Algebraic Geometry Seminar Spring 2016

The seminar meets on Fridays at 2:25 pm in Van Vleck B113.

The schedule for the previous semester is here.

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Spring 2016 Schedule

date	speaker	title	host(s)
January 22	Tim Ryan (UIC)	Moduli Spaces of Sheaves on \PP^1 \times \PP^1	Daniel
January 29	Jay Yang (Wisconsin)	Random Toric Surfaces	Local
February 5	Botong Wang (Wisconsin)	Topological Methods in Algebraic Statistics	Local
February 12	Jay Yang (Wisconsin)	Random Toric Surfaces	Local
February 19	Daniel Erman (Wisconsin)	Supernatural Analogues of Beilinson Monads	Local
February 26	TBD
March 4	Claudiu Raicu (Notre Dame)	Characters of equivariant D-modules on spaces of matrices	Steven
March 11	Eric Ramos (Wisconsin)	Local Cohomology of FI-modules	Local
March 18	Roman Fedorov (Kansas State)	Motivic classes of moduli spaces of vector bundles with connections	Dima
March 25	Vasily Dolgushev (Temple)	The Intricate Maze of Graph Complexes	Andrei
April 1
April 8	TBD
April 15	DJ Bruce (Wisconsin)	Noether Normalization in Families	Local
April 22	No seminar
April 29	David Anderson (Ohio State)	Old and new formulas for degeneracy loci	Steven
May 6	Dima Arinkin (Wisconsin)	Geometric approach to linear ODEs	Local

Abstracts

Tim Ryan

Moduli Spaces of Sheaves on \PP^1 \times \PP^1

In this talk, after reviewing the basic properties of moduli spaces of sheaves on P^1 x P^1, I will show that they are $\mathbb{Q}$-factorial Mori Dream Spaces and explain a method for computing their effective cones. My method is based on the generalized Beilinson spectral sequence, Bridgeland stability and moduli spaces of Kronecker modules.

Botong Wang

Topological Methods in Algebraic Statistics

In this talk, I will give a survey on the relation between maximum likelihood degree of an algebraic variety and it Euler characteristics. Maximam likelihood degree is an important constant in algebraic statistics, which measures the complexity of maximum likelihood estimation. For a smooth very affine variety, June Huh showed that, up to a sign, its maximum likelihood degree is equal to its Euler characteristics. I will present a generalization of Huh's result to singular varieties, using Kashiwara's index theorem. I will also talk about how to compute the maximum likelihood degree of rank 2 matrices as an application.

Daniel Erman

Supernatural analogues of Beilinson monads

First I will discuss Beilinson's resolution of the diagonal and some of the applications of that construction including the notion of a Beilinson monad. Then I will discuss new work, joint with Steven Sam, where we use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the diagonal. I will discuss some applications of these new constructions.

Claudiu Raicu

Characters of equivariant D-modules on spaces of matrices

I will explain how to compute the characters of the GL-equivariant D-modules on a complex vector space of matrices (general, symmetric, or skew-symmetric), and describe some applications to calculations of local cohomology and Bernstein-Sato polynomials.

Eric Ramos

The Local Cohomology of FI-modules

Much of the work in homological invariants of FI-modules has been concerned with properties of certain right exact functors. One reason for this is that the category of finitely generated FI-modules over a Noetherian ring very rarely has sufficiently many injectives. In this talk we consider the (left exact) torsion functor on the category of finitely generated FI-modules, and show that its derived functors exist. Properties of these derived functors, which we call the local cohomology functors, can be used in reproving well known theorems relating to the depth, regularity, and stable range of a module. We will also see that various facts from the local cohomology of modules over a polynomial ring have analogs in our context. This is joint work with Liping Li.

Roman Fedorov

Motivic classes of moduli spaces of vector bundles with connections

For an Artin stack of finite type, one can define its motivic class in a certain localization of the K-ring of varieties. We calculate the motivic class of a moduli stack of vector bundles with connections on a smooth projective curve. We also discuss generalizations to parabolic bundles with singular connections. This is a joint project with Alexander Soibelman and Yan Soibelman.

Vasily Dolgushev

The Intricate Maze of Graph Complexes

In the paper “Formal noncommutative symplectic geometry”, Maxim Kontsevich introduced three versions of cochain complexes GCCom, GCLie and GCAs “assembled from” graphs with some additional structures. The graph complex GCCom (resp. GCLie, GCAs) is related to the operad Com (resp. Lie, As) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes GCCom, GCLie and GCAs (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces. In my talk, I will describe the links between these graph complexes (and their modifications) to the cohomology of the moduli spaces of curves, the group of outer automorphisms Out(Fr) of the free group Fr on r generators, the absolute Galois group Gal(Qbar/Q) of rationals, finite type invariants of tangles, and the homotopy groups of embedding spaces.

DJ Bruce

Noether Normalization in Families Classically, Noether normalization says that any projective (resp. affine) variety of dimension n over a field admits a finite surjective morphism to P^n (resp. A^n). I will discuss whether we can generalize such theorems to other bases like Z, C[t], etc. This is based on joint work with Daniel Erman appearing in [1].

David Anderson

Old and new formulas for degeneracy loci

A very old problem asks for the degree of a variety defined by rank conditions on matrices. The story of the modern approach begins in the 1970’s, when Kempf and Laksov proved that the degeneracy locus for a map of vector bundles is given by a certain determinant in their Chern classes. Since then, many variations have been studied — for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. I will describe recent extensions of these results — beyond determinants and Pfaffians, and beyond ordinary cohomology — including my joint work with W. Fulton, as well as work of several others.

Dima Arinkin

Geometric approach to linear ODEs

There is a classical correspondence between systems of n linear ordinary differential equations (ODEs) of order one and linear ODEs of order n. (The correspondence may be viewed as a kind of canonical normal form for systems of ODEs.) The correspondence can be restated geometrically: given a Riemann surface C, a vector bundle E on C, and a connection [math]\displaystyle{ \nabla }[/math] on E, it is possible to find a rational basis of E such that [math]\displaystyle{ \nabla }[/math] is in the canonical normal form.

All of the above objects have a version for arbitrary semisimple Lie group G (with the case of systems of ODEs corresponding to G=GL(n)): we can consider differential operators whose `matrices' are in the Lie algebra of G, and then try to `change the basis' so that the `matrix' is in the `canonical normal form'. However, the statement turns out to be significantly harder. In my talk, I will show how the geometric approach can be used to prove the claim for any G.

The talk is based on my paper Irreducible connections admit generic oper structures.