Department of

Mathematics


Seminar Calendar
for events the day of Wednesday, December 3, 2014.

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Wednesday, December 3, 2014

Yangians, Quantum Loop Algebras and Elliptic Quantum Groups

Sachin Gautam   [email] (Department of Mathematics, Columbia University)

Abstract: This talk will focus on three affine quantum groups associated to a simple Lie algebra, namely the Yangian, quantum loop algebra and elliptic quantum group. These correspond respectively to rational, trigonometric and elliptic solutions of the Yang–Baxter equation, and arise in their simplest guises as symmetries of Heisenberg XXX, XXZ and XYZ lattice models. In previous work, we showed how to construct representations of the quantum loop algebra from those of the Yangian, and vice versa, thus settling the long standing problem of relating their represen- tations. I will explain this construction and its extension to the case of elliptic quantum groups. This talk is based on a joint work with V. Toledano Laredo.

4:00 pm in 245 Altgeld Hall,Wednesday, December 3, 2014

Differentiation in metric measure spaces

Jeremy Tyson (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: This talk will be an introduction to (first-order) analysis on metric measure spaces. The traditional definition of the derivative as a linear operator acting between tangent spaces presupposes that both source and target are equipped with smooth structure. I will reformulate the definition of the derivative in a way which naturally makes sense for metric source spaces. This leads to the concept of a 'differentiable structure' on a metric measure space. 'Differentiable' here refers only to the first derivative; at present we do not have a clear understanding of how to make sense of higher-order derivatives in metric spaces. Smooth manifolds provide the canonical example of spaces with differentiable structure. I will briefly describe several examples of such structures on nonsmooth spaces. Which spaces support differentiable structures? An answer to this question was given by Jeff Cheeger (GAFA, 1999): the Cheeger-Rademacher differentiation theorem is one of the cornerstones of modern analysis in metric measure spaces. I will conclude with a brief description of this theorem and its significance.