UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 21, 2017

939 Evans Hall

3:45PM: Power structure over the Grothendieck ring of varieties and its applications

Sabir Gusein-Zade

A power structure over a ring is a method to give sense to an expression of the form $(1 + a_1 t + a_2 t^2 + ...)^m$ where $a_i$ and $m$ belong to the ring so that all the standard properties of the exponential function hold. One has a natural power structure over the Grothendieck ring $K_0(\mathcal{V}_{\mathbb{C}})$ of complex quasi-projective varieties with an interesting geometric description. It can be used both for formulation of some known statements in a short form and for proving new ones. As examples of applications one can give, e.g., an equation for the generating series of classes of Hilbert schemes of zero-dimensional subschemes in a smooth complex quasiprojective variety of any dimension, a Macdonald-type formula for the generating series of generalized higher order orbifold Euler characteristics (with values in the Grothendieck ring $K_0(\mathcal{V}_{\mathbb{C}})$ extended by the rational powers of the complex affine line) of the wreath products of a smooth quasiprojective variety with an action of a finite group.

5:00PM: Betti tables of graph ideals

Justin Chen

Given a graph G, one can associate many (often monomial) ideals to it, as invariants of G. This talk will focus on a few specific classes of ideals, which are less well-known than the standard examples. The commutative algebra of these ideals - in particular, the data of their free resolutions - display fascinating patterns, and have connections to chip-firing, lattice point counting and other combinatorial topics. I will sketch some of these connections for certain classes of graphs; e.g. low-genus graphs and complete graphs.

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