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.
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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 | |||
| March 6 | Matt Satriano (Johns Hopkins) | TBD | Max |
| March 13 | Jose Rodriguez (Notre Dame) | TBD | Daniel |
| March 17 | Dima Arinkin (Wisconsin) | Smooth categorical representations of reductive groups | |
| March 27 | Joerg Schuermann (Muenster) | Chern classes and transversality for singular spaces | Max |
| April 10 | Byeongho Lee (Purdue) | TBA | Andrei |
| April 17 | Lee McEwan (OSU, Mannsfield) | TBA | Max and Gonzalez Villa |
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.
Jose Rodriguez
TBA
Dima Arinkin
Smooth categorical representations of reductive groups
Let G be a complex reductive group. Consider categories equipped with smooth (sometimes called strong) action of G. Natural and important examples of such categories arise from geometry: if X is a variety equipped with an action of G, for instance, X=G itself, or X=G/B is the flag space of G, then the category of D-modules on X carries a smooth action of G. We view such categories as (smooth) categorical representations of G.
The theory of smooth categorical representations of G is similar to the representation theory of a reductive group over a finite field, I will discuss this similarity in my talk. The surprising twist (and the main result of this talk) is that the theory of smooth categorical representations is simpler than its classical counterpart: there are no cuspidal representations!
Joerg Schuermann
Chern classes and transversality for singular spaces
Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be closed complex subvarieties in an ambient complex manifold [math]\displaystyle{ M }[/math]. We will explain the intersection formula [math]\displaystyle{ c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y) }[/math] for suitable notions of Chern classes and transversality for singular spaces. If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] 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 [math]\displaystyle{ M }[/math] with respect to [math]\displaystyle{ X\times Y }[/math]. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.