# Colloquia

## Zhi-Hong Chen

Cycles of claw-free graphs and Catlin's reduction method

In the study of cycles of claw-free graphs,

Ryj\'{a}\u{c}ek developed a very nice closure concept: For a connected claw-free graph $H$, there is a $K_3$-free graph $G$ such that its closure $cl(H)=L(G)$ and for each cycle $C_0$ in $L(G)$, there exists a cycle $C$ in $H$ with $V(C_0)\subseteq V(C)$. A theorem by Harary and Nash-Williams shows that a line graph $L(G)$ has a Hamiltonian cycle if and only if $G$ has a dominating closed trail.

Thus, to find a cycle in a claw-free graph $H$ can be reduced to finding a closed trail in the preimage $G$ of the line graph of $L(G)$, where $L(G)=cl(H)$.

In this talk, I will discuss applications of Catlin's reduction method to the study of cycles in claw-free graphs and discuss new results and solutions of some conjectures on claw-free graphs that we obtained recently.

Date: 10/14/2016

Time: 3:45PM-4:45PM

Place: 315 Armstrong Hall

## Liang Hong

Credibility Theory without Tears

After a brief introduction of the actuarial science program at Robert Morris University, I will introduce the basic framework of Bayesian statistics and classical theory of credibility theory, a topic heavily tested on SOA's Exam C.

The last part of the talk will give the latest development in credibility theory; this work was done by my colleague (Dr. Ryan Martin, at North Carolina State University) and me, and our project was jointly sponsored by the Casualty Actuarial Society and Society of Actuaries through 2015 Individual Grant. The technical details are kept to a minimum. Therefore, the talk is accessible to anyone with a background in calculus-based probability theory, that is, the basic topics on Exam P.

Dr. Hong received his PhD in mathematics from Purdue University. He has received grants from the Casualty Actuarial Society (CAS), Society of Actuaries (SOA), and State Farm Insurance Company; and he has given research talks at several Center for Actuarial Excellence (CAE) schools including Drake University, Georgia State University, Temple University, University of Waterloo, University of Wisconsin, Madison.

Date: 9/23/2016

Time: 12:30PM-1:20PM

Place: 121 Armstrong Hall

## Hao Li

On the g-extra connectivity of $3$-ary $n$-cube networks

Let $G$ be a connected graph and let $F$ be a set of vertices.

The $g$-extra connectivity of $G$ is the cardinality of a minimum set

$F$ such that $G-F$ is disconnected and each component of $G-F$ has at

least $g+1$ vertices. The $g$-extra connectivity is an important

parameter to measure the reliability and fault tolerance ability of

large interconnection network of parallel computing systems.

In 2011, the $g$-extra connectiviy for $g=1,2$ of 3-ary $n$-cubes are

gotten by Zhu et al. The 3-extra connectivity of 3-ary $n$-cubes are

given by Gu and Hao in 2014. Here, we determine the $g$-extra

connectivity of 3-ary $n$-cubes for $0\le g \le 2n$.

Date: 8/25/2016

Time: 3:45PM-4:45PM

Place: 315 Armstrong Hall

## Xiao-Dong Zhang

Some Extremal Graph Results with Degree Sequences

As it is said in the book \Extremal Graph Theory" by Bollobas, in extremal graph theory one is interested in the relations between the various graph variants, such as order, size, connectivity, minimum degree, maximum degree, chromatic number and diameter, and also in the values of these in-variants which ensure that the graph has certain properties. In this talk, we introduce some progresses and results on graph variants, such as spectral radius, Laplacian spectral radius, the average distance, and the number of sub-trees for a class of graphs with given degree sequences, and we also introduce some related open problems.

Professor Xiao-Dong Zhang's Website

Date: 8/23/2016

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

## Professor Zijian Diao

Trigonometry Revisited or: If Fourier Had a Quantum Computer

Is sine of 1 degree rational? At a glance this question seems light years away from quantum computing, a cutting-edge research area where computer science meets quantum mechanics. Surprisingly, this question matters not only in the mathematical world, but also in the quantum realm. Its answer and many other rudimentary facts of trigonometry have found their way into the study of various research problems in quantum computing. In this talk we will explore these intriguing connections through the interplay of quantum algorithms and topics from number theory and classical analysis. Along the way, we will solve a long standing puzzle in quantum search and provide a quantum approach to the centuries-old Basel problem.

