# Colloquia

## Dr. Leobardo Rosales 2/24/2012

The single and two-valued minimal surface equation.

Date: 2/24/2012

Time: 3:30-4:30 PM

Place: 315 Armstrong Hall

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

The two-valued minimal surface equation is a degenerate PDE

used to produce examples of stable minimal immersions with branch

points. In this talk, drawing analogies from the minimal surface

equation, we investigate questions of existence, regularity, and

rigidity of solutions.

## Dr. Kevin Milans 2/22/2012

Sparse Ramsey Hosts

Date: 2/22/2012

Time: 3:30-4:30 PM

Place: 422 Armstrong Hall

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

In Ramsey Theory, we study conditions under which every partition of a large structure yields a part with additional structure. For example, Van der Waerden's theorem states that every s-coloring of the integers contains arbitrarily long monochromatic arithmetic progressions, and the Hales--Jewett Theorem guarantees that every game of tic-tac-toe in high dimensions has a winner. Ramsey's Theorem implies that for any target graph G, every s-coloring of the edges of some sufficiently large host graph contains a monochromatic copy of G. In Ramsey's Theorem, the host graph is dense (in fact complete). We explore conditions under which the host graph can be sparse and still force a monochromatic copy of G.

We write H→sG if every s-edge-coloring of H contains a monochromatic copy of G. The s-color Ramsey number of G is the minimum k such that some k-vertex graph H satisfies H→sG. The degree Ramsey number of G is the minimum k such that some graph H with maximum degree k satisfies H→sG. Chvátal, Rödl, Szemerédi, and Trotter proved that the Ramsey number of bounded-degree graphs grows only linearly, sharply contrasting the exponential growth that generally occurs when the bounded-degree assumption is dropped. We are interested in the analogous degree Ramsey question: is the s-color degree Ramsey number of G bounded by some function of s and the maximum degree of G? We resolve this question in the affirmative when G is restricted to a family of graphs that have a global tree structure; this family includes all outerplanar graphs. We also investigate the behavior of the s-color degree Ramsey number as s grows. This talk includes results from three separate projects that are joint with P. Horn, T. Jiang, B. Kinnersley, V. Rödl, and D. West.

## Dr. Flor Espinoza Hidalgo 2/21/2012

Analysis of the Organization and Dynamics of Proteins in Cell Membranes

Date: 2/21/2012

Time: 3:30-4:30 PM

Place: 315 Armstrong Hall

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

Cells communicate with the outside world through membrane receptors that recognize one of many possible stimuli (hormones, antibodies, peptides) in the extracellular environment and translate this information to intracellular responses. Changes in the organization and composition of the plasma membrane are critical to transmembrane signal transduction, so there is great interest in understanding the organization of membrane proteins in resting cells and in tracking their dynamic reorganization during signaling. Problems in signaling networks are important in understanding many diseases including cancer and asthma.

Our protein of interest is the IgE high affinity receptor FceRI, found in mast cells and basophils. The activation of this receptor starts when IgE bound to FceRI is crosslinked by multivalent antigens, initiating a tyrosine kinase signaling cascade that triggers histamine release and other preformed inflammatory mediators that are stored in cytoplasmic granules.

In this talk, we present some results of our analysis of biological data on the distribution and mobility of this receptor during signaling. The data analyzed are from Janet Oliver’s Lab (STMC). First, we will present the results of our clustering analysis of high resolution electron microscopy images (static data). We focus on the analysis of the organization of the IgE-FceRI after crosslinking with the multivalent antigen, DNP-BSA. The data were generated using gold particles of size 5nm as labels to identify the location of the receptors in RBL-2H3 mast cell membranes at fixed times after stimulation. In the clustering analysis we used the dendrogram command from Matlab in our hierarchical clustering and dendrogram algorithm (HCDA). This algorithm gives an intrinsic distance number, that provides the distance for the maximum number of clusters in the biological data. Then, we compare this number to the number provided by randomly generated data for the same number of receptors in each experiment. This ratio is called the clustering ratio. It is this ratio that quantifies clustering. The HCDA algorithm also provides, number of clusters and sizes, giving more detailed information about the data. Next, we will present the analysis of real time fluorescence microscopy data (dynamic data), that track the temporal behavior of IgE-FceRI after being stimulated by different doses of the multivalent antigen, DNP-BSA. These

data were generated using quantum dots (QD) of sizes 5-10nm as labels to track the positions of the receptors in time in RBL-2H3 mast cell membranes. One of the restrictions of QDs is that they blink. As a result, the data sets have missing positions. Our analysis of dynamic data algorithm (ADDA) takes cares of this limitation. For these data, we analyzed changes in the standard deviation of the jump lengths and quantified changes in jump lengths with different stimulus.

## 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.

## 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.

## 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.

## 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.

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