Geometry, Combinatorics, and Integrable Systems  

  Autumn 2018

Time: Thursdays 3-4pm
Location: MA 317


August 30  
Thurs, 3pm 
Rachel Karpman 
Triangulations and Soliton Graphs for the totally positive Grassmannian
September 6  
Thurs, 3pm 
Yuji Kodama 
On the rational solutions and the solitons of the KP hierarchy
September 13  
Thurs, 3pm 
McCabe Olsen 
Ehrhart Theory and Lecture Hall Partitions
October 11  
Thurs, 3pm 
(fall break) 
October 18  
Thurs, 3pm 
Chris Eur 
Divisors on matroids and their volumes
October 25  
Thurs, 3pm 
Takeshi Ikeda 
K-theory Schubert calculus of the maximal orthogonal Grassmannian and set-valued decomposition tableaux
November 1  
Thurs, 3pm 
Melissa Sherman-Bennett 
Combinatorics of cluster structures in Schubert varieties


(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.

(Eur): The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, give a complete combinatorial formula, and show that it is a valuation under matroid polytope subdivisions. For a realizable matroid, we thus obtain an explicit formula for the classical volume polynomial of the associated wonderful compactification; in particular, we obtain another formula for volumes of generalized permutohedra. We then introduce a new invariant called the volume of a matroid as a particular specialization of its volume polynomial, and discuss its algebro-geometric and combinatorial properties in connection to graded linear series on blow-ups of projective spaces.

Notes are available here.

(Ikeda): We formulate a new combinatorial rule to express the Schubert structure constants for K-theory of maximal orthogonal Grassmannian. Our approach is to use the combinatorial notion of set-valued decomposition tableaux, SVDT for short, whose non-set-valued version was introduced by Serrano. The rule in general is still a conjecture apparently different from the ones previously given by Clifford-Thomas-Yong '14, and Pechenik-Yong '16. We proved a special case called the Pieri-rule which was first proved by Buch-Ravikumar '12. Our arguments comes from previous works on K-theory version of Schur P-functions. I will discuss related open problems in combinatorics of SVDTs. This talk is based on joint work with S. Cho and M. Nakasuji.

(Sherman-Bennett): The (affine cone over the) Grassmannian is a prototypical example of a variety with "cluster structure"; that is, its coordinate ring is a cluster algebra. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs. I will discuss recent work with K. Serhiyenko and L. Williams proving this conjecture. The proof uses a result of Leclerc, who shows that many Richardson varieties in the full flag variety have cluster structure using cluster-category methods, and a construction of Karpman to build plabic graphs for each Schubert variety.

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Past Seminars

Spring 2018

Autumn 2017

Spring 2017