Department of

Mathematics


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

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     January 2014          February 2014            March 2014     
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           1  2  3  4                      1                      1
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Wednesday, February 19, 2014

3:00 pm in 145 Altgeld Hall,Wednesday, February 19, 2014

The combinatorics of non-commutative discrete integrable systems

Philippe Di Francesco (UIUC)

Abstract: Discrete integrable systems are systems of recursion relations describing evolution in a discrete time variable, with a suitable number of independent conservation laws. We concentrate on the examples of $A_1$ Q- and T-systems, both part of cluster algebras, respectively of rank 2 and infinity. As such they enjoy the positive Laurent property: the solutions may be expressed in terms of the initial data as Laurent polynomials with non-negative integer coefficients. We then formulate non-commutative analogues of these systems defined on a non-commutative algebra A, and prove the non-commutative positive Laurent property for their solutions. The proof relies on the existence of a GL_2(A) flat connection on the solutions of these systems, a manifestation of their discrete non-commutative integrability. The solutions may be interpreted combinatorially as partition functions of paths on networks and/or dimers on graphs, with non-commutative weights. (based on arXiv:0909.0615 and arXiv:1402.2851)

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

Dynamics on a Network: Stability, Polytopes, and Fireflies

Jared Bronski (Department of Mathematics, University of Illinois at Urbana-Champaign)