# 2019-20 colloquium

Fall 2019

- September 13, Chi-Kwong Li (William & Mary) Error correlation schemes for fully correlated quantum channels protecting both quantum and classical information

Abstract: We study efficient quantum error correction schemes for the fully correlated channel on an n-qubit system with error operators that assume the form $\sigma_x^{\otimes n}$, $\sigma_y^{\otimes n}$, $\sigma_z^{\otimes n}$. In particular, when 2k+1 is odd, we have a quantum error correction scheme using one arbitrary qubit $\sigma$ to protect the data state $\rho$ in the 2k-qubit system. When n=2k+2 is even, we have a hybrid quantum error correction scheme that protects a 2k-qubit state $\rho$ and 2-classical bits. The scheme was implemented using Matlab, Mathematica and the IBM's quantum computing framework qiskit. Note: Problems and results will be described in terms of elementary matrix theory. No quantum mechanics background is needed. Co-authors: Seth Lyles, and Yiu-Tung Poon - Septmeber 20, Fan Ge (William & Mary) Moments of zeta and L-functions

Abstract: The study of moments of the Riemann zeta-function is important and has been very active. For example, fine estimates of these moments would yield the Lindelof Hypothesis, which is one of the central unsolved problems in number theory. In this talk, I will discuss the history and some recent progress of moments. - Septmber 27, Saeed Nasseh (Georgia Southern University) Modules over finite dimensional algebras: Application to a conjecture in commutative algebra.

Abstract: When F is a field and R an F-algebra, it is often important to classify the R-module structures carried by some fixed F-vector space V. We will describe, in down-to-earth terms, a classical approach to that problem, which involves linear algebra, group theory, algebraic geometry, and homological algebra. Using tools from algebraic topology, we will then sketch new developments of these techniques and their application to the proof of a result in commutative algebra, which completely answers a conjecture posed by Vasconcelos in 1974. - October 4, Stephen Trefethen (William & Mary) Frobenius--Schur indicators of finite exceptional groups.

Abstract: Let G be a finite group. The Frobenius--Schur indicator of an irreducible character $\chi$, denoted $\varepsilon(\chi)$, is defined as $\varepsilon(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^2)$. It is known that $\varepsilon(\chi)=1, -1,$ or $0$, where $\varepsilon(\chi) = 0$ precisely when $\chi$ is not real-valued. When $\chi$ is real-valued, $\varepsilon(\chi) = 1$ if $\chi$ is afforded by a representation that may be defined over the real numbers, otherwise $\varepsilon(\chi) = -1$. In this talk we outline a computational method used to prove that the exceptional groups $F_4(q)$, $E_7(q)_{\textrm{ad}}$, and $E_8(q)$ have no irreducible characters with Frobenius--Schur indicator -1. - October 18, Mathematics Career Panel (homecoming special event) In this special mathematics colloquium, several recent mathematics alumni will discuss their career path and how mathematics has helped them to achieve their goals. Mathematics Career Panelists: Kelvin Abrokwa-Johnson '17 (Square), Sean Dalby '09 (World Bank), Maggie Liu '10 (Smith College), Timothy McDade '12 (Duke University), Margaret Swift '17 (Duke University)
- October 25, Pierre Clare (William & Mary) Symmetries in Harmonic Analysis

It is a fundamental principle in Analysis that taking into account the symmetries of an object simplifies the study of functions on that object. Starting from elementary examples and the classical theory of Fourier series, we will explore various instances of this principle and its natural connections to the theory of groups and their representations. Following Alain Connes' philosophy of noncommutative geometry, certain spaces of representations will be studied by means of algebras of linear operators, leading to the characterization of a new class of representations, of particular topological relevance. Unexpected symmetries will be shown to appear in that context. - November 1, Punit Gandhi (Virginia Commonwealth University) Water Transport in Models of Dryland Vegetation Patterns

Abstract: Reaction-advection-diffusion models that capture the interactions between plants, surface water and soil moisture can qualitatively reproduce community-scale vegetation patterns that are observed in dryland ecosystems. On gently sloped terrain, these patterns often appear as bands of vegetation growth alternating with bare soil. The vegetation bands can be tens of meters thick with spacing on the order of a hundred meters, and form a regular striped pattern that often occupies tens of square kilometers on the landscape. I will focus on aspects of the surface/subsurface water dynamics within these models. Capturing these hydrological processes on appropriate timescales may allow us to better utilize observational data as we work to identify the dominant mechanisms underlying the formation of dryland vegetation patterns and understand how environmental factors influence pattern characteristics. - November 8, Gexin Yu (William & Mary) Highly connected minors in k-chromatic graphs

