## abstracts for 2015-16 seminars

(The seminar webpage is here.)

Mihnea Popa (Northwestern University)
September 25, 2015

Positivity for Hodge modules and geometric applications

M. Saito’s theory of Hodge modules provides a powerful generalization of classical Hodge theory that is beginning to find basic applications to birational geometry. One of the main reasons for this is that generalizations of Hodge bundles and de Rham complexes arising in this context satisfy analogues of well-known vanishing and positivity theorems. I will review a number of such results that have been obtained recently, and how they can be applied to deduce new statements regarding holomorphic one-forms and families of varieties of general type. Much of this work is joint with C. Schnell. This talk is intended for a wide algebro-geometric audience, and will contain a review of necessary background from Hodge theory.

Tony Pantev (University of Pennsylvania)
October 2, 2015

Symplectic geometry, foliations, and potentials

I will explain how Lagrangian foliations in derived algebraic symplectic geometry give rise to global potentials. I will also give natural constructions of isotropic foliations on moduli
spaces and will discuss the associated potentials. I will give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.  This talk is intended for a general algebro-geometric audience, and no advanced background will be assumed.

Alex Perry (Harvard)
October 30, 2015

Categorical joins

Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of
classical projective duality. I will describe a categorical version of the classical join of two projective varieties, and its relation to homological projective duality. I will discuss some applications to the structure of the derived categories of Fano varieties and to derived equivalences of Calabi-Yau varieties. This is work in progress with Alexander Kuznetsov.

Zijun Zhou (Columbia)
November 13, 2015

Relative orbifold Donaldson-Thomas theory and local gerby curves

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.

Vikraman Balaji (Chennai Mathematical Institute)
November 20, 2015, 2:30-3:30

Degeneration of moduli of Higgs bundles on curves

In this this talk I will discuss the construction of a degeneration of the moduli space of Higgs bundles on smooth curves, as the smooth curve $X$ degenerates to a nodal curve with a single node. As an application we get new compactifications of the Picard variety for smooth curves degenerating to irreducible nodal curves (with multiple nodes) which have analytic normal crossing singularities.

Jose Rodriguez (University of Chicago)
November 20, 2015

Numerically computing Galois groups with Bertini.m2

Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile.

Yukinobu Toda (Institute for the Physics and the Mathematics of the Universe, University of Tokyo)
December 4, 2015, 2:30-3:30 (because of the department party)

Non-commutative deformations and Donaldson-Thomas invariants

In this talk, I will show the existence of certain global non-commutative structures on the moduli spaces of stable sheaves on algebraic varieties, whose formal completion at a closed point gives the pro-representable hull of the non-commutative deformation functor of the sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. I will then introduce the generating series through integrations over Hilbert schemes of points on these NC structures. When the underlying variety is a Calabi-Yau 3-fold, and the moduli space of stable sheaves satisfy some assumptions, this generating series admits a product expansion described by generalized DT invariants. This formula explains the dimension formula of Donovan-Wemyss’s contractions algebras for floppable curves on 3-folds in terms of genus zero Gopakumar-Vafa invariants.

Aaron Landesman (Harvard)
Tuesday, January 5, 2016 in 384-H (Student Algebraic Geometry Seminar)

Interpolation of Projective Varieties

In this talk, we discuss interpolation of projective varieties through points. It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. More recently, it was shown that one can always find nonspecial curves through the expected number of general points. We consider the generalization of this question to varieties of all dimensions and explain why rational normal scrolls satisfy interpolation. Time permitting, we’ll also discuss joint work with Anand Patel on interpolation for del Pezzo surfaces and present several interesting open interpolation problems. We’ll place particular emphasis on explaining the standard techniques used to solve interpolation problems: deformation theory, specialization, degeneration, and the gale transform.

Tonghai Yang (Wisconsin)
January 15, 2016, 2:30-3:30 pm (joint with Number Theory)

Generating series of arithmetic divisors on a Shimura variety of unitary type $(n-1, 1)$

In this talk we will define a generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1), and give a rough idea how to prove that it is modular. This can be viewed as generalization of modular form of weight 2 of Heeger divisors used in Gross-Zagier formula and its generalization Yuan-Zhang-Zhang formula. If time permits, I will mention its application to a special case of Colmez conjecture and `integral’ generalization of the Gross-Zagier and Yuan-Zhang-Zhang formula. This is a joint work with Bruinier, Howard, Kudla, and Rapoport.

Sheldon Katz (UIUC)
January 15, 2016

Stable pair invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms

I explain the conjectural description of the generating function of stable pair invariants of elliptically fibered Calabi-Yau threefolds with fixed base class and variable fiber class in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base class.  This talk is based on joint work with Albrecht Klemm and Minxin Huang.

