Geometry, Combinatorics, and Integrable Systems  

  Spring 2019

Time: Thursdays 3-4pm
Location: MA 317


January 17  
Thurs, 3pm 
CH 240
Matt Baker 
(Georgia Tech)  
Hyperfields, Ordered Blueprints, and Moduli Spaces of Matroids
February 7  
Thurs, 3pm 
MW 154
Cesar Cuenca 
q-analogues of representation-theoretic measures
February 28  
Thurs, 3pm 
Ben Wormleighton 
Approaching quasi-period collapse via orbifold singularities
March 7  
Thurs, 3pm 
Rahul Singh 
The Conormal Variety of a Cominuscule Schubert Variety
March 21  
Thurs, 3pm 
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March 28  
Thurs, 3pm 
Steven Karp 
Combinatorics of the amplituhedron
April 4  
Thurs, 3pm 
Fu Liu 
(UC Davis)  
Combinatorics of nested Braid fans
April 11  
Thurs, 3pm 
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April 18  
Thurs, 3pm 
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(Baker): In tropical geometry, linear spaces are in 1-1 correspondence with "valuated matroids". I will discuss a unified theory which views linear subspaces, ordinary matroids, and valuated matroids as special cases of "matroids over hyperfields". I will then discuss the geometric construction of a "universal Grassmannian" which, via base change, yields (fine) moduli spaces for matroids over hyperfields. The universal Grassmannian is constructed using Oliver Lorscheid's theory of ordered blueprints, which is also useful for understanding the relationship between Berkovich spaces, skeletons, and tropicalizations. This is joint work with Nathan Bowler and Oliver Lorscheid.

(Cuenca): I will survey results from my joint work with Grigori Olshanski, about a remarkable family of probability measures, arising from the theory of orthogonal polynomials. The measures lead to discrete, q-analogues of continuous processes from mathematical physics (log-gas systems, eigenvalues of random matrices, etc). It is conjectured that the probability measures in question are related to the representation theory of quantum groups.

(Wormleighton): Ehrhart theory seeks to study the number of lattice points in dilates of a given polytope. Ehrhart famously proved that when P is a lattice polytope, the function counting lattice points in integer dilates of P is given by a polynomial. When one allows P to be a rational polytope, this polynomial is replaced by a 'quasi-polynomial': a polynomial L(n) with coefficients that are periodic functions of the input n. The lcm of the periods of these coefficient functions is called the 'quasi-period' of P. There is a natural upper bound for the quasi-period of a rational polytope that is often but not always equal to the quasi-period. When the quasi-period is smaller than expected, we say that P undergoes 'quasi-period collapse'. Polytopes arising naturally in algebraic and symplectic geometry, representation theory, and combinatorics all suffer quasi-period collapse yet there is no satisfying characterisation of when and why it occurs. After setting the scene for rational Ehrhart theory, I will describe recent work with Al Kasprzyk on studying quasi-period collapse in two dimensions via toric algebraic geometry, where the deformation theory of orbifold singularities allows us to provide a new characterisation of quasi-period collapse and to systematically produce interesting examples.

Slides are available here.

(Singh): Conormal varieties of Schubert varieties show up as important objects in the representation theory of reductive groups. While the combinatorics of these objects is somewhat well-understood, not much is known about their local geometry. In this talk, we will present a resolution of singularities of the conormal variety of a cominuscule Schubert variety. Further, we use this to construct a system of defining equations in types A, B, and C; we also discuss the current obstruction to extending this to type D, namely the lack of a combinatorial description of the degeneracy loci of flagged vector bundles in type D.

(Karp): The tree amplituhedron A(n,k,m) is a geometric object introduced by Arkani-Hamed and Trnka in 2013, in their study of scattering amplitudes in high-energy physics. I will introduce the amplituhedron and describe how it generalizes objects such as cyclic polytopes, cyclic hyperplane arrangements, and positive Grassmannians. I will discuss the problem of triangulating amplituhedra, which involves the combinatorics of Catalan objects and, conjecturally, plane partitions. This is joint work with Lauren Williams and Yan Zhang. I will also discuss recent work about the topology of amplituhedra and positive Grassmannians, which is joint with Pavel Galashin and Thomas Lam.

(Liu): Generalized permutohedra are defined as polytopes obtained from usual permutohedra by moving facets without passing vertices; or equivalently they are polytopes whose normal fan coarsens the Braid fan. We wanted to generalize this construction by allowing "passing some vertices". In joint work with Castillo, we consider a refinement of the Braid fan, called the nested Braid fan, and construct generalized nested permutohedra which have the nested Braid fan refining their normal fan. We extend many results on generalized permutohedra to this new family of polytopes.

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Past Seminars

Autumn 2018

Spring 2018

Autumn 2017

Spring 2017