# Colloquia

## 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).

## Professor Hao Shen 10/29/2015

Resolvable group divisible designs and (k,r)-colorings of complete graphs

**Date:** 10/29/2015**Time:** 4:30PM-5:30PM**Place:** 315 Armstrong Hall

Hao Shen

Abstract: Let k and r be given positive integers, a

(k,r)-coloring of a complete graph K is a coloring of the edges of K with r colors such that all monochromatic connected subgraphs have at most k vertices. The Ramsey number f(k,r) is defined to be the smallest u such that the complete graph with u vertices does not admit a (k,r)-coloring.

A group divisible design is called resolvable if all the blocks can be partitioned into parallel classes. In this talk, we will introduce the known results on the existence of resolvable group divisible designs and their applications in the study of (k,r)-colorings of complete graphs.

## Professor Carsten Conradi 10/1/2015

Steady states of polynomial ODEs arising in biology with application

to multisite phosphorylation

**Date:** 10/1/2015**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Carsten Conradi

Abstract:Polynomial Ordinary Differential Equations are an important tool in many areas of quantitative biology. Due to high measurement uncertainty, few experimental repetitions and a limited number of measurable components, parameters are subject to high uncertainty and can vary in large intervals. One therefore effectively has to study families of parametrized polynomial ODEs. In this talk a class of ODEs is discussed, where the steady states can be parametrized by solutions of parameter independent linear inequality systems. To this class belong, for example, multisite phosphorylation systems. For a special instance of this subclass, one can formulate parameter conditions that guarantee the existence of three steady states.

## Professor Alan Rendall 9/29/2015

Sustained oscillations in phosphorylation cascades

**Date:** 9/29/2015**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Alan Rendall

Abstract:Signalling networks are sets of chemical reactions used to transmit information

in living cells. One pattern frequently encountered in this context is that of a

phosphorylation cascade, where phosphate groups are added to proteins in

successive stages. In this talk I report on work with Juliette Hell on the existence

of periodic solutions in systems of ODE modelling a key example of

a cascade of this type, the MAP kinase cascade. The mathematical tools used

for this are bifurcation theory and geometric singular perturbation theory.

I will also describe the relation of these results to the idea that oscillations

are often related to negative feedback loops, where the feedback may arise

in an implicit way due to sequestration effects.

## Professor Martha Alibali 9/28/2015

Defining and Measuring Conceptual Knowledge of Mathematics

**Date:** 9/28/2015**Time:** 3:30PM-4:30PM**Place:** 121 Armstrong Hall

Martha Alibali

Abstract:Both researchers and educators recognize the importance of conceptual knowledge in mathematics. However, it has proven difficult to identify and measure conceptual knowledge in many mathematical domains. This talk provides an overview of research on conceptual knowledge in the literature on mathematical thinking. I discuss (1) how conceptual knowledge is defined in the mathematical thinking literature, broadly speaking, and (2) how conceptual knowledge is defined, operationalized, and measured in three specific mathematical domains: equivalence, cardinality, and inversion. This review uncovers several shortcomings in this body of literature, most notably a lack of consistency in definitions of conceptual knowledge and a lack of alignment between definitions and measures. To address these issues, I propose a general framework that divides conceptual knowledge into two facets: knowledge of general principles and knowledge of the principles underlying procedures.

## Professor Bing Wei 9/25/2015

Hamiltonian properties, branch number and k-tree related graphs

**Date:** 9/25/2015**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Bing Wei

Abstract: Download Here

## Professor Suohai Fan 8/13/2015

On $r$-hued colorings

of graphs

**Date:** 8/13/2015**Time:** 2:30PM-3:20PM**Place:** 315 Armstrong Hall

Suohai Fan

Abstract:For integers $k, r > 0$, a $(k, r )$-coloring of a graph $G$

is a proper $k$-coloring $c$ such that for any vertex $v$ with degree

$d(v)$, $v$ is adjacent to at least

min$\{d(v),r\}$ different colors. Such coloring is also called as an $r$-hued

coloring. The {\it $r$-hued chromatic number} of $G$, $\chi_{r}(G)$, is the least integer

$k$ such that a $(k, r )$-coloring of $G$ exists. In this talk, we will present some

of the progresses in this area.

## Michael Wester 4/29/2015

Determining the Parameters in Spatially Resolved Models of the

Motion of Proteins in the Membranes of Stimulated Cells

**Date:** 4/29/2015**Time:** 3:30PM-4:30PM**Place:** 315 Armstrong Hall

Michael Wester

Abstract: We show how to compute a dimerization rate from stimulated cell diffusion

data which can be used in a spatially resolved stochastic simulator to

accurately reproduce the data. For our data and for strong stimuli,

the time dependent diffusion coefficient rapidly transitions from the

diffusion constant for unstimulated cells to a significantly smaller

value. The diffusion data is generated using sparse labeling with

quantum dots which allows us to analyze using a non-spatial system

of two linear differential equations. We then use a closely related

system of two non-spatial nonlinear differential equations to compute a

spatially resolved reaction rate to use in our simulation code. Using

these reaction rates we successfully reproduce the biological data.

The framework developed here can be extended to analyze other time

dependent diffusion data.

## John Thompson 4/15/2015

Investigating student understanding and application of mathematics needed in physics: Definite integrals and the Fundamental Theorem of Calculus

**Date:** 4/15/2015**Time:** 4:30PM-5:30PM**Place:** 422 Armstrong Hall

John Thompson

Abstract: Learning physics concepts often requires the ability to interpret and manipulate the underlying mathematical representations and formalism (e.g., equations, graphs, and diagrams). Physics students are expected to be able to apply mathematics concepts to find connections between various physical quantities that are related via derivatives and/or integrals. Our own research into student conceptual understanding of physics has led us to investigate how students think about and use prerequisite, relevant mathematics, especially calculus, to solve physics problems. This is a rapidly growing research area in physics education.

Based on responses to questions administered in thermodynamics, we developed or adapted questions related to definite integrals and the Fundamental Theorem of Calculus (FTC), specifically with graphical representations, that are relevant in physics contexts, including some integrals that result in a negative quantity. Questions were administered in written form and in individual interviews; some questions had parallel versions in both mathematics and physics. Eye-tracking experiments provided additional information on visual attention during problem solving. Our findings are consistent with much of the literature in undergraduate mathematics education; we also have identified new difficulties and reasoning in students’ responses to the given problems.

## 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.

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