Department of

Mathematics


Seminar Calendar
for events the day of Wednesday, September 4, 2013.

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Wednesday, September 4, 2013

3:00 pm in 143 Altgeld Hall,Wednesday, September 4, 2013

Integrability of generalized pentagram maps and cluster algebra

Michael Shapiro (Michigan State University)

Abstract: The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras.

We extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems. Our main tool is Poisson geometry of weighted directed networks on surfaces.. The ingredients necessary for complete integrability -- invariant Poisson brackets, integrals of motion in involution, Lax representation -- are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter $k>1$. The case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. If $k=2$ and the ground field is $\mathbb C$, we give a geometric interpretation in terms of circle patterns.

(joint with M.Gektman, S.Tabachnikov, and A.Vainshtein)

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

Representations via geometry

Thomas Nevins (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Representations of groups (and Lie algebras) naturally emerge from studying physical systems with symmetry. I will explain one path by which an algebraic geometer approaches the study of representations via the example of the group of 2x2 matrices of determinant one. I hope to briefly indicate connections not only to algebraic geometry and mathematical physics but also to symplectic geo