Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, November 19, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Wednesday, November 19, 2014

2:00 pm in 441 Altgeld Hall,Wednesday, November 19, 2014

#### A Primer on Reductions and Integral Closure of Ideals

###### Matthew Mastroeni (UIUC Math)

Abstract: The aim of this talk is to give a basic introduction to reductions and the integral closure of ideals, two important tools in the study of Hilbert-Samuel multiplicity. I will explain how both topics arise naturally from the integral closure of a Rees ring $R[It]$, and then I will focus on what can be said when $R$ is Noetherian local ring with infinite residue field.

3:00 pm in 347 Altgeld Hall,Wednesday, November 19, 2014

#### Representations of complex semisimple Lie algebras via geometry

###### Tom Nevins (UIUC Math)

Abstract: Following up on my IRT talk from earlier this semester, I’ll give an introduction to the realization of representations of a complex semisimple Lie algebra g via the associated flag variety. The goal will be to explain the Beilinson-Bernstein localization theorem, that offers precise control over the realization of representations. The talk will assume a bit of background on Lie algebras but will not assume attendance at the prior talk.

4:00 pm in 245 Altgeld Hall,Wednesday, November 19, 2014

#### Some problems involving matrices

###### Timur Oikhberg (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: We discuss two families of problems involving matrices. (1) The "almost versus near" problem. Suppose a pair of matrices (A,B) almost satisfies a certain equation -- that is, p(A,B) has small norm, where p is a polynomial (for instance, we can take p(A,B) = AB-BA - then the pair (A,B) almost commutes). Can we approximate (A,B) by a pair (X,Y) so that p(X,Y) = 0? (2) The preserver problem. Suppose a map T on the space of n by n matrices preserves a certain property (such as invertibility, or rank). What can we deduce about the general form of T? Time permitting, other open questions will be mentioned.