# Colloquia

## 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.

## Hung P. Tong-Viet 3/15/2016

Derangements in primitive permutation groups and applications

**Date:** 3/15/2016**Time:** 3:30PM-5:00PM**Place:** 315 Armstrong Hall

Hung P. Tong-Viet

**Abstract:** A derangement (or fixed-point-free permutation) is a permutation with no

fixed points. One of the oldest theorems in probability, the Montmort

limit theorem, says that the proportion of derangements in finite

symmetric groups Sn tends to e−1 when n tends to infinity. A classical

theorem of Jordan implies that every finite transitive permutation group

of degree greater than 1 contains derangements. This result has many

applications in number theory, topology, game theory, combinatorics, and

repre- sentation theory. There are several interesting questions on the

order and the number of derangements that have attracted much attention

in recent years. In this talk, I will discuss some of these questions

and I will report on recent results on finite primitive permutation

groups with some restriction on derangements (joint with T.C. Burness)

and some application to modular representation theory (joint with M.L.

Lewis).

## RUME Colloquium

Opportunity to learn from lectures in advanced mathematics

**Date:** 3/11/2016**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Tim Fukawa-Connelly

**Abstract:** In this report, we synthesize studies that we have conducted on how students interpret mathematics lectures. We present a case study of a lecture in which students in an advanced mathematics lecture did not comprehend the points that their professor intended to convey. We present three accounts for this: students’ note-taking strategies, their beliefs about proof, and their understanding of the professor’s colloquial mathematics. Finally, we explore via a larger-scale study, lecturing practices and student-note-taking behaviors. We refute claims that mathematicians do not present intuitive or conceptual explanations, and demonstrate that students are unlikely to take meaning away from these more informal aspects of lecture.

## Professor Michael Schroeder 3/10/2016

One Row, One Column,

One Symbol

**Date:** 3/10/2016**Time:** 3:45PM-4:45PM**Place:** 315 Armstrong Hall

**Abstract:** Let n be a positive integer and and r,c,s each be integers in {1,2,...,n}. A partial latin square P satisfies the RCS property if for every ordered triple (x,y,z) belonging to P, either x=r, y=c, or z=s. Partial latin squares of this type were introduced by Casselgren and Haggkvist in a 2013 paper, in which they show that some infinite families of partial latin squares with the RCS property are completable. In this talk, we classify when any partial latin square with the RCS property is completable. This is joint work with Jaromy Kuhl of the University of West Florida.

## Professor Maria Emilianenko 3/4/2016

Kinetic Modeling of

Coarsening in Polycrystals

**Date:** 3/4/2016**Time:** 1:30PM-2:30PM**Place:** 315 Armstrong Hall

**Abstract:** When microstructure of polycrystalline materials undergoes coarsening driven by the elimination of energetically unfavorable crystals, a sequence of network transformations, including continuous expansion and instantaneous topological transitions, takes place. This talk will be focused on recent advances related to the mathematical modeling of this process. Two types of approaches will be discussed, one aimed at simulating the evolution of individual crystals in a 2-dimensional system via a vertex model focused on triple junction dynamics, and one providing a kinetic Boltzmann-type description for the evolution of probability density functions. Predictions based on the new kinetic mesoscale model will be discussed and contrasted with those obtained by large-scale simulations for several classes of interfacial energies.

## Professor Lianying Miao 2/22/2016

On the extremal values of the

eccentric distance sum of trees

**Date:** 2/22/2016**Time:** 3:30PM-5:00PM**Place:** 315 Armstrong Hall

Lianying Miao

**Abstract:**Let G = (V,E) be a simple connected graph. The eccentricity ε(v) of a vertex v is the maximum distance from v to any other vertex. The eccentric distance sum of G is deﬁned as ξd(G) = Pv∈V ε(v)D(v), where D(v) = Pu∈V d(u,v) is the sum of all distance from the vertex v.

In this paper, we continue to study the eccentric distance sum of trees. The trees of order n with domination number at most dn

3e are characterized.

## Professor Mokshay Madiman 1/27/2016

Entropy and the additive combinatorics of probability densities

on locally compact abelian groups

**Date:** 1/27/2016**Time:** 3:30PM-4:50PM**Place:** 315 Armstrong Hall

Mokshay Madiman

**Abstract:** Additive number theory contains a number of so-called

"sumset inequalities" that relate the cardinalities of various finite

subsets of an abelian group G, for instance, the sumset A+A and the

difference set A-A of a finite subset A of G. It also contains

“inverse" results such as Freiman’s theorem, which asserts that sets A

such that A+A is relatively small must have some "additive structure".

Motivated by considerations coming from multiple directions including

probability theory, combinatorics, information theory, and convex

geometry, we explore probabilistic analogues of such results in the

general setting of locally compact abelian groups.

For instance, we show that for independent, identically distributed random variables X

and X’ whose distribution has a density with respect to Haar measure

on a locally compact abelian group G, the entropies of X+X' and X-X'

strongly constrain each other. We will also discuss stronger

statements that can be made for specific groups of interest, such as

R^n, the integers, and finite cyclic groups.

Based on (multiple) joint works with Ioannis Kontoyiannis (Athens Univ. of Economics),

Jiange Li (Univ. of Delaware), Liyao Wang (Yale Univ.), and Jaeoh Woo

(Univ. of Texas, Austin).

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