Jacob Tsimerman (Toronto)
January 22, 2016

Bounding 2-torsion in class groups

(joint with Bhargava, Shankar, Taniguchi, Thorne, and Zhao) Zhang’s conjecture asserts that for fixed positive integers $m$, $n$, the size of the m-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a handful of cases, the best known result towards this conjecture is the ”convex” bound given by the Brauer-Siegel Theorem. We make progress on this conjecture by giving a `”subconvex” bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of $n$. The proof is surprisingly elementary, and we give several applications of this result stemming from the case of cubic fields, including improved bounds on the number of $A_4$ fields, and on the number of integer points an elliptic curve can have. Along the way, we prove a surprising result on the shape of the lattice of the ring of integers of a number field. Namely, we show that such a lattice is very limited in how ‘skew’ it can be.

Vivek Shende (Berkeley)
January 29, 2016

Cluster varieties from Legendrian knots

We give a uniform geometric explanation for the existence of cluster structures on the positroid strata, character varieties, wild character varieties, etc.

The point is that such spaces can be identified as the moduli spaces of sheaves with microsupport in an appropriate Legendrian knot. In general, an exact Lagrangian with asymptotics in the knot gives rise to an algebraic torus chart on the moduli space. We construct such fillings for alternating knots, so get charts in correspondence with alternating representatives of the knot in question. Such can be enumerated by bicolored graphs, and we recover the various aspects of the bicolored graph theory — the boundary measurement map of Postnikov; the cluster structure on the positroid varieties, the analogous construction on the character varieties due to Fock and Goncharov — by computing the Floer homology between our fillings.

This talk presents joint work with David Treumann, Harold Williams, and Eric Zaslow.

Izzet Coskun (University of Illinois at Chicago)
April 15, 2016

Birational geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent developments in the birational geometry of moduli spaces of sheaves on surfaces motivated by Bridgeland stability conditions. After reviewing joint work with Arcara, Bertram, Huizenga and Woolf in the case of the projective plane, I will describe how to use Bridgeland stability conditions to construct nef divisors on moduli spaces of sheaves on surfaces. I will illustrate the theory with examples on the projective plane, the quadric surface and some general type surfaces. This is joint work with Jack Huizenga.

Zhiwei Yun (Stanford)
April 29, 2016

Intersection numbers and higher derivatives of L-functions for function fields

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under some unramifiedness assumptions. Our formula relates the self-intersection number of certain cycles on the moduli of Drinfeld Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

Shouwu Zhang (Princeton)
Thursday, May 12, 2016, 4:30 pm, 380-W

Lecture 1 of Distinguished Lecture Series: Rational points: ABC conjecture and Szpiro’s conjecture

I will talk about ABC conjecture and its relation to Szpiro’s conjecture, Milnor’s proof of Szpiro’s inequality (before it was known as Szpiro’s conjecture) and its relation to hyperbolic geometry, and relation between Szpiro’s conjecture and Landau—Siegel conjecture on zeros of L-functions.

Shouwu Zhang (Princeton)
Friday, May 13, 2016, 2:30 pm, 383-N

Lecture 2 of Distinguished Lecture Series: Torsion points and preperiodic points: Manin—Mumford’s conjecture and its dynamical analogue

I will talk about techniques used in different proofs of Manin-Mumford’s conjecture and its analogue in dynamical system: p-adic rigid geometry (Raynaud), o-minimality geometry (Pila—Zannier), Arakelov geometry (Ullmo—Zhang), and perfectoid geometry (Xie).

Shouwu Zhang (Princeton)
Monday, May 16, 2016, 4 pm, 384-H

Lecture 1 of Distinguished Lecture Series: CM points and derivatives of L-dunctions: Andre—Oort’s conjecture and Colmez’ conjecture

I will talk about recent work of Tsimerman about reducing Andre—Oort’s conjecture to an averaged version of Colmez’ conjecture, and some related work on derivatives of L-functions by Zhiwei Yun and Wei Zhang using Drinfeld’s moduli of Shtukas, and by Xinyi Yuan using Shimura curves.

Ronen Mukamel (Rice)
May 20, 2016 (joint with Informal Geometry and Topology Seminar)

Equations for real multiplication in genus two and applications to Teichmuller curves

I will describe methods for computing with genus two curves whose Jacobians admit real multiplication and discuss applications of these methods to Teichmuller curves. This is joint work with Abhinav Kumar.

Colleen Robles (Duke)
May 27, 2016

Degenerations of Hodge structures

Two motivating questions from algebraic geometry are: how can a smooth projective variety degenerate? and what are the “relations” between two such degenerations? One way to gain insight into these questions is to ask the analogous questions of invariants associated with the smooth projective varieties. In this case, the invariant that we have in mind is a polarized Hodge structure. (And indeed detailed analysis of degenerations of polarized Hodge structures can be used to better understand degeneration of smooth projective varieties, and moduli spaces and their compactifications.)

I will explain how the work of Cattani, Kaplan and Schmid allows us to view a polarized limiting mixed Hodge structure (PLMHS) as a degeneration of a polarized Hodge structure. There is a notion of “polarized relation” between PLMHS that encodes information on how varieties may degenerate within a family. I will give a classification of PLMHS and their polarized relations in terms of Hodge diamonds (discrete data associated with a PLMHS), effectively answering the Hodge-theoretic analogs of the two motivating questions above. This is joint work with Matt Kerr.