Date 
Speaker and Institution 
Title and Abstract 
9/13 
Diana Davis (Williams College)

Lattice surfaces and the Modulus Miracle
A square has reflectional and rotational symmetries. The square torus surface has an additional symmetry, a shear. I will explain what this is good for, and I will show you that amazingly, many other surfaces also have these three symmetries.

9/20 
Jacob Matherne (UMass)

A combinatorial Fourier transform for quiver representation varieties in type A
There is a geometric version of the Fourier transform that we meet in analysis. We will study this geometric Fourier transform for certain sheaves on quiver representation varieties. I will spend some time talking about the basics of quiver representations, compute some examples of the Fourier transform there, and explain how we can do it all combinatorially for quivers which are linearly ordered type A Dynkin diagrams. This is joint work (in progress) with Pramod Achar and Maitreyee Kulkarni.

9/27 
Raj Mehta (Smith College)

Integration of Constant Courant algebroids
Courant algebroids come up in the context of constrained mechanical systems, and they also can be used to construct 3dimensional topological field theories. There are reasons to expect that Courant algebroids should "integrate" to symplectic 2groupoids, in a similar sense to how Lie algebras integrate to Lie groups; however, there are very few explicit examples of such integrations. I will describe a simple class of symplectic 2groupoids and explain how they correspond to a simple class of Courant algebroids. 
10/4 
Gufang Zhao (UMass)

From points on the punctured plane to Higgs bundles on an elliptic curve
The affine Hecke algebra is related to points configurations on the punctured plane as well as representations of padic groups; the double affine Hecke algebra is related to points configurations on an elliptic curve as well as representations of padic loop groups. In this talk I will introduce a missing link between these two algebras, called the elliptic affine Hecke algebra. It has a construction using equivariant elliptic cohomology. Its irreducible representations are parameterized by nilpotent Higgs bundles on an elliptic curve. This talk is based on my joint work with Changlong Zhong. 
10/18 
No seminar 
Conn Valley Math/Stats Colloquium at Amherst Roger Howe (Yale) speaking 
10/25 
Jordan Tirrell (Mt Holyoke) 
Title Combinatorics of Orthogonal Polynomials and Enumeration of Young Tableaux
We will introduce the combinatorial theory of orthogonal polynomials developed by Viennot (1985) and Flajolet (1980) and then see how it can be applied to the enumeration of skew Young tableaux with at most three rows. In particular, a sequence is the moment sequence for some orthogonal polynomials if and only if it enumerates Motzkin paths with certain weights. In the simple unweighted case, the polynomials are a variant of the Chebyshev polynomials. We will use insights from the combinatorial theory of orthogonal polynomials to give a combinatorial enumeration of skew Young tableaux with at most three rows and fixed skew part as a linear combination of Motzkin numbers. We obtain a simple combinatorial proof of a refinement of earlier results. This follows work by Zeilberger (2006), Regev (2009), SenPeng Eu (2010) and Jong Hyun Kim (2011). 
11/1 
No seminar 
Abstract TBA 
11/8 
No seminar 
Abstract TBA 
11/15 
Julianna Tymoczko (Smith College) 
Splines and a conjecture of Strang's
Splines are an essential tool in analysis and applied math to deal with approximations and interpolation. We describe a general algebraic/combinatorial construction of splines that coincides with the traditional analytic definition in cases that arise naturally. It also coincides with a construction of GoreskyKottwitzMacPherson of the equivariant cohomology of many varieties of interest. We then discuss a conjecture of Strang's about the lower bound of the dimension of certain spline spaces, with some new equivalences and results. 
11/29 
Alex Woo (U Idaho) 
Inversion arrangements and coessential sets
Given a permutation (or more generally an element in a finite reflection
group) w, one can define a hyperplane arrangement called the inversion
arrangement. On the symmetric group (or any finite reflection group), one
can define a partial order known as Bruhat order. Hultman showed that the
number of chambers of the inversion arrangement is always at most the
number of elements less than or equal to w in Bruhat order, and gave a
condition on the Bruhat graph (a graph related to Bruhat order) for when
equality occurs.
This result of Hultman generalizes work of Hultman, Linusson, Shareshian,
and Sjostrand in the case of permutations. In this case, they show
equality occurs precisely when w pattern avoids the 4 permutations 4231,
35142, 42513, and 351624. This set of permutations was earlier studied in
a different context by Gasharov and Reiner, who characterized these
permutations as those whose lower Bruhat intervals are defined by
inclusion relations. I will talk about a potential generalization of the
GasharovReiner conditions defining this set, which I can prove is
equivalent to the Hultman condition for type B (which is the
hyperoctahedral group, the symmetry group of the ncube). The type B
elements can also be characterized by a list of 31 pattern avoidance
conditions. 
12/6 
Hiraku Abe (Osaka City University and McMaster University) 
Flat families of Hessenberg varieties with an application to NewtonOkounkov bodies
Hessenberg varieties are subvarieties of the full flag variety. In this talk, I will concentrate on Lie type A. I will talk about a flat degeneration of a regular semisimple Hessenberg variety to a regular nilpotent Hessenberg variety, and I will explain how we can use this flat family to compute some NewtonOkounkov bodies of the Peterson variety of dimension 2. Along the way, we will also see that any regular nilpotent Hessenberg variety is a local complete intersection; this is a generalization of a result in Erik Insko’s PhD thesis. This is a joint work with Lauren DeDieu, Federico Galetto, and Megumi Harada. 
12/13 
Tamar Friedmann (Smith College) 
Title TBA
Abstract TBA 