Seminar

Title: In Memoriam: Tom Coleman’s Contributions to Applied Mathematics and Optimization

Description:

Thomas F. Coleman—a leader in optimization and scientific computing, professor at the University of Waterloo, and a SIAM Fellow—passed away on April 20, 2021. Tom served as the Director of the Theory Center at Cornell and then as Dean of the Faculty of Mathematics at the University of Waterloo. His research spanned continuous optimization, combinatorial scientific computing, automatic differentiation, financial optimization, mathematical software, etc. In this session, his wife and collaborator, Yuying Li, and five of his students and colleagues will describe the pioneering contributions that Tom made to these fields in his research.

Title: Asymptotic quantum state transfer using two loops with large weights

Abstract:

We study the question of asymptotic transfer strength between two nodes of a graph when large weight loop edges are located at these nodes.  It turns out that the limiting strength can be exactly computed and depends only on the extended neighborhoods of the nodes.

Title: The $k$-Independence Number

Abstract:

An independent set, also known as a stable set or coclique, in a graph is a set of vertices, no two of which are adjacent. The size of a largest independent set is called the independence number. Two classical eigenvalue bounds on the independence number, proved using eigenvalue interlacing are the Hoffman's ratio bound and the Cvetkovi\'{c}'s inertia bound.

Title: State transfer for paths with weighted loops at the end vertices

Abstract:

We consider a continuous time quantum walk on an unweighted path on n vertices, to which a loop of weight w has been added at each end vertex. Let f(t) denote the fidelity of state transfer from one end vertex to the other at time t. In this talk we give upper and lower bounds on f(t) in terms of w, n and t; further, given a > 0 we discuss the values of t for which f(t) > 1-a.

Title: Average Mixing Matrices of Trees and Stars

Abstract:

We define the average mixing matrix (AMM) of a continuous-time quantum walk on a graph using the orthogonal projections onto the eigenspaces of the adjacency matrix A. From there, one of the properties that has been studied is the rank of the AMM. This is easiest to do if the eigenvalues of A are simple, and we’ll review some of the results on this from Coutinho et. al. (2018).

Title: Arctic curves for groves

Abstract:

The limit shape phenomenon is a "law of large numbers" for random surfaces: the random surface looks macroscopically like the "average surface". The first result of this kind was the celebrated arctic circle theorem for domino tilings of the aztec diamond. The limit shape has macroscopic regions with different qualitative behavior, and the arctic curve is the boundary separating these regions.

Title: From low probability to high confidence in stochastic convex optimization

Abstract:

Standard results in stochastic convex optimization bound the number of samples that an algorithm needs to generate a point with small function value in expectation. More nuanced high probability guarantees are rare, and typically either rely on “light-tail” noise assumptions or exhibit worse sample complexity. In this work, we show that a wide class of stochastic optimization algorithms can be augmented with high confidence bounds at an overhead cost that is only logarithmic in the confidence level and polylogarithmic in the condition number.

Title: Lattice Walk Enumeration: Analytic, algebraic and geometric aspects

Abstract:

This talk will examine the rich topic of lattice path enumeration. A very classic object of combinatorics, lattice walks withstand study from a variety of perspectives. Even the simple task of classifying the two dimensional walks restricted to the first quadrant has brought into play a surprising diversity of techniques from algebra to analysis to geometry. We will consider walks under a few different lenses.

Title: Counting Antichains in the Boolean Lattice

Abstract:

How many antichains are there in the Boolean lattice P(n)? Sperner's theorem (1928) tells us that the largest antichain in P(n) has size A = (n choose n/2). A subset of an antichain is an antichain, so there are at least 2^A antichains in P(n). Interestingly, it turns out that this is close to the total, as Kleitman (1969) showed that the number of antichains is 2^(A(1+x)) where x goes to zero as n goes to infinity.

Title: Identities for ninth variation Schur Q-functions

Abstract:

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity.