Algebra and Algebraic Geometry Seminar Spring 2019

The seminar meets on Fridays at 2:25 pm in room B235.

Here is the schedule for the previous semester, for the next semester, and for this semester.

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Spring 2019 Schedule

date	speaker	title	host(s)
January 25	Daniel Smolkin (Utah)	Symbolic Powers in Rings of Positive Characteristic	Daniel
February 1	Juliette Bruce	Asymptotic Syzgies for Products of Projective Spaces	Local
February 8 (B135)	Isabel Vogt (MIT)	Low degree points on curves	Wanlin and Juliette
February 15	Pavlo Pylyavskyy (U. Minn)	Zamolodchikov periodicity and integrability	Paul Terwilliger
February 22	Michael Brown (Wisconsin)	Chern-Weil theory for matrix factorizations	Local
March 1	Chris Eur (UC Berkeley)	Chow rings of matroids, ring of matroid quotients, and beyond	Daniel
March 8	Jay Kopper (UIC)	Stable restrictions of vector bundles on projective varieties	Daniel
March 15	Singularities	No regular meeting	Max
March 22	No Meeting	Spring Break	TBD
March 29		TBD	Daniel
April 5 (at 11:00 in B329)	Eric Canton (Michigan)	Log canonical thresholds, Kahler seminorms, and normalized volume	Michael
April 5	Alexsandra Sobieska (Texas A&M)	TBD	Daniel
April 12	Botong Wang	Lyubeznik numbers of irreducible projective varieties	local
April 19	Eloísa Grifo (Michigan)	Symbolic Powers and the (Stable) Containment Problem	Daniel
April 26	TBD	TBD	TBD
May 3	Hannah Larson (Stanford)	Vector bundles on P^1 bundles	Melanie

Abstracts

Daniel Smolkin

Symbolic Powers in Rings of Positive Characteristic

The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!

Juliette Bruce

Title: Asymptotic Syzygies for Products of Projective Spaces

I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.

Isabel Vogt

Title: Low degree points on curves

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.

Pavlo Pylyavskyy

Zamolodchikov periodicity and integrability

T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin.

Michael Brown

Chern-Weil theory for matrix factorizations

This is joint work with Mark Walker. Classical algebraic Chern-Weil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. In this talk, I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical Chern-Weil theory.

Chris Eur

Chow rings of matroids, ring of matroid quotients, and beyond

We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Connor Simpson and Spencer Backman.

Jay Kopper

Stable restrictions of vector bundles on projective varieties

Stable vector bundles---and more generally, stable sheaves---play a role in the classification of algebraic vector bundles analogous to that of simple groups in group theory. Recent developments in this subject have extended the notion of stability to the entire derived category of sheaves. This broader perspective can be used to study the classical moduli space. In this talk I will discuss these ideas in the context of restriction theorems: situations in which a stable vector bundle remains stable when restricted to a subvariety. I will conclude with some applications to higher-rank Brill-Noether theory. This is joint work with S. Feyzbakhsh.

Shamgar Gurevich

Harmonic Analysis on GLn over finite fields, and Random Walks

There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:

$$ trace(\rho(g))/dim(\rho), $$

for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).

Eric Canton

Log canonical thresholds, Kahler seminorms, and normalized volume

The log canonical threshold of a closed subscheme Y of an algebraic variety X gives some real number that measures the singularities of Y. This is, in turn, defined in terms of the "amount" of a given divisor that must be inserted to make X\Y a smooth variety relatively compact (i.e. proper) over X; this "amount" goes by the name of the log discrepancy of that divisor on Y. Already, the study of log discrepancies is subtle when X is a complex variety, but without the guarantee of a smooth compactification in positive characteristics, effective results can seem remote. In this talk, I present an approach to effective results in positive characteristics from my thesis. This approach is described in terms of the Berkovich analytic space associated to X, realizing the log discrepancy as a natural seminorm to put on the sheaf of Kahler differentials of X, when X is normal. I'll finish by discussing new directions related to K-stability.

Botong Wang

Lyubeznik numbers of irreducible projective varieties

Lyubeznik numbers are invariants of singularities that are defined algebraically, but has topological interpretations. In positive characteristics, it is a theorem of Wenliang Zhang that the Lyubeznik numbers of the cone of a projective variety do not depend on the choice of the projective embedding. Recently, Thomas Reichelt, Morihiko Saito and Uli Walther constructed examples of reducible complex projective varieties whose Lyubeznik numbers depend on the choice of projective embeddings. I will discuss their works and a generalization to irreducible projective varieties.

Eloísa Grifo

Symbolic powers and the (stable) containment problem

Given a variety X in C^d, corresponding to an ideal I, which polynomials vanish up to order n along X? The polynomials in the n-th symbolic power of I, which are often not the same as the polynomials in the n-th ordinary power of I.

In trying to compare symbolic and ordinary powers, Harbourne conjectured that a famous containment by Ein--Lazersfeld--Smith, Hochster--Huneke and Ma--Schwede could be improved. Harbourne's Conjecture is a statement depending on n that unfortunately has been disproved for particular values of n. However, recent evidence points towards a stable version of Harbourne's conjecture, where we substitute all n by all n large enough. Some of that evidence is joint work with Craig Huneke and Vivek Mukundan.

Hannah Larson

Vector bundles on P^1 bundles

Every vector bundle on P^1 splits as a direct sum of line bundles. Given a vector bundle E on a P^1 bundle PW --> B, the base B is stratified by subvarieties defined by the condition that the restriction of E to the fibers has a certain splitting type. It is natural to ask for the classes of the closures of these strata in the Chow ring of B. We answer this question through a study of the moduli stack of vector bundles on P^1 bundles. In describing this moduli space, we also discover an algebraic version of Bott periodicity. This is joint work with Ravi Vakil.