# Difference between revisions of "Algebraic Geometry Seminar Spring 2015"

The seminar meets on Fridays at 2:25 pm in Van Vleck B135.

The schedule for the previous semester is here.

## Algebraic Geometry Mailing List

• Please join the Algebraic Geometry Mailing list to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## Fall 2014 Schedule

date speaker title host(s)
January 30 Manuel Gonzalez Villa (Wisconsin) Motivic infinite cyclic covers
February 20 Jordan Ellenberg (Wisconsin) Furstenberg sets and Furstenberg schemes over finite fields I invited myself
February 27 Botong Wang (Notre Dame) TBD Max
March 6 Matt Satriano (Johns Hopkins) TBD Max
March 13 Jose Rodriguez (Notre Dame) TBD Daniel
March 27 Joerg Schuermann (Muenster) Chern classes and transversality for singular spaces Max

## Abstracts

### Manuel Gonzalez Villa

Motivic infinite cyclic covers (joint work with Anatoly Libgober and Laurentiu Maxim)

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) an element in the Grothendieck ring, which we call motivic infinite cyclic cover, and show its birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.

### Jordan Ellenberg

Furstenberg sets and Furstenberg schemes over finite fields (joint work with Daniel Erman)

We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer.

TBA

### Joerg Scuermann

Chern classes and transversality for singular spaces

Let $X$ and $Y$ be closed complex subvarieties in an ambient complex manifold $M$. We will explain the intersection formula $$c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y)$$ for suitable notions of Chern classes and transversality for singular spaces. If $X$ and $Y$ intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If $X$ and $Y$ are "splayed" in the sense of Aluffi-Faber, then this formula holds for the Fulton-(Johnson-) Chern classes, and is conjectured for the MacPherson Chern classes. We explain, that the version for the MacPherson Chern classes is true under a micro-local "non-characteristic" condition for the diagonal embedding of $M$ with respect to $X\times Y$. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.