out X in Pn. A classical result of Ein-Lazarsfeld implies that this bundle is stable when X is a curve embedded in Pn by a complete linear series of large degree. In this talk, I will discuss joint work with Lawrence Ein and Robert Lazarsfeld which generalizes this result to surfaces.

Sep. 13: Daniel Erman (Michigan), Syzygies and Boij--Soederberg Theory .
Abstract:
For a system of polynomial equations, it has long been known that the relations (or syzygies) among the polynomials provide powerful insights into the properties and invariants of the corresponding projective varieties. Boij--Soederberg Theory provides a powerful perspective on syzygies, and in particular reveals a surprising duality between syzygies and cohomology of vector bundles. I will describe some new results on this duality and on the properties of syzygies.

Abstracts for Spring 2012

Apr. 26: David Hansen (BC)
What is an eigenvariety, and how big are they?
Abstract:
Eigenvarieties are universal parameter spaces for overconvergent p-adic modular forms, and are profound examples of Tate's rigid analytic spaces. I'll explain what these words mean, how to use some beautiful ideas of Ash and Stevens in order to avoid knowing what an overconvergent modular form is, and some new tools for analyzing the dimension and local geometry of eigenvarieties.

Apr. 12: Sug-Woo Shin (MIT)
Fields of rationality for automorphic representations.
Abstract:
Loosely speaking, the field of rationality (or coefficient field) for a normalized modular form (eigenform) is the extension field of Q generated by the coefficients in its q-expansion. I will explain 1. how the notion of field of rationality generalizes to automorphic representations 2. its arithmetic significance and 3. the growth of field of rationality in an infinite family of automorphic representations of increasing level. Joint with Nicolas Templier.

Mar. 29: Yu-jong Tzeng (Harvard),
Counting curves with higher singularities on surfaces.
Abstract:
A famous problem in classical algebraic geometry is how many r-nodal curves are there in a linear system |L| on an algebraic surface S. If the line bundle L is sufficiently ample, Gottsche conjectured that the number of r-nodal curves is a universal polynomial of Chern numbers of L and S for any r. This conjecture was proven independently by Tzeng and Kool-Shende-Thomas. In this talk we will generalize Gottsche's conjecture and show the numbers of curves with any number of arbitrary isolated singularity on surfaces are also given by universal polynomials. Moreover these polynomials can be combined to form a huge generating series and we will discuss its properties.

Thu. Mat. 15: Nir Avni (Harvard),
Groups in positive characteristic and groups in characteristic zero. .
Abstract:
I will talk about several connections between the representation theories of algebraic groups over rings, when you vary the ring.

Thu. Feb. 23: Lei Zhang (BC),
Automorphic period on (Sp_{4n}(F), Sp_2n(E)).
Abstract:
In this talk, I will discuss two topics. First, considering a symmetric pairs (Sp_{4n}(F), Sp_2n(E)), we will discuss the automorphic period over this symmetric pair and local analogues of this global period integral. Second, I will discuss the construction of automorphic construction which is discussed by Dihua Jiang in BC-MIT seminar for some lower rank cases. It is work in progress with Dihua Jiang.

Thu. Feb. 16: Lior Rosenzweig (Hebrew University)
Galois group of random elements of linear groups.
Abstract:
Let A be a finitely generated subgroup of GLn(k), where k is a finitely generated field of characteristic zero. In the talk we will discuss what type of groups can occur as Gal(k_g/k), where g is an element of A, and k_g is the splitting field of the characteristic polynomial of g. In particular, we will show that if the Zariski closure of A does not contain a central torus (e.g semisimple), then given a random walk on A, the behaviour of Gal(k_g/k) is generic with respect to connected components of the Zariski closure. This is joint work with Alex Lubotzky

Abstracts for Fall 2011

Thu. Dec. 08: Liang Xiao (U. of Chicago))
Computing log-characteristic cycles using rami cation theory.
Abstract:
There is an analogy among vector bundles with flat connections, overconvergent F-isocrystals, and lisse l-adic sheaves. Given one of the objects, the property of being clean says that the ramification is controlled by the ramification along all generic points of the ramified divisors. In this case, one expects that the Euler characteristics may be expressed in terms of (subsidiary) Swan conductors; and (in first two cases) the log-characteristic cycles may be described in terms of the so-called refined Swan conductors.

Tue. Nov. 29: Bianca Virai (Brown)
On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell-Lang conjecture.
Abstract:
Let F : P^n --> P^n be a morphism of degree d > 1 defined over a number field K. Broadly speaking, arithmetic dynamics is the study of the orbit O_F(P) of a K-point P. Many questions in arithmetic dynamics are analogous to questions from diophantine geometry. The focus of the talk will be the dynamical Mordell-Lang conjecture which, as the name suggests, is analogous to the classical Mordell-Lang conjecture about intersections of subvarieties of abelian varieties with finitely generated subgroups. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_F(P) and a subvariety X of P^n is usually finite. We consider the number of linear subvarieties L in P^n such that the intersection of O_F(P) and L is "larger than expected." When F is the d'th-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are "super-spanned" by O_F(P), and further that the number of such subvarieties is bounded by a function of n, independent of the point P or the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in O_F(P)\S are in linear general position in P^n. No background in arithmetic dynamics is needed for this talk.

Thu. Nov. 10: Dawei Chen (BC)
Flat surfaces and Teichmueller curves.
Abstract:
A holomorphic 1-form defines a flat structure on a Riemann surface, such that it can be visualized as a plane polygon. Varying the shape of the polygon induces a natural action on the moduli space of 1-forms. On some rare occasions, an orbit under this action forms a complex curve, and we call it a Teichmueller curve. A special type of Teichmueller curves arises from a branched cover construction. Using it as example, I will illustrate an interesting interplay between polygon billiards, counting branched covers and the intersection theory on moduli space. The talk will be accessible (perhaps more accessible) to geometers as well as to graduate students.

Tue. Nov. 01: Michael Volpato (BC)
Generalizations of some results of Deuring.
Abstract:
We generalize certain results of Deuring to superspecial PEL abelian varieties. Specifically, we generalize his correspondence between supersingular elliptic curves and quaternionic ideal classes, the mass of supersingular elliptic curves as well as his lifting theorem. We give applications to bounding the number of mod-p eigensystems of (geometrically defined) automorphic forms, and to counting the number of irreducible components of the corresponding moduli space.

Tue. Oct. 11: Jared Weinstein (BU)
Maximal varieties over finite fields.
Abstract:
This is joint work with Mitya Boyarchenko. We construct a special hypersurface X over a finite field, which has the property of "maximality", meaning that it has the maximum number of rational points relative to its topology. Our variety is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups. As a consequence, the cohomology of X can be shown to realize a piece of the local Langlands correspondence for certain wild Weil parameters of low conductor.

Thu. Oct 06: Dubi Kelmer (BC)
Logarithm laws for unipotent flows.
Abstract:
The notion of a logarithm law is related to the rate excursion of generic orbits into the cusps, and it originates from problems in Diophantine approximations. I will describe joint work with Amir Mohammadi, establishing a logarithm law for certain homogenous spaces. In particular I will discuss the case of X=&Gamma\ SL(2,C). The main new ingredient in the proof is a uniform bound on the L2 norm of theta functions (also known as incomplete Eisenstein series) in this setting.