Mathematics & Statistics Colloquium

Joe Kraisler (Amherst College): Real space quantum optics in periodic media
October 2, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: In quantum physics, the allowable energy levels of a system are described by the spectrum of a certain linear operator called the Hamiltonian. First, we will introduce the Schrodinger equation which describes the motion of an electron through an external potential and survey some results in the case that this potential is periodic. Then we will introduce a relatively new model which describes the energy exchange between a collection of cold atoms with two distinct states, and an electromagnetic field. While this model is quite different physically from studying the motion of an electron, there are many mathematical analogies that can be made. Finally we will consider the case that the atoms are distributed periodically and discuss the structure of the spectrum of the Hamiltonian for this system.
Joe Krasisler's Website

Theo Douvropoulos (Brandeis University): Hyperplane arrangements in combinatorics, representation theory, and geometry
October 16, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: Combinatorialists love to turn their favorite objects (graphs, posets, permutations) into hyperplane arrangements and their regions. The Braid, Shi, and Catalan arrangements, and their analogues for Weyl groups, have been particularly popular examples with remarkable numerological and structural properties: their regions can be labeled by trees, their characteristic polynomials factor with positive integer roots, they can be extended to free filtrations of the affine Weyl arrangements.
Representation theorists and geometers love them all the same: These three types of arrangements are used to study Kazhdan-Lusztig cells and GIT stability conditions, and the polynomial vector fields tangent to them form free modules with a very rigid structure.
We will present a new large family of generalizations of these arrangements, with surprisingly good behavior, that we introduced in recent work studying restrictions on arbitrary flats. We will prove numerological and structural results for them and relate them to the representation theory of parking spaces and rational Cherednik algebras, and discuss work joint with Olivier Bernardi, where we give refined combinatorial models for their regions. We will finish with a connection to periodic and aperiodic tilings and discuss open problems.
Theo Douvropoulos' Website

Hanwen Zhang (Yale University): Finding scattering resonances via generalized colleague matrices
October 30, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: Locating scattering resonances is a standard task in certain areas of physics and engineering. This often can be reduced to finding zeros of complex analytic functions. In this talk, I will discuss joint work with Vladimir Rokhlin on a scheme for finding all roots of a complex analytic function in a square domain. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on an interval by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” Our extension to the complex domain is based on several observations that enable the construction of polynomial bases that satisfy three-term recurrences and are reasonably well-conditioned, giving rise to "generalized colleague matrices." We also introduce a special-purpose QR algorithm for finding eigenvalues of the resulting structured matrices stably and efficiently. I will demonstrate the effectiveness of our approach to locating scattering resonances and reflectionless states by coupling it to a highly accurate integral equation solver.
Hanwen Zhang's Website

William Dugan (UMass Amherst): On the f-vector of flow polytopes for complete graphs
November 6, 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: The Chan-Robbins-Yuen polytope ($CRY_n$) of order $n$ is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow (1,0,0, ... , 0, -1). The volume and lattice points of this polytope have been actively studied, however its face structure has been studied less. We give explicit formulas and generating functions for the $f$ vector of $CRY_n$ by using Hille's (2007) result bijecting faces of a flow polytope to certain graphs, as well as Andresen-Kjeldsen's (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes over the complete graph having arbitrary (non-negative) netflow vectors and study the face lattice of $CRY_n$.
William Dugan's Website

Guangming Jing (UMass Lowell): Edge Coloring and Its Applications
November 18 (Monday), 11 a.m. - Noon
Location: Southwick Hall 350W

Abstract: In graph theory, a proper edge coloring of a graph is an assignment of “colors” to the edges so that no two incident edges share the same color. The chromatic number for a graph is the minimum number of colors required to achieve a proper edge coloring. In this talk, we will introduce several applications of edge coloring and discuss recent advances in this field.