Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, September 18, 2013.

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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, September 18, 2013

3:00 pm in 143 Altgeld Hall,Wednesday, September 18, 2013

#### $sl_2$ Whittaker functions and their relations through intertwiners

###### Bolor Turmunkh (UIUC Math)

Abstract: We've introduced the Whittaker vectors and Whittaker functions of $sl_2$ and showed that Whittaker functions are the eigenfunctions of the quantum open Toda Hamiltonian. We will pick up where we left off and prove further relations of Whittaker functions using a specific intertwiner of $sl_2$-modules. This is an explicit example that demonstrates the power of intertwiners and their use in the theory of special functions. This talk does not require representation theory background and should be accessible to everyone.

4:00 pm in 245 Altgeld Hall,Wednesday, September 18, 2013

#### Planar Maps: geometry, combinatorics and integrability

###### Philippe Di Francesco (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: We discuss the enumeration of planar maps, i.e. graphs made of vertices, edges and faces embedded in the plane. Maps appear in various contexts: in physics as toy models of quantum gravity; in algebraic geometry as cell decompositions of moduli spaces of curves; in probability theory as discrete formulations of Brownian surfaces; and as combinatorial objects per se. We show how the enumeration can be performed by cutting the maps into trees. The method allows to keep track of geodesic distances between marked points on the maps, and leads to discrete integrable equations, that can be solved by means of discrete solitons.