Future students

Title: P\'olya enumeration theorems in algebraic geometry

Abstract:

We will start by comparing Macdonald's formula of the generating function for the symmetric powers of a compact complex manifold and Grothendieck's formula of the zeta series of a projective variety over a finite field, an explicit version of Dwork's rationality result.

Title: Weighted Maximum Multicommodity Flows over time

Abstract:

In various applications, flow does not travel instantaneously through a network, and the amount of flow traveling on an edge may vary over time. This temporal dimension is not captured by the classic static network flow models but can be modeled using flows over time. 

Title: Continuous Quantum Walks on Graphs

Abstract:

A quantum walk is a (rather imperfect analog) of a random walk on a graph. They can be viewed as gadgets that might play a role in quantum computers, and have been used to produce algorithms that outperform corresponding classical procedures.

Title: Data-Driven Sample-Average Approximation for Stochastic Optimization with Covariate Information

Abstract:

We consider optimization models for decision-making in which parameters within the optimization model are uncertain, but predictions of these parameters can be made using available covariate information.  We consider a data-driven setting in which we have observations of the uncertain parameters together with concurrently-observed covariates.  Given a new covariate observation, the goal is to choose a decision that minimizes the expected cost conditioned on this observation.  We investigate a data-driven framework in which the outputs from a machine learning prediction model are directly used to define a stochastic programming sample average approximation (SAA). 

Title: Semidefinite programming representations for separable states

Abstract:

The set of separable (i.e., non-entangled) bipartite states is a convex set that plays a fundamental role in quantum information theory. The problem of optimizing a linear function on the set of separable states is closely related to polynomial optimization on the sphere. After recalling the sum-of-squares hierarchy for this problem, I will show bounds on the rate of convergence of this SDP hierarchy; and prove that the set of separable states has no SDP representation of finite size.

Title: Chord diagrams, colours, and QED

Abstract:

Feynman graphs in quantum electrodynamics are essentially chord diagrams with photon edges taking the role of chords attached to lines or cycles given by electron edges. The associated Feynman integrals involve traces of Dirac gamma matrices whose computation leads to large sums of scalar Feynman integrals (cf. the earlier talk by O. Schnetz).

Title: A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case

Abstract:

Given a connected undirected graph G on n vertices, and non-negative edge costs c, the 2ECM problem is that of finding a 2-edge connected spanning multisubgraph of G of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the Traveling Salesman Problem on the metric closure of G, gives a lower bound on the optimal cost.

Title: Group Theory and the Erd\H{o}s-Ko-Rado (EKR) Theorem

Abstract: 

Group theory can be a key tool in sovling problems in combinatorics; it can provide a clean and effective proofs, and it can give deeper understanding of why certain combinatorial results hold. My research has focused on the famous Erd\H{o}s-Ko-Rado (EKR) theorem.

Title: Dynamics of plane partitions

Abstract:

Consider a plane partition P in an a X b X c box. The rowmotion operator sends P to the plane partition generated by the minimal elements of its complement. We show rowmotion resonates with frequency a+b+c-1, in the sense that each orbit size shares a prime divisor with a+b+c-1. This confirms a 1995 conjecture of Peter Cameron and Dmitri Fon-Der-Flaass. (Based on joint works with Kevin Dilks & Jessica Striker and with Becky Patrias.)

Title: Two unsolved problems: Birkhoff--von Neumann graphs and PM-compact graphs

Abstract:

A well-studied object in combinatorial optimization is the {\it perfect matching polytope} $\mathcal{PMP}(G)$ of a graph $G$ --- the convex hull of the incidence vectors of all perfect matchings of $G$. A graph $G$ is {\it Birkhoff--von Neumann} if $\mathcal{PMP}(G)$ is characterized solely by non-negativity and degree constraints, and $G$ is {\it PM-compact} if the combinatorial diameter of $\mathcal{PMP}(G)$ equals one.