Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, October 8, 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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30


Wednesday, October 8, 2014

2:00 pm in 441 Altgeld Hall,Wednesday, October 8, 2014

#### Analogues of Cyclotomic Extensions in Function Fields

###### Ravi Donepudi (UIUC Math)

Abstract: The extensions of the rational numbers obtained by adjoining a root of unity are called cyclotomic extensions. Every abelian extension of the rational numbers is contained in a cyclotomic extension. Since the field of rational functions over a finite field shares many properties with the rational numbers, it is natural to ask whether a similar phenomenon happens in this case. Carlitz (1938) found a class of function field extensions with properties strikingly similar to the cyclotomic extensions. Indeed, these extensions turn out to contain all the abelian extensions of the rational function field. In our talk we will focus on highlighting the similarities between these fields and how the geometry of the function fields often simplifies results imported from the rational numbers.

3:00 pm in 347 Altgeld Hall,Wednesday, October 8, 2014

#### A low-tech introduction to localization for representations of Lie algebras

###### Tom Nevins (Department of Mathematics, University of Illinois)

Abstract: Geometric representation theory aims to use geometry of interesting spaces to analyze representations and uncover new structures. I will give an introduction to a fundamental tool in the subject, Beilinson-Bernstein localization, that realizes representations of complex semisimple Lie algebras using flag varieties. Despite those possibly ambitious-sounding hopes, I will confine almost the entire talk to the case of sl(2), where almost everything can be done with vector calculus and linear algebra. Thus, the talk should be fairly accessible to anyone who knows those undergraduate subjects.

4:10 pm in 245 Altgeld Hall,Wednesday, October 8, 2014

#### Double Feature: Hilbert's 17th Problem and the Stern Sequence

###### Bruce Reznick (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Hilbert's 17th Problem involves real polynomials in several variables which only take non-negative values, and their representations as a sum of squares of rational functions. The Stern sequence is a simple recursively-defined integer sequence with a wide range of unexpected properties. The two subjects are related by their styles, rather than their subject area. The talk assumes no graduate prerequisite, and the speaker hopes you will leave the room having whistling the proofs.