Seminar Calendar
for events the day of Wednesday, October 1, 2008.

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Wednesday, October 1, 2008

Algebra, Geometry and Combinatorics Seminar
3:00 pm   in 345 Altgeld Hall,  Wednesday, October 1, 2008
 Del Edit Copy
Submitted by ayong.
 Philippe Di Francesco (Institut de Physique Theoretique)Q-systems, Heaps, Paths and the Cluster Positivity ConjectureAbstract: Q-systems are discrete integrable equations introduced in the context of the proof of completeness of the Bethe Ansatz for quantum spin chains. Cluster algebras are universal dynamical systems introduced initially for investigating total positivity of the Grassmannian. We consider the cluster algebra associated to the Q-system for $A_r$, as a tool for relating Q-system solutions to initial data. We show that the conserved quantities of the Q-system are partition functions for hard particles on particular target graphs, determined by the initial data. This allows to interpret the fundamental solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions that leave their final value unchanged. Finally, the remaining solutions of the Q-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices, namely paths with nearest neighbour exclusion rules. This displays all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ Q-system.

Math 499: Introduction to Graduate Mathematics
4:00 pm   in 245 Altgeld Hall,  Wednesday, October 1, 2008
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Submitted by seminar.
 Iwan Duursma (Department of Mathematics, University of Illinois)Using algebraic geometry to share a secret