Geometry, Combinatorics, and Integrable SystemsAutumn 2018Time: Thursdays 34pmLocation: MA 317 

August 30
Thurs, 3pm  Rachel Karpman
(OSU) 
Triangulations and Soliton Graphs for the totally positive Grassmannian 
September 6
Thurs, 3pm  Yuji Kodama
(OSU) 
On the rational solutions and the solitons of the KP hierarchy 
September 13
Thurs, 3pm  McCabe Olsen
(OSU) 
Ehrhart Theory and Lecture Hall Partitions 
September 20
Thurs, 3pm 
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September 27
Thurs, 3pm 
( ) 

October 4
Thurs, 3pm 
( ) 

October 11
Thurs, 3pm  (fall break)


October 18
Thurs, 3pm 
( ) 

October 25
Thurs, 3pm 


November 1
Thurs, 3pm  Melissa ShermanBennett
(Berkeley) 

November 8
Thurs, 3pm 


November 15
Thurs, 3pm 


November 22
Thurs, 3pm  (Thanksgiving)


November 29
Thurs, 3pm 

(Karpman): The KP equation is a nonlinear dispersive wave equation which gives an excellent model for resonant interactions of shallowwater waves. Regular soliton solutions of the KP equation may be constructed from points in the totally nonnegative Grassmannian of Nplanes in Mdimensional space. For the positive Grassmannian of 2planes in Mspace, Kodama and Williams showed that soliton graphs are in bijection with triangulations of the Mgon. We extend this result to Gr(N,M) when N = 3 and M = 6, 7, or 8. In each case, we show that soliton graphs are in bijection with Postnikov's plabic graphs, which play a key role in the combinatorial theory of the positive Grassmannian.
Slides are available here.
(Kodama): It is well known that the Schur polynomials satisfy the Hirota bilinear equations of the KP hierarchy, and that each Schur polynomial can be parametrized by a unique Young diagram. We also know that the KP solitons (exponential solutions) can be parametrized by certain decomposition of the Grassmannians. In the talk, I will explain the connection between the rational solutions and the KP solitons in terms of the Young diagrams. More explicitly, I will show how one gets a rational solution from a KP soliton. I will also discuss a connection between quasiperiodic solutions (theta or sigma functions) and the KP solitons. The rational solutions then give theta divisors of certain algebraic curves. The talk will be an elementary introduction of the Sato theory of the KP hierarchy.
(Olsen): Ehrhart theory is the study of lattice point enumeration in convex rational polyhedra. In this talk, we will provide a brief overview of some concepts, definitions, and properties of interest in Ehrhart theory. We will then discuss results on polytopes arising from lecture hall partitions, which are a particularly rich and interesting family of combinatorial objects.