# Colloquia

## Dr. Hehui Wu 2/20/2012

Longest Cycles in Graphs with Given Independence Number and Connectivity.

Date: 2/20/2012

Time: 3:30-4:30 PM

Place: 422 Armstrong Hall

*Refreshments will be served at 3:00PM in 310 Armstrong Hall.

A vertex cut of a connected graph is a set of vertices whose removal renders the graph disconnected. The connectivity of a

graph is the size of the smallest vertex cut. An independent set of a graph is a set of vertices such that between any two vertices in the set,

there is no edge connecting them. The independence number is the size of the largest independent set. A cycle is spanning or Hamiltonian

if it visits all the vertices.

The Chv\'atal--Erd\H{o}s Theorem states that every graph whose connectivity

is at least its independence number has a spanning cycle. In 1976, Fouquet and

Jolivet conjectured an extension: If $G$ is an $n$-vertex $k$-connected graph

with independence number $a$, and $a \ge k$, then $G$ has a cycle of length

at least $\frac{k(n+a-k)}{a}$. We prove this conjecture. This is joint work with Suil O and Douglas B. West.

## Dr. Tuoc Phan 2/15/12

Navier-Stokes Equations in Critical Spaces: Existence and Stability of Steady State Solutions

Date: 2/15/2012

Time: 3:30-4:30 PM

Place: 315 Armstrong Hall

*Refreshments will be served at 3:00PM in 310 Armstrong Hall.

Abstract. In this talk, I will briefly derive the Navier-Stokes equations which is the most fundamental equations in fluid mechanics. I then discuss my recent results on the uniqueness existence of solutions to the stationary Navier-Stokes equations with small singular external forces belonging to a critical space. To the best of my knowledge, this is the largest critical space that is currently available for this kind of existence result. The stability of the steady state solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup. Some related results concerning the Cauchy problem for the non-stationary Navier Stokes equations will be also addressed.

The talk is based on the joint work with N. C. Phuc (LSU).

## Dr. Rong Luo 2/14/2012

Dr. Luo will present Map-coloring, Edge-coloring and Vizings Conjectures.

Date: 2/14/2012

Time: 1:30-2:30 PM

Place: 422 Armstrong Hall

*Refreshments will be served at 3:00PM in 310 Armstrong Hall.

A graph is a set of vertices and a set of edges that connect pairs of vertices. An edge coloring of a graph is an assignment of colors to the edges of the graph so that any two edges sharing a common endvertex receive different colors. Edge coloring was first studied by Tait in 1880 as an approach to attack the well-known Map 4-Coloring conjecture. Vizing’s theorem classifies the simple graphs into two classes, Class one graphs and Class two graphs. However, it is NP-hard to determine whether a graph is in Class one or two. In late 1960s, Vizing proposed several conjectures to study the “barely” Class two graphs (critical graphs). Those conjectures are fundamental problems in the area of edge coloring. In the last ten years, there are lots of progresses on those conjectures. In this talk, I will first talk about the relation between Map Coloring and Edge Coloring and then survey the progresses on Vizing’s conjectures.

## Professor Slepcev 2/13/2012

CMU Professor, Dr. Slepcev, will host a colloquium and

all are invited to attend.

Title: **Global-in-time weak measure solutions, finite-time aggregation
and confinement for nonlocal interaction equations**

Date: 2/13/2012

Time: 3:30-4:30PM

Place: 315 Armstrong Hall

I will talk about well-posedness theory for weak measure

solutions of the Cauchy problem for a family of nonlocal interaction

equations. These equations are continuum models for interacting particle

systems with attractive/repulsive pairwise interaction potentials. The main

phenomenon of interest is that, even with smooth initial data, the

solutions can concentrate mass in finite time. I will discuss the

existence, uniqueness and stability of solutions which hold even after

the blow-up time in the classical norms.

In particular, in the case of sufficiently attractive potentials,

the solutions collapse in finite time onto a single point.

Finally, compensation between the attraction at large distances

and local repulsion of the potentials, and conditions to have

global-in-time confined systems will be discussed. The approach is

based on the theory of gradient flows in the space of probability

measures endowed with the Wasserstein metric.

## Ms. Krista Toth 2/16/12

Student Understanding of the Definite Integral When Applied to Finding Volumes of Solids

Date: 2/16/12

Time: 2:30-3:30PM

Place: 422 Armstrong Hall

Abstract: Past research has shown that students struggle when solving definite integral application problems, but little has been done to examine the sources of their difficulties. This study aims to more thoroughly examine student misconceptions about definite integrals and develop new curricula to address these issues. Participants are second-semester calculus students enrolled at WVU. Past exam problems required students to sketch approximating slices of given solids, and set up a corresponding volume integral. Students’ written solutions were analyzed for common mistakes and misconceptions. Although some students solved the problems correctly, a majority exhibited major deficiencies in their understanding of how to apply the definite integral. Most surprising was students’ widespread failure to make a connection between the sketch and the set up of the integral. Further research is currently under way that aims to expose sources of students’ faulty thought processes when using definite integrals to solve volume problems.

