Geometry, Combinatorics, and Integrable Systems  

  Spring 2018

Time: Thursdays 3-4pm
Location: MA 317


January 11 
Thurs, 3pm 
Dan Corey 
Initial degenerations of Grassmannians
January 18 
Thurs, 3pm 
Aurelien Sagnier 
(Ecole Polytechnique) 
An arithmetic site of Connes-Consani type for Gaussian integers
February 1 
Thurs, 3pm 
Anna Weigandt 
Prism Tableaux and Alternating Sign Matrices
February 15 
Thurs, 3pm 
Chris Fraser 
Webs, dimers, and total positivity
March 15  
Thurs, 3pm 
(Spring Break) 
March 29 
Thurs, 3pm 
Martha Precup 
The cohomology of abelian Hessenberg varieties
April 26  
Thurs, 3pm 
Kaie Kubjas 
Geometry and maximum likelihood estimation of the binary latent class model


(Corey): Let Gr0(d,n) denote the open subvariety of the Grassmannian Gr(d,n) consisting of d-1 dimensional subspaces of Pn-1 meeting the toric boundary transversely. We prove that Gr0(d,n) is schoen in the sense that all of its initial degenerations are smooth. The main technique we will use is to express the initial degenerations of Gr0(3,7) as a inverse limit of thin Schubert cells. We use this to show that the Chow quotient of Gr(d,n) by the maximal torus H in GL(n) is the log canonical compactification of the moduli space of 7 lines in P2 in linear general position.

(Sagnier): We are used to seeing integers with the usual structure of an ordered ring. A.Connes and C.Consani proposed in 2014 to look at them with another structure which is an idempotent semiring with an action of the positive integers by multiplication. With the eyes of algebraic geometry, it is a semiringed topos whose points are linked with Riemann zeta function. The hope in the long term is that this new framework coming from algebraic geometry could help translating ideas of the demonstration of the analog of the Riemann hypothesis for zeta functions associated to smooth projective curves over a finite field to the actual Riemann zeta function. I will explain A.Connes' and C.Consani's point of view in the first part of my talk. However, this point of view heavily relies on the natural order on R compatible with addition and multiplication so one may wonder if, for Gaussian integers, where nothing of this sort exists, one can adapt the ideas and the methods of A.Connes and C.Consani. This is what I have done in my Ph.D. thesis and what I will explain in the second part of my talk.

(Weigandt): A prism tableau is an overlay of semistandard tableaux. In joint work with A. Yong, prism tableaux were used to provide a formula for Schubert polynomials. This expression directly generalizes the tableau rule for Schur polynomials. We will discuss fillings of more general prism shapes. The resulting polynomials are multiplicity free sums of Schubert polynomials. The terms in this sum are determined by an associated alternating sign matrix. We use this connection to give a prism formula for the multidegrees of alternating sign matrix varieties.

(Fraser): I will formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. This map sends a network to a linear combination of SLr tensor invariants known as webs, and is defined using the r-fold dimer model on the network. When r equals 1, our map is Postnikov's boundary measurement used to coordinatize positroid strata. The main result is that the higher rank map factors through Postnikov's map. As a corollary, we deduce a complete set of diagrammatic relations between SLr webs, reproving a result of Cautis-Kamnitzer-Morrison in the classical (q=1) setting. We establish compatibility between our map and restriction to positroid strata. This is joint with Thomas Lam and Ian Le.

(Precup): Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. These varieties have gained recent attention due to a conjecture of Shareshian and Wachs relating the chromatic quasisymmetric function of the incomparability graph of a unit interval order to the dot action representation on the cohomology of an associated regular semisimple Hessenberg variety. In this talk, I will report on recent joint work with M. Harada in which we prove an inductive formula for the Betti numbers of certain regular Hessenberg varieties called abelian Hessenberg varieties. Using a theorem of Brosnan and Chow, this formula yields an inductive description of the dot action representation.

(Kubjas): The binary latent class model consists of binary tensors of nonnegative rank at most two inside the standard simplex. We characterize its boundary stratification with the goal to use this stratification for exact maximum likelihood estimation in statistics. We explain two different approaches for deriving the boundary stratification: by studying the geometry of the model and by using the fixed points of the Expectation-Maximization algorithm. In the case of 2x2x2 tensors, we obtain closed formulas for the maximum likelihood estimates. This talk is based on the joint work with Elizabeth Allman, Hector Banos Cervantes, Robin Evans, Serkan Hosten, Daniel Lemke, John Rhodes and Piotr Zwiernik.

Past Seminars

Autumn 2017

Spring 2017