Date: 4/26/2016

Time: 3:00PM-4:00PM

Place: 315 Armstrong Hall

## Professor Megan Wawro

Research on the Learning and Teaching of Diagonalization and Eigentheory

My research focuses on the learning and teaching of linear algebra. In addition to characterizing student reasoning at both the individual and collective levels, I work to develop curriculum and instructor supports for inquiry-oriented linear algebra courses, as well as investigate student understanding of mathematics in physics. In this talk, I will highlight the synergy of this body of work by summarizing both an analysis of student symbolizing through reinvention of the diagonalization equation and the associated curriculum on diagonalization and eigentheory. I will conclude by sharing preliminary results regarding student reasoning about and symbolization of eigentheory in quantum physics.

Date: 4/22/2016

Time: 3:30PM-4:30PM

Place: 306 Armstrong Hall

## Professor David Offner

Characterizing Cop-Win Graphs and Their Properties Using a Corner Ranking Procedure

Cops and Robber is a two-player vertex pursuit game played on graphs. Some number of cops and a robber occupy vertices of a graph, and take turns moving between adjacent vertices.

If a cop ever occupies the same vertex as the robber, the robber is caught. If one cop is sufficient to catch the robber on a given graph, the graph is called cop-win, and the maximum number of moves required to catch the robber is called the capture time. In this talk we introduce a simple “corner ranking” procedure that assigns a rank to every vertex in a graph.

The results of this procedure can be used to determine many properties of the graph, for example, if it is cop-win, and if so, what the capture time is. We show how corner rank can be used to describe optimal strategies for the cop and robber, and to characterize those cop-win graphs with a given number of vertices which have maximum capture time. Though cop-win graphs and their capture time have previously been characterized, in many cases corner rank gives more streamlined proofs, and allows us to extend known results.

Date: 4/21/2016

Time: 4:00PM-5:00PM

Place: 315 Armstrong Hall

## Professor Truyen Nguyen 4/14/2016

Gradient estimates for nonlinear degenerate parabolic systems

**Date:** 4/14/2016**Time:** 3:45PM-4:45PM**Place:** 313 Armstrong Hall

Truyen Nguyen

**Abstract:** We study nonlinear degenerate parabolic systems of the form $u_t = \mbox{div}\, \mathbf{A}(x,t,u,\nabla u) + \mathbf{B}(x,t, u,\nabla u )$, which include those of $p$-Laplacian type. The system is degenerate if $p\geq 2$ and singular if $1

.

## Professor Chiranjib Mukherjee 3/17/2016

Compactness, Large Deviations

and Statistical Mechanics

**Date:** 3/17/2016**Time:** 3:30PM-5:00PM**Place:** 315 Armstrong Hall

**Abstract:** In a reasonable topological space, large deviation estimates essentially deal with probabilities of events that are asymptotically (exponentially) small, and in a certain sense, quantify the rate of these decaying probabilities. In such estimates, lower bound for open sets and upper bound for compact sets are essentially local estimates. However, upper bounds for all closed sets often require compactness of the ambient space or stringent technical assumptions (e.g., exponential tightness), which is often absent in many interesting problems which are motivated by questions arising in statistical mechanics (for example, distributions of occupation measures of Brownian motion in the full space Rd).

Motivated by problems that carry certain shift-invariant structure, we present a robust theory of “translation-invariant compactiﬁcation” of orbits of probability measures in Rd. This enables us to prove a desired large deviation estimates on this “compactiﬁed” space. Thanks to the inherent shift-invariance of the underlying problem, we are able to apply this abstract theory painlessly to solve a long standing problem in statistical mechanics, the mean-ﬁeld polaron problem.

This is based on joint work with S. R. S. Varadhan (New York).

## Abbey Bourdon 3/16/2016

Rational Torsion on CM

Elliptic Curves

**Date:** 3/16/2016**Time:** 3:30PM-5:00PM**Place:** 315 Armstrong Hall

**Abstract:** Let E be an elliptic curve defined over a number field F. By a classical

theorem of Mordell and Weil, the collection of points of E with

coordinates in F form a finitely generated abelian group. We seek to

understand the subgroup of points with finite order. In particular,

given a positive integer d, we would like to know precisely which

abelian groups arise as the torsion subgroup of an elliptic curve

defined over a number field of degree d, and we would like to know how

the size of the torsion subgroup grows as d increases. After providing a

brief introduction to elliptic curves and summarizing prior results, I

will discuss recent progress on these problems for the special class of

elliptic curves with complex multiplication (CM).

Abbey is a candidate for a position in our department.

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