Geometry, Combinatorics, and Integrable Systems  

  Autumn 2018

Time: Thursdays 3-4pm
Location: 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 
 
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October 4  
Thurs, 3pm 
 
( )  
October 11  
Thurs, 3pm 
(fall break) 
 
October 18  
Thurs, 3pm 
 
( )  
October 25  
Thurs, 3pm 
 
 
November 1  
Thurs, 3pm 
Melissa Sherman-Bennett 
(Berkeley)  
November 8  
Thurs, 3pm 
 
 
November 15  
Thurs, 3pm 
 
 
November 22  
Thurs, 3pm 
(Thanksgiving)  
 
November 29  
Thurs, 3pm 
 
 



Abstracts


(Karpman): The KP equation is a nonlinear dispersive wave equation which gives an excellent model for resonant interactions of shallow-water waves. Regular soliton solutions of the KP equation may be constructed from points in the totally nonnegative Grassmannian of N-planes in M-dimensional space. For the positive Grassmannian of 2-planes in M-space, Kodama and Williams showed that soliton graphs are in bijection with triangulations of the M-gon. 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 quasi-periodic 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.



(Sherman-Bennett):



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Past Seminars

Spring 2018

Autumn 2017

Spring 2017