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for events the day of Wednesday, October 22, 2014.

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Wednesday, October 22, 2014

2:00 pm in 441 Altgeld Hall,Wednesday, October 22, 2014

Introduction to Cluster Algebras and some of their geometric incarnations.

Bolor Turmunkh (UIUC Math)

Abstract: Cluster Algebras were invented by Fomin and Zelevinsky in order to better understand Lusztig's canonical basis. However, the Cluster Algebra structure turned out to be a rather ubiquitous thing in various areas of Mathematics. They appear in Algebraic Geometry, Integrable Combinatorics, Representation Theory and the theory of Teichmüller spaces to name a few. We will introduce Cluster algebras and state their important properties and results through examples. Finally, we will expose an old friend from Algebraic Geometry to have been a Cluster Algebra all along.

3:00 pm in 347 Altgeld Hall,Wednesday, October 22, 2014

Tiling a square with triangles

Aaron Abrams (Washington and Lee)

Abstract: If you want to cut a square into triangles, what restrictions are there on the areas of the triangles? It turns out that for each combinatorial type of tiling, there's a polynomial relation that must be satisfied by the areas of the triangles in any tiling of the given type. I will describe some new results about this polynomial invariant. (Joint work with J. Pommersheim)

3:00 pm in 243 Altgeld Hall,Wednesday, October 22, 2014

Geometric stability of the Coulomb energy

Almut Burchard   [email] (U of Toronto)

Abstract: I will discuss new work with Greg Chambers on the Coulomb energy in the context of recent geometric stability results (due to Christ, Figalli, Jerison and others) for functionals that describe non-local interactions. The Coulomb energy of an electrostatic charge distribution is given by the double integral of the Newton potential against the charge density. It is known that the energy of a positive charge distribution increases under symmetrization: The physical reason is that interaction energy between the charges grows as the typical distance between them shrinks. The energy increases strictly, unless the distribution is already radially decreasing about some point. Is this characterization of equality cases "stable"? In other words, must near-maximizers be close to radially decreasing? Greg and I answer this question for charge distributions that are uniform on a set of finite positive volume. Specifically, we bound the difference of a set from a suitably translated ball in terms of the difference in Coulomb energy. Time permitting, I will sketch the proof and mention some open problems.

4:00 pm in 245 Altgeld Hall,Wednesday, October 22, 2014

Non-commutative tori

Florin Boca (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: This talk will introduce non-commutative tori (an intensively studied class of C*-algebras), describe some of their main features, and discuss some open problems.