Department of

Mathematics


Seminar Calendar
for events the day of Friday, October 24, 2014.

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Friday, October 24, 2014

10:00 am in AH 447,Friday, October 24, 2014

CANCELLED

Xiuxiong Chen (SUNY Stony Brook)

4:00 pm in 343 Altgeld Hall,Friday, October 24, 2014

Cohomology classes of rank varieties and a counterexample to a conjecture of Liu

Brendan Pawlowski   [email] (University of Minnesota)

Abstract: Given any diagram (a finite collection of boxes on a grid), one can define an associated symmetric function. In many cases, these symmetric functions contain interesting and nontrivial information related to the diagram: for Young diagrams one obtains Schur functions; for skew diagrams, skew Schur functions; for permutation diagrams, Stanley symmetric functions, which describe reduced words. Liu defined a collection of subvarieties of the Grassmannian indexed by diagrams, and conjectured that their cohomology classes are represented by the corresponding diagram symmetric functions. I will give a counterexample to Liu's conjecture, along with results limiting how badly it can fail in the case of permutation diagrams. I will also discuss a connection to rank varieties (a special case of Knutson-Lam-Speyer's positroid varieties), and some new results on their cohomology classes.

4:00 pm in 243 Altgeld Hall,Friday, October 24, 2014

A brief introduction to Hodge theory

Shiyu Shen (UIUC Math)

Abstract: Hodge's theorem asserts that on a compact Riemannian manifold, the space of harmonic q-forms is isomorphic to the q-dimension cohomology. The application of this principle to projective complex variety (smooth) shows that its cohomology can be decomposed into direct sum of classes represented by (p,q) forms. We will talk about some consequences of this theorem, calculate some examples, and talk briefly about how to put a generalization of this structure, called the mixed Hodge structure, onto singular varieties.

4:00 pm in 345 Altgeld Hall,Friday, October 24, 2014

Borel cardinalities of bounded invariant equivalence relations

Krzysztof Krupinski (University of Wroclaw)

Abstract: Lascar strong types play an important role in model theory. The relation of having the same Lascar strong type is the finest bounded, invariant equivalence relation on a given sort (or product of sorts) of a monster model of a given theory. For a bounded, type-definable equivalence relation, its set of classes equipped with the so-called logic topology forms a compact Hausdorff topological space. However, for relations which are only invariant but not type-definable, the logic topology is not necessarily Hausdorff, so it is not so useful. The question arises how to measure the complexity of the "spaces'' of classes of such relations. One of the ideas is to investigate Borel cardinalities of such relations, which was formalized in my joint paper with A. Pilllay and S. Solecki. During the talk, I will give an overview of the progress which has been made so far in this direction.