Events

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.

Title: Non-realizability of polytopes via linear programming

Abstract: A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction.

Title: Stochastic Knapsack Problem

Abstract: The classical NP-hard Knapsack problem takes as input a set of items with some fixed values and weights. The goal is to compute a subset of items of maximum total value, subject to the constraint that the total weight of these elements is less than or equal to a given limit. In this talk we will review a paper by Dean, Goemans and Vondrák, in which an stochastic variation of this problem is considered. 

Title: Equiangular lines and eigenvalue multiplicities

Abstract: Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle.

Title: Conditional Value-at-Risk

Abstract: The mean and variance of a probability distribution may not reflect what one wants from a scenario involving uncertainty.In particular, such measures fall short of expressing risk in a way suitable for financial and similar applications.