**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.

*Title: ***GKZ -Systems and Mixed Hodge Modules.**

**January 31st: Uli Walther (Purdue
University)**

*Title: ***GKZ -Systems and Mixed Hodge
Modules.**

**February 7th: ****Andras Lorincz (Purdue University)**

*Title: ***On Categories of equivariant D-modules.**

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

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

*Title: ***Parametric behavior of A-hypergeometric solutions.**

*Title: *F-rationality of Rees algebras

for Rees rings to be F-rational. This is joint work with M. Koley.

*Title: ***On stratified vector bundles in characteristic p.**

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 stratified vector

bundle is trivial, as a stratified bundle.

*Title: ***Toric vector bundles and buildings.**

*Title: *

*Title: *

Title:

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.**