Events

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: 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.

Title: On sesqui-regular graphs with fixed smallest eigenvalue

Abstract: Let λ ≥ 2 be an integer. For strongly regular graphs with parameters (v, k, a, c) and fixed smallest eigenvalue −λ, Neumaier gave two bounds on c by using algebraic property of strongly regular graphs. Subsequently, we studied a new class of regular graphs called sesqui-regular graphs, which contains strongly regular graphs as a subclass, and proved that for a given sesqui-regular graph with parameters (v, k, c) and smallest eigenvalue −λ, if k is very large, then either c ≤ λ² (λ − 1) or v − k − 1 ≤ (λ−1)²/4 + 1. This is joint work with Jack Koolen, Brhane Gebremichel and Jae Young Yang

Title: Stochastic Probing with Applications

Abstract:  We will explore a stochastic probing problem. Given a set of elements which have weights and independent probabilities of being "active," the goal is to construct a subset of active elements of maximum weight. To form such a set, we must "probe" elements sequentially to determine whether they are active.

Title: Cheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues

Abstract:

The classical Cheeger's inequality relates the edge conductance $\phi$ of a graph and the second smallest eigenvalue $\lambda_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $\psi^2 / \log |V| \lesssim \lambda_2^* \lesssim \psi$ connecting the vertex expansion $\psi$ of a graph $G=(V,E)$ and the maximum reweighted second smallest eigenvalue $\lambda_2^*$ of the Laplacian matrix. In this work, we first improve their result to $\psi^2 / \log d \lesssim \lambda_2^* \lesssim \psi$ where $d$ is the maximum degree in $G$, which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. 

Title: Essential Covers of the Cube by Hyperplanes

Abstract:  An essential cover of the vertices of the n-cube $\{0,1\}^n$ by hyperplanes is a minimal covering where no hyperplane is redundant, and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a construction of an essential cover with $\lceil \frac{n}{2} \rceil + 1$ hyperplanes and showed that $\Omega(\sqrt{n})$ hyperplanes are required.

Title: Graphical Designs and Gale Duality