Events

Title: The k-independence number of graph products

Abstract: The k-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k, generalizing the standard independence number. In this talk, I will discuss well-known sharp bounds on the independence number of graph products, and extend some of these bounds to the k-independence number. Specifically, we will cover the Cartesian, tensor, strong, and lexicographic products.

Joint work with Aida Abiad.

Title: An identity in the group algebra of the symmetric group

Abstract: Come with me on a magical journey into the mysterious world of inverse Wronskians.

Title: Stochastic Optimization

Abstract:  While deterministic optimization problems are very useful in practice, often times the assumption that all data is known in advance does not hold true. One possible way to relax this assumption is to assume that the data depends on random variables.

Title: Positivity and sums of squares in products of free algebras

Abstract: A noncommutative polynomial is said to be positive relative to some constraints if plugging matrices (or more generally, operators on a Hilbert space) satisfying the constraints into the polynomial always yields a positive operator. It is a natural problem to determine whether or not a given polynomial is positive, and if it is, to find some certificate of positivity. This problem is closely connected with noncommutative polynomial optimization, where we want to find matrices or operators that maximize the operator norm of some polynomial, subject to the constraint that some other polynomials in the operators are positive or vanish. When the algebra cut out by the constraints is a free algebra, free group algebra, or similar algebra, it's well-known that a polynomial is positive on operators satisfying the constraints if and only if it's a sum of Hermitian squares in the algebra.

Title: New Characterizations of Distance-Biregular Graphs

Abstract: Fiol, Garriga, and Yebra introduced the notion of pseudo-distance-regular vertices, and were able to use this notion to come up with a characterization of when a graph is distance-regular. Subsequently, Fiol and Garriga were able to use pseudo-distance-regular vertices and a bound on the excess of a vertex to come up with another characterization of distance-regular graphs. We will present an overview of their results, as well as recent extensions to distance-biregular graphs.

Title: High-Rank Matroids and Unavoidable Flats

Abstract: We discuss a variety of questions and results pertaining to conjectures of Geelen from 2021 on the unavoidable flats in matroids of sufficiently high rank. We will also explore the differences in how such questions are posed for various classes of matroids, why such differences are necessary, and how they could potentially be reconciled. A result for the class of binary matroids and an outline of its proof will be discussed in detail. Joint work with Jim Geelen.

Title: Data-Driven Chance Constrained Programs over Wasserstein Balls

Abstract: In many real-world applications, precise problem data is not available to the decision maker. One way to handle this uncertainty is by using chance-constraints, where the probability that at least one constraint is violated is bounded above by some parameter. However, such an approach assumes that the decision maker has access to the true probability distribution which governs the data behavior.

Title: Strongly regular graphs with a regular point 

Abstract:  Arising from Hoffman and Singleton's study of Moore graphs, strongly regular graphs play an important role in algebraic graph theory. Strongly regular graphs can be construct from geometric objects, such as generalized quadrangles and certain geometric properties, such as having a regular point, can be studied in the context of graphs.

Title: Thresholds

Abstract: Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will present recent progress on this topic. Based on joint work with Huy Tuan Pham.

Title: Stochastic Load Balancing on Unrelated Machines

Abstract: We will take a look at the stochastic load balancing problem. The goal is to assign tasks to machines, so that the maximum amount of time taken by any machine to complete all its assigned tasks is minimized. The stochastic twist to this problem is that now the time required to complete each task is a random variable sampled from some known distribution. For the stochastic version, we need to minimize the maximum time taken by any machine in expectation. We will look at a constant factor approximation algorithm for this problem that appeared in a recent paper by Gupta, Kumar, Nagarajan and Shen.