## Dr. Charis Tsikkou 2/10/12

Dr. Tsikkou will present Conservation Laws with no Classical

Riemann Solutions: Existence of Singular Shocks.

Date: 2/10/2012

Time: 4:30-5:30 PM

Place: 315 Armstrong Hall

*Refreshments will be served at 4:00PM in 310 Armstrong Hall.

Conservation laws are the most fundamental principles of continuum

mechanics. The basic tool in the construction of solutions to the Cauchy problem for

conservation laws with smooth initial data is the Riemann problem. It consists of

piecewise constant initial data having a single discontinuity at the origin.

In this talk I will review the results obtained for the solutions to the Riemann

problem and present a system of two equations derived from isentropic gas dynamics

with no classical solution. I will then use the blowing-up approach to geometric

singular perturbation problems to show that the system exhibits unbounded solutions

(singular shocks) with Dafermos profiles.

## Dr. Nicole Engelke 12/12/2011

Dr. Engelke will present "Student Difficulties in the Production of Combinatorial Proofs"

Date: 12/12/2011

Time: 4:00-5:00 PM

Place: 315 Armstrong Hall

*Refreshments will be served at 3:30PM in 310 Armstrong Hall.

Combinatorial proof, the art of counting a set in two distinct ways to prove a statement, is a technique which emphasizes conceptual understanding of a problem and encourages creative thinking. We identify four categories of student difficulties with this type of proof, and introduce the term pseudo-semantic proof production to describe the attempt to write a combinatorial proof by relying on the syntax of previously encountered proofs. We illustrate the categories of student difficulties and pseudo-semantic proof production with three case studies drawn from a preliminary study of combinatorial proofs written by students in an upper-division combinatorics course and a graduate-level discrete mathematics course.

## Dr. Yi-Yin (Winnie) Ko 12/9/2011

Dr. Ko will present, "Undergraduate Mathematics Majors’ Performance Constructing Proofs".

Date: 12/9/2011

Time: 4:00-5:00 PM

Place: 315 Armstrong Hall

*Refreshments will be served at 3:30PM in 310 Armstrong Hall.

Current reforms highlight the importance of teaching and learning proof in undergraduate mathematics education. Undergraduate mathematics majors, including pre-service secondary mathematics teachers, are expected to have mastered the skills required to produce proofs. However, the corpus of existing literature suggests that many undergraduate mathematics students still have considerable difficulty with proof. In this talk, I will present results concerning undergraduate mathematics majors’ performance constructing proofs in the domains of algebra, analysis, geometry, and number theory. Implications of this study’s findings for undergraduate proof courses and for pre-service secondary mathematics teachers, including directions of future research, will also be addressed.

## Professor Jeremy Edwards 12/8/2011

University of New Mexico Department of Molecular Genetics and Microbiology

Professor to give a presentation titled, "Membrane Organization and the Quantitative Understanding of Cell Signaling".

Date: 12/8/2011

Time: 3:45-4:45 PM

Place: 315 Armstrong Hall

Prof. Jeremy Edwards has a remarkable record of achievement in

quantitative systems biology. He obtained his PhD (1999) in

Bioengineering at UC San Diego with Prof. Bernhard Palsson. He was

first author on a number of seminal papers in the field of metabolic

network analysis. After a postdoc in the Church Lab at Harvard, he

joined the faculty at the University of Delaware, where he started a

new line of investigation, in collaboration with Prof. Dionysios

Vlachos. They applied ideas and computational methods from catalysis

modeling to the spatial dynamics of membrane-bound receptors which in

turn modulate the first steps in cell signaling. The cell membrane is

a complex system whose structural and functional details are at the

edge of current experimental capabilities, and "big science"-grade

computational models are the only way to bring together the emerging

large and diverse sets of data.

In 2005 Prof. Edwards moved back to his home town of Albuquerque to

join the University of New Mexico (UNM) where he serves as co-director

for mathematical modeling at the New Mexico Center for Spatio-Temporal

Modeling of Cell Signaling (STMC). The STMC, one of a few NIH-NIGMS

funded Centers for Systems Biology, brings together a set of

world-class experimental labs and modelers dedicated to

the spatio-temporal aspects of cell signaling. Dr. Edwards is also a

member of the NCI-designated Cancer Center affiliated with UNM.

In addition to his work on the modeling side, Prof. Edwards is a major

figure in next generation DNA sequencing, where his lab holds several

patents. He is the main author of the method of polymerase colony

orpolony sequencing and is principal investigator on a large NIH grant

aimed at developing the "$1000 genome".

His talk will discuss some of the significant progress on

understanding signal initiation, obtained through computational

methods ranging from large scale Monte Carlo simulations to

small dynamical simulations and mathematical insight, in conjunction

with cutting edge experimental data collected at the STMC.

## Professor David Klinke 10/27/2011

West Virginia University Professor to give presentation....

Date: 10/27/2011

Time: 4:00 PM- 5:00 PM

Place: 315 Armstrong Hall

Title: In silico model-based inference: a contemporary approach for

hypothesis testing in cell signaling research

Below you can download the files that will accompany the talk.

The following compatibility pack for older versions of Microsoft Word

will allow you to open and view the DOCX file.

## Pages