Events - 2020

Title: Decomposing discrete quantum walks into continuous quantum walks

Abstract:

The Grover walk is a discrete quantum walk inspired by Grover's search algorithm. It takes place on the arcs of a graph, and alternates between "coin flips" and "arc reversal". In this talk, I show that for a distance regular graph X with diameter d and intertible A(X), the Grover walk on X can be "decomposed" into at most d "commuting" continuous quantum walks.

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: 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: 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.

Title: On the flip graph on perfect matchings of complete graphs and sign reversal graphs

Abstract:

In this talk, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper bound on its chromatic number.

Title: Formulas for Macdonald polynomials arising from the ASEP

Abstract:

The asymmetric simple exclusion process (ASEP) is a one-dimensional model of hopping particles that has been extensively studied in statistical mechanics, probability, and combinatorics. It also has remarkable connections with orthogonal symmetric polynomials in many variables such as Macdonald and Koornwinder polynomials.

Title: Number-theoretic methods in quantum computing

Abstract:

An important problem in quantum computing is the so-called \emph{approximate synthesis problem}: to find a quantum circuit, preferably as short as possible, that approximates a given target operation up to given $\epsilon$. For nearly two decades, from 1995 to 2012, the standard solution to this problem was the Solovay-Kitaev algorithm, which is based on geometric ideas. This algorithm produces circuits of size $O(\log^c(1/\epsilon))$, where $c$ is a constant approximately equal to $3.97$. It was a long-standing open problem whether the exponent $c$ could be reduced to $1$.

Title: Discrete diffusion on graphs and real hyperplane arrangements

Abstract:

In 2016, Duffy, Lidbetter, Messinger, and Nowakowski introduced the following variation of a chip-firing model on a graph. At time zero, there is an integer number of "chips" at each vertex. Time proceeds in discrete steps.  At each step, each edge is examined (in parallel) -- one chip is moved from the greater end to the lesser end if the ends are not equal.

Title: Widths in even-hole-free graphs

Abstract:

Historically, the study of even-hole-free graphs is motivated by the analogy with perfect graphs. The decomposition theorems that are known for even-hole-free graphs are seemingly more powerful than the ones for perfect graphs: the basic classes and the decompositions are in some respect more restricted. But strangely, in an algorithmic perpective, much more is known for perfect graphs. 

Title: Graph coloring of graphs with large girth is hard for the Nullstellensatz

Abstract:

In this talk we will discuss a method to solve combinatorial problems using hierarchies of systems of linear equations using Hilbert's Nullstellensatz. In particular, we will study the behaviour of these hierarchies for deciding the non-$k$-colorabilty of graphs.