Department of

# Mathematics

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

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2014          February 2014            March 2014
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4                      1                      1
5  6  7  8  9 10 11    2  3  4  5  6  7  8    2  3  4  5  6  7  8
12 13 14 15 16 17 18    9 10 11 12 13 14 15    9 10 11 12 13 14 15
19 20 21 22 23 24 25   16 17 18 19 20 21 22   16 17 18 19 20 21 22
26 27 28 29 30 31      23 24 25 26 27 28      23 24 25 26 27 28 29
30 31


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