# Colloquia

## Gexin Yu 4/13/2015

On path cover of

regular graphs

**Date:** 4/13/2015**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Gexin Yu

Abstract: A path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number p(G) of graph G is the cardinality of a path cover with minimum number of paths. Reed conjectured that a 2-connected 3-regular graph has path cover number at most $\lceil n/10\rceil$. In this paper, we confirm this conjecture.

## Arthur T. Benjamin 4/9/2015

Combinatorial Trigonometry (and a method to DIE for)

**Date:** 4/9/2015**Time:** 2:30PM-3:30PM**Place:** 315 Armstrong Hall

Arthur T. Benjamin

Abstract: Many trigonometric identities, including the Pythagorean theorem, have combinatorial proofs. Furthermore, some combinatorial problems have trigonometric solutions. All of these problems can be reduced to alternating sums, and are attacked by a technique we call D.I.E.

(Description, Involution, Exception). This technique offers new insights to identities involving binomial coefficients, Fibonacci numbers, derangements, and Chebyshev polynomials.

## Guantao Chen 3/6/2015

Lovasz-Plummer Conjecture on Spanning Halin Subgraphs

**Date:** 3/6/2015**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Guantao Chen

Abstract: Here

## Lijiang Wu 2/4/2015

Non-local Interaction Equations in Heterogeneous Environment With Boundary

Date: 2/4/2015

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Lijiang Wu

Abstract:We discuss biological aggregation in a heterogeneous environment and on noncovex, nonsmooth domains. We model the heterogeneous environment as a Riemannian manifold with boundary and develop gradient flow approach in the space of probability measures on the manifold endowed with Riemannian 2-Wasserstein metric. We use the gradient flow structure to show the well-posedness of a class of nonlocal interaction equations. We discuss how heterogeneity of the environment leads to new dynamical phenomena. We also present a result on generalizing the well-posedness of weak measure solutions to a class of nonlocal interaction equations on nonconvex and nonsmooth domains. We use particle approximations, solve the discrete ODE systems and pass to the continuum limit by stability property. The novelty here is that under mild regularity conditions on space(i.e. prox-regularity), we can show the well-posedness of the ODE systems and the stability properties with explicit dependence on the geometry of the space(prox-regular constant). The talk is based on collaborations with J. A. Carrillo and D. Slepcev

## Seyfi Turkelli 1/14/2015

Expander graphs and

arithmetic applications

Date: 1/14/2015

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor Seyfi Turkelli

Abstract: In this talk, I will first introduce expander graphs, gonality of curves and the arithmetic objects that we are interested in. Then, I will talk about several recent expansion results and their applications to number theory.

## RUME Seminar 12/4/2014

Viewing teaching and learning mathematics in K-12 schools through the lens of the Common Core State Standards

Date: 12/4/2014

Time: 4:00PM

Place: 315 Armstrong Hall

Johnna Bolyard, Associate Professor of Mathematics Education

Matthew Campbell, Assistant Professor of Mathematics Education

Abstract:

The Common Core State Standards for Mathematics put forth the recommended mathematical content and discipline-specific proficiencies to guide K-12 education. In the midst of what has become a hot-button political issue around assessment, school funding and accountability, and government control in schools, the Standards themselves provide the next step in a long line of research and policy on students' mathematical development and the educational environments that promote that development. In this talk we will provide some background on the Common Core Standards using examples from elementary and high school, highlighting both the content of the Standards as well as the focus on "mathematical practice". We will discuss implications of these new standards on K-12 mathematics teaching, teacher development and support, and on students' preparedness for college mathematics and mathematics-focused careers.

## David Swigon 11/6/2014

Dynamics of price and wealth in a

multi-group asset flow model

Date: 11/6/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor David Swigon

Abstract: A recently developed model of asset flow dynamics allows one to use

the tools of nonlinear dynamics to study various market conditions and

trading scenarios. I will present the results of two such studies. In the

first we focused on the stability of market equilibria in cases in which

investor groups follow commonly used trading strategies, such as

fundamentalist or trend-based. We show that a market comprised of

fundamental traders is always stable while the presence of trend-based (also

called momentum) traders destabilizes market equilibria and potentially

leads to market bubbles or flash crashes. In the second study we analyzed

the constant rebalanced portfolio (CRP) strategy, in which investor divides

his wealth equally between different types of assets and maintains those

proportions constant as the price changes. We show that CRP strategy is

optimal in that it minimizes the potential losses incurred during price

fluctuations in the market. We also show that any other trading strategy can

be taken advantage by other investors in the market and lead to a loss of

wealth.

## Richard Coultas 11/5/2014

K-Crossing

Free Antichains

Date: 11/5/2014

Time: 3:45PM-4:45PM

Place: 315 Armstrong Hall

Professor Richard Coultas

Abstract:

A set S of vectors in Z^w is an antichain if no vector in S has all coordinates less than or equal to the coordinates of some other vector in S, and is k-crossing free if a certain condition on the maximum difference between corresponding coordinates of different vectors is satisfied. This paper considers how large a k-crossing free antichain of vectors in Z^w can be. The problem is solved for w=3, and some partial results are given for w>3.

## C.S. Aravinda 9/30/2014

Dynamics of geodesic

conjugacies

Date: 9/30/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor C. S. Aravinda

abstract:

The question of whether a time-preserving geodesic conjugacy

determines a closed, negatively curved Riemannian manifold up to an

isometry is one of the central problems in Riemannian geometry. While an

answer to the question in this generality has yet remained elusive, we

discuss some available affirmative results.

## Dr. Dejan Slepcev 9/26/2014

Introduction to optimal transportation and Wasserstein gradient flows

Two Different Times.

1. Date: 9/26/2014

1. Time: 10:00AM-10:50AM

1. Place: Mountaineer Room, Mountainlair

2. Date: 9/26/2014

2. Time: 2:35PM-3:25PM

2. Place: Mountaineer Room, Mountainlair

Professor Dejan Slepcev

abstract:

I will cover some of the basics of the theory of optimal transportation and gradient flows in spaces of measures endowed with Wasserstein metric. The goal it to provide the background material for the geometric approaches to the PDE the workshop focuses on. In the first part I will introduce the notion of optimal transport, the Monge and Kantorovich formulations and discuss some properties of the optimal transportation maps. I will then discuss the geometry of the space of probability measures, in particular the McCann interpolation, its connection to pressureless Euler equation and the Benamou-Brenier characterization of the Wasserstein distance.

In the second part I will discuss the notion of a gradient flow in the spaces of probability measures, and in particular the characterization of some PDE like the heat equation, porous medium equation and nonlocal-interaction (a.k.a aggregation) equation as such gradient flows. Finally I will discuss the applications of this viewpoint to existence, uniqueness and stability of the solutions.

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