Current students

Title: Shorter Zero-Knowledge Proofs from MPC

Abstract:

In this talk I will review the MPC-in-the-head approach to constructing zero-knowledge proofs, then talk about some recent research results to make the proofs shorter.

In a zero-knowledge proof system, a prover wants to convince a verifier that they know a secret value, without revealing it. A common case involves a one-way function, where the prover wants to convince a verifier that they know a secret input corresponding to a public output.

Title: The (3+1)-free conjecture of chromatic symmetric functions

Abstract:

The chromatic symmetric function, dating from 1995, is a generalization of the chromatic polynomial. A famed conjecture on it, called the Stanley-Stembridge (3+1)-free conjecture, has been the focus of much research lately. In this talk we will be introduced to the chromatic symmetric function, the (3+1)-free conjecture, new cases and tools for resolving it, and answer another question of Stanley of whether the (3+1)-free conjecture can be widened. This talk requires no prior knowledge.

Title: The results of a search for small association schemes with noncyclotomic eigenvalues

Abstract:

Title: On complete classes of valuated matroids

Abstract:

Valuated matroids were introduced by Dress and Wenzel in 1992. They are a central object in discrete convex analysis, and play important roles in other areas such as mathematical economics and tropical geometry. Finding a constructive characterization, i.e., showing that all valuated matroids can be derived from a simple class by some basic operations has been a natural question proposed in various contexts.

Title: State transfer on graphs with twin vertices

Abstract:

In this talk, we discuss algebraic and spectral properties of graphs with twin vertices that are important in quantum state transfer. We give a characterization of strong cospectrality between twin vertices, and characterize some types of state transfer that occur between them.

Title: Four families of polynomials in spectral graph theory

Abstract:

 In this talk we describe four families of polynomials related to the spectrum of a graph. First, some known main results involving such polynomials, such as the spectral excess theorem characterizing distance-regularity, are discussed. Second, some new results giving bounds for the $k$-independence number $\alpha_k$ of a graph are presented. In this context, we comment on some relationships between the inertia (Cvetkovi\`c) and ratio (Hoffman) bounds of $\alpha_k$.

Title: Induced subgraphs and treewidth

Abstract:

Treewidth, introduced by Robertson and Seymour in the graph minors series, is a fundamental measure of the complexity of a graph. While their results give an answer to the question, “what minors occur in graphs of large treewidth?,” the same question for induced subgraphs is still open. I will talk about some conjectures and recent results in this area.

Joint work with Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Sepehr Hajebi, Pawel Rzazewski, Kristina Vuskovic.

Title: Structured (In)Feasibility: Nonmonotone Operator Splitting in Nonlinear Spaces

Abstract:

Several challenging optimization problems in power networks involve operational decisions, non-linear models of the underlying physics described by the network as well as uncertainty in the system parameters. However, these networks exhibit a nice structure. This talk provides an overview of approaches that combine recent advances in robust optimization and conic relaxations of polynomial optimization problems along with exploiting the structure of the underlying problem. These approaches are demonstrated on applications arising in power networks.

Title: Semidefinite Optimization Approaches for Reactive Optimal Power Flow Problems

Abstract:

The Reactive Optimal Power Flow (ROPF) problem consists in computing an optimal power generation dispatch for an alternating current transmission network that respects power flow equations and operational constraints. Some means of voltage control are modelled in ROPF such as the possible activation of shunts, and these controls are modelled using discrete variables. The ROPF problem belongs to the class of nonconvex MINLPs, which are NP-hard problems. We consider semidefinite optimization approaches for solving ROPF problems and their integration into a branch-and-bound algorithm.

Title: Newell-Littlewood numbers

Abstract:

The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities.