ALGEBRAIC GEOMETRY SEMINAR

Time: Wednesday, 3:25-4:25 pm

Location: MATH 731

Speakers/Abstracts:

January 17th: Isabel Leal (University of Chicago)

Title: Generalized Hasse-Herbrand $\psi$-functions.

Abstract: The classical Hasse-Herbrand $\psi$-function is an important object in ramification theory, related to higher ramification groups. In this talk, I will discuss generalizations of the  Hasse-Herbrand function and go over some of their properties. These generalized $\psi$-functions are defined for extensions $L/K$ of complete discrete valuation fields where the residue field $k$ of $K$ is perfect of characteristic $p>0$ but the residue field $l$ of $L$ is possibly imperfect.

January 24th: Uli Walther (Purdue University)

Title: GKZ -Systems and Mixed Hodge Modules.

Abstract: I will define GKZ-systems, and talk a little about their properties from the algebraic, analytic, and combinatorial point of view. Then I will discuss a theorem of Gelfand et al, and a sharpening by Mathias Schulze and myself, on the question which GKZ-systems arise as (D-module-)direct image of a natural D-module on a torus. In such cases, the GKZ-system inherits a mixed Hodge module structure. I will then explain work with Thomas Reichelt that computes this MHM structure on a class of GKZ-systems that comes up naturally in mirror symmetry. Very few of such explicitly computed structures seem to be known.

January 31st: Uli Walther (Purdue University)

Title: GKZ -Systems and Mixed Hodge Modules.

Abstract: Continuation of the first talk.

February 7th: Andras Lorincz (Purdue University)

Title: On Categories of equivariant D-modules.

Abstract: Let X be an algebraic variety with the action of an algebraic group G. In this talk I will discuss various results on G-equivariant coherent D-modules on X, such as descriptions of the space of global sections as G-representations, and descriptions of the categories of equivariant D-modules. When G acts on X with finitely many orbits, this category is isomorphic to the category of finite-dimensional representations of a quiver with relations. We describe explicitly these categories in some special cases when the quivers are of finite representation type (i.e. with finitely many indecomposable representations). At the end I will mention some applications to local cohomology. This is joint work with Uli Walther.

February 14th: Botong Wang (Univ. of Wisconsin, Madison)

Title: Hodge theory in the enumeration of points, lines, planes, etc.

Abstract: Given $n$ points on the plane, by connecting each pair of them, one obtains either one line or at least $n$ lines. This is an old theorem in enumerative combinatorial geometry due to de Bruijn and Erdos. In the first part of the talk, we will present a higher dimensional generalization of this theorem, which confirms a "top-heavy" conjecture of Dowling and Wilson in 1975. The key idea is to relate some combinatorial quantities to the cohomology ring of algebraic varieties and to use the hard Lefschetz theorem of intersection cohomology groups. In the second part, I will discuss some work in progress further generalizing the result to non-realizable matroids. This is joint work with Tom Braden, June Huh, Jacob Matherne and Nick Proudfoot.

February 21st: Daxin Xu (Cal. Tech)

Title: Frobenius descent for convergent isocrystals and a conjecture of Berthelot

Abstract: Let k be a field of characteristic p > 0 and W the ring of Witt vectors of k. In this talk, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over k relative to W. This proof allows us to deduce an analogue of the de Rham complexes comparaison theorem of Berthelot without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot's conjecture on the preservation of convergent F-isocrystals under the higher direct image by a smooth proper morphism of k-varieties.

February 28th: Christine Berkesch (Univ. of Minnesota)

Title: Parametric behavior of A-hypergeometric solutions.

Abstract: A-hypergeometric systems are the D-module counterparts of toric ideals, and their behavior is linked closely to the combinatorics of toric varieties. I will discuss recent work that aims to explain the behavior of the solutions of these systems as their parameters vary. In particular, we stratify the parameter space so that solutions are locally analytic within each (connected component of a) stratum. This is joint work with Jens ForsgŒrd and Laura Matusevich.

March 7th: Manoj Kummini (Chennai Math. Institute)

Title:  F-rationality of Rees algebras

Abstract: We will discuss some necessary and some sufficient conditions
for Rees rings to be F-rational. This is joint work with M. Koley.

March 21st: V. Srinivas (TIFR)

Title: On stratified vector bundles in characteristic p.

Abstract: This is a report on some work with Helene Esnault, motivated by a
conjecture of Gieseker, which was proved earlier by Esnault and Mehta.

For a smooth quasi-projective variety $X$ over $\bar{\F}_p$, with trivial
etale $\pi_1$, such that $X$ has a projective normal compactification with
codimension 2 boundary, we show that all stratified vector bundles on $X$
are trivial.

Another result of ours is the following: if a  morphism of smooth
projective varieties in char. p induces the trivial map
on Žtale fundamental groups, then the pullback of any stra
tified vector
bundle is trivial, as a stratified bundle.

March 28th: Kiumars Kaveh (Univ. of Pittsburgh)

Title: Toric vector bundles and buildings.

Abstract: A toric variety is a variety equipped with an action of algebraic torus T with an open orbit. The geometry of toric varieties is intimately connected to piecewise linear convex geometry and they are classified by collections of convex cones called fans. Klyachko famously classified (T-linearized) vector bundles on a toric variety by certain data of filtrations associated to the rays in the fan. We give an interpretation of Klyachko's data as a piecewise linear map'' and make a connection with the notion of building'' from representation theory. This leads us to a generalization of Klyachko's classification to principal G-bundles on toric varieties for semisimple groups G other than GL(n). This is joint work in progress with Chris Manon.

April 4th: No Seminar

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April 11th: No Seminar

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April 18th:
Title:
On the VC-density of definable families over valued fields
Abstract: We give tight bounds on the Vapnik-Chervonenkis density (VC-density) for definable families over any algebraically closed valued field in the language of valued fields. This result improves and generalizes prior results of Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko (Trans. of AMS, 2015).

Our result on the VC-density follow from more general results on bounding the individual Betti numbers of certain semi-algebraic subsets of Berkovich analytic spaces. The talk will (hopefully) exhibit an interesting interplay between combinatorics (Sauer-Shelah lemma), logic (model theory), topology (basic properties of sheaf cohomology), andalgebraic geometry (geometry of Berkovich spaces). (Joint work with Deepam Patel).

April 25th: Helene Esnault (FU Berlin)

Title: Rigidity and F-isocrystals.

Abstract: We explain that rigid connections yields F-overconvergent isocrystals (in the projective case) and discuss a few points concerning the p-curvature conjecture. (Joint work with Michael Groechenig)