abstracts for 2018-19 seminars

(The seminar webpage is here.)

June Huh (Institute for Advanced Study)

October 26, 2018

Distinguished Lecture 2 of 3: Kazhdan-Lusztig theory for matroids

There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration), based on their combinatorial cohomology theories. After giving an overview of the similarity, I will report on a cohomological approach to some conjectures in enumerative combinatorics. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.

David Jensen (Kentucky)

November 2, 2018, 3:15-4:15 pm

Linear Systems on General Curves of Fixed Gonality

The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Sam Payne (UT Austin and MSRI)

November 2, 2018, 4:30-5:30 pm

Tropical methods for the Strong Maximal Rank Conjecture

I will present joint work with Dave Jensen using tropical methods on a chain of loops to prove new cases of the Strong Maximal Rank Conjecture of Aprodu and Farkas that are relevant to computing the Kodaira dimensions of the moduli spaces M_22 and M_23.  As time permits, I will also discuss relations to an analogous approach via limit linear series on chains of genus 1 curves, developed in the work of Liu, Osserman, Teixidor, and Zhang.

Margaret Bilu (Courant)

April 19, 2019, 2:30-3:30 pm

Motivic Euler products and motivic height zeta functions

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin’s conjecture and on its motivic analog: the latter predicts the behavior of moduli spaces of curves of large degree on some algebraic varieties. It may be formulated in terms of the generating series of the classes of these moduli spaces in the Grothendieck ring, called the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition and how this can be used to give a proof of the motivic version of Manin’s conjecture for equivariant compactifications of vector groups.

Ronno Das (University of Chicago)

April 19, 2019, 4-5 pm

Points and lines on cubic surfaces

The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a ‘universal family’ of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I’ll also explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the average smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

Chenyang Xu

Thursday, May 23, 2019

Department Colloquium: Compact moduli spaces of varieties

Moduli space is ubiquitous in modern algebraic geometry. In this talk, I will discuss the recent progress on the construction of compact moduli spaces parametrising varieties whose Chern classes are positive. Ideas from differential geometry play an essential role.

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