Abstract: One of the deepest problems in graph theory is Hadwiger's Conjecture, which asserts that every k-chromatic graph contains a clique minor of order k for each positive integer k. The conjecture is confirmed to be true when k is at most 6, and in fact, it is equivalent to the Four Color Theorem when k equals 5 or 6, and remains wide open for k at least 7. In this talk, we will explore a weaker version of the conjecture: what's the function f(k) so that an f(k)-chromatic graph contains a k-connected minor? No prior knowledge in graph theory will be assumed in this talk. This is based on joint work with Runrun Liu and Martin Rolek. - November 15, Xiaofeng Gu (University of Western Georgia) Rigidity and applications in graph theory

Abstract: Rigidity, arising from mechanics, is the property of a framework that does not flex. A combinatorial characterization of rigidity in the Euclidean plane has been obtained by Laman in 1970. In this talk, we will present sufficient conditions of rigidity as well as some applications in graph theory. - November 22, Yuesheng Xu (Old Dominion University) Sparse Machine Learning

Abstract: We shall present recent development in sparse machine learning. In particular, we shall elaborate the needs of developing sparse machine learning methods with an illustration by simulation results. Machine learning methods in reproducing kernel Banach spaces will be discussed. Numerical results will be presented to demonstrate the effectiveness of sparse learning methods.

Spring 2020

- February 28, Wei Meng (William & Mary): Linear connectivity for tournaments to be highly linked

Abstract: A digraph is k-linked if for any two disjoint sets of vertices x_1,..., x_k and y_1,...,y_k there are vertex disjoint paths P_1,...,P_k such that P_i is directed from x_i to y_i for i=1, ..., k. Pokrovskiy in 2015 proved that every strongly 452k-connected tournament is k-linked. In this paper, we significantly reduce this connectivity bound and show that any (24k-19)-connected tournament is k-linked. This is joint work with Martin Rolek, Yue Wang, and Gexin Yu.

Yue Wang (William & Mary): Enhancing Erdos-Lovasz Tihany conjecture for line graphs of multigraphs

Abstract: let s,t be integers. A graph G is (s,t)-splittable if the vertices of G can be partitioned into two sets S and T such that the chromatic number of subgraph induced by S (denoted by G[S]) is at least s and the chromatic number of subgraph induced by T (denoted by G[T]) is at least t. The well-known Erdos-Lovasz Tihany Conjecture from 1968 states that every graph G whose chromatic index equals s+t-1 and larger than its clique number is (s,t)-splittable. This conjecture is hard, and only known to be true for line graphs, quasi-lines, and graphs with independent number 2. In this paper, we prove an enhanced version of the conjecture for line graphs of multigraphs. - March 6, Xiaoyuan Chang (Harbin University of Science and Technology)

Title: Analysis of a stoichiometric bacteria-grazer reaction-diffusion model for organic matter biodegradation

Abstract: The dynamical behaviors of a bacteria-carbon-nitrogen reaction-diffusion system with grazers and without grazers are investigated in this paper. It is shown that the bacteria death rate and the grazer death rate play a decisive role in the dynamics of the system. The existence, stability and persistence of the solutions are proved by applying the theorem of existence of eigenvalue, the local and global bifurcation theory, and the abstract persistence theory. Through numerical simulations, the transient dynamic and the oscillation phenomenon are obtained.

Junping Shi (William & Mary)

Title: Modeling animal movement with memory with partial differential equations with time-delay

Abstract: Animal populations often self-organize into territorial structure from movements and interactions of individual animals. Memory is one of cognitive processes that may affect the movement and navigation of the animals. We will review several mathematical approaches of animal spatial movements, and also introduce our recent work using partial differential equations with time-delay to model and simulate the memory-based movement. We will show the bifurcation and pattern formation for such models. It is based on joint work with Chuncheng Wang, Hao Wang, Xiangping Yan, Qingyan Shi and Yongli Song.