# Colloquia

## Professor Matthew Johnston 2/6/2014

Correspondence of Standard and

Generalized Mass Action Systems

Professor Matthew Johnston

University of Wisconsin-Madison

Date: 2/6/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract:

Correspondence of Standard and Generalized Mass Action Systems

Under suitable modeling assumptions, the dynamical behavior of interacting chemical systems can be modeled by systems of polynomial ordinary differential equations known as mass action systems. It is a surprising result of the analysis of such systems that many dynamical properties often follow from the structure of the reaction graph of the network alone. That is to say, we may often conclude things about the system's long-term behavior, steady state properties, and persistence of chemical species without even writing down the governing differential equations.

In this talk, we will investigate systems where such graph-based correspondence of dynamics may not be made directly, but for which the original mass action system may be corresponded to a "generalized" mass action system satisfying certain properties. In particular, the constructed generalized mass action system will contain different monomials than implied by the chemistry of the system, but will have a "well-structured" reaction graph. We will also discuss some of the newest results regarding the algorithmic construction of such generalized mass action systems.

## Professor Helge Kristian Jenssen 2/4/2014

Global solutions of

conservative hyperbolic systems

Professor Helge Kristian Jenssen

The Pennsylvania State University

Date: 2/4/2014

Time: 2:30PM-3:30PM

Place: G27 Eiesland Hall

Abstract:

We discuss the problem of providing an existence theory

for the initial value problem for systems of conservation laws in one

spatial dimension.

We shall first review the two methods currently available for general

systems:

(1) wave interactions and BV compactness (Glimm's theorem);

(2) compensated compactness (applicable to systems of two equations).

Neither of these cover "large'' initial data for systems of three or more

equations, such as the Euler system for compressible gas dynamics.

We will report on recent works that illustrate obstructions for large

data results for the specific case of isentropic gas flow.

## Dr. Stefan Mueller 11/21/2013

Sign conditions for injectivity and

surjectivity of generalized

polynomial maps

Dr. Stefan Mueller

Austrian Academy of Sciences

Date: 11/21/2013

Time: 3:30 - 4:30

Place: 315 Armstrong Hall

Abstract:

We characterize the injectivity of families of generalized polynomial maps in terms of sign vectors. The term "generalized" indicates that we allow polynomials with real exponents, which define maps on the positive orthant. Our work relates to and extends existing injectivity conditions expressed in terms of determinants. Moreover, we provide sign conditions for surjectivity which allow a generalization of Birch's theorem.

As one application, we give conditions for precluding multiple steady states in chemical reaction networks with power-law kinetics, for precluding multiple "special" steady states, for guaranteeing two distinct special steady states in some compatibility class for some rate constants, and, finally, for existence and uniqueness of special steady states in every compatibility class for all rate constants.

## Professor Maya Mincheva 11/19/2013

Dynamic instabilities of

biochemical reaction networks

Dr. Maya Mincheva

Northern Illinois University

Date: 11/19/2013

Time: 2:30 - 3:30

Place: 315 Armstrong Hall

Abstract:

Interactions of complex networks of genes, proteins and enzymes play a central role in modern cellular biology.

The talk will focus on mathematical models for biochemical reaction networks, which are usually modeled as

large systems of coupled nonlinear dierential equations with many unknown parameters. First we will give

overview of the history of the problem. Then we will describe current eorts to relate the dynamic behavior

of the dierential equations models to the topology of the corresponding biochemical networks. Examples

of models of multistability, oscillations and Turing instability (pattern formation) will be given.

## Dr. Rosemary Braun 10/29/2013

Hearing the Shape of Life: A Spectral Graph Theoretic Approach

for the Analysis of Gene Regulatory Networks

Dr. Rosemary Braun

Date: 10/29/2013

Time: 3:30 - 4:30

Place: 315 Armstrong Hall

Abstract:

High-throughput genome-wide assays have become a ubiquitous tool

of modern biology, yielding detailed measurements of the genetic

sequence, epigenetic modifications, and expression level of thousands

of genes in each sample. However, because most phenotypes of

interest are complex (i.e., driven by networks of interactions

rather than individual genes), there is a need for analytical

techniques that can articulate differences between samples based

on multi-gene expression profiles and reveal systems-level differences

in gene regulation dynamics in the context of known interaction

networks (pathways).

In this talk, I describe a novel method for identifying altered

gene regulatory networks by overlaying experimental measurements

onto the putative topology of the pathway of interest and computing

the graph Laplacian for the resulting graph. Just as the Laplacian

of a physical system (eg, the shape of a drumhead) can be used to

infer dynamical properties (its sound), spectral decomposition of

the graph Laplacian provides a means by which dynamical properties

of the network may be summarized; here, we use the spectrum of the

pathway network to summarize its bulk co-expression behavior.

Spectral graph theory is further used to characterize the effect

of individual gene expression values on the network dynamics. I

will describe the method in detail and demonstrate how our spectral

pathway analysis approach can identify significantly altered systems

without relying on single-gene association statistics, thereby

enabling more accurate classification of samples and yielding

detailed characterizations of the interaction networks that play a

role in the phenotypes of interest. I will also discuss how this

approach may be used to infer dynamical properties of biological

pathways based on "snapshot" gene expression measurements.

## Professor Mike Plummer 10/10/2013

Matching Extension in Graphs

Embedded in Surfaces

Professor Mike Plummer

Date: 10/10/2013

Time: 4:30pm -5:30pm

Place: 315 Armstrong Hall

## Dr. Danut Arama 10/7/2013

Absolute minimizers for some

Optimal Transport problems

Dr. Danut Arama

Abstract: The Monge-Kantorovich optimal transportation problem is to move a hill into an excavation using the minimum amount of energy. Therefore, the cost functional is usually represented by an integral. However, there is a practical interest in the study of transportation problems with a cost functional of the "ess-sup" type. In my talk I will present such a problem and some of the progress we made.

Date: 10/7/2013

Time: 3:30pm -4:30pm

Place: 315 Armstrong Hall

## Professor Petronela Radu 9/18/2013

Math in the City – A model for a project

based Learning Experience

Professor Petronela Radu

Date: 9/18/2013

Time: 4:00pm -5:00pm

Place: 315 Armstrong Hall

Abstract: The Math in the City course designed at University of Nebraska-Lincoln provides students with an opportunity to see how mathematics is used outside academia, and also to become more engaged with activities of businesses and research centers around the city of Lincoln, NE. During the seven offerings of the course we have seen that such experiences are beneficial to undergraduates on professional as well as personal levels; they learn how to set up and analyze mathematical models based on real world data, they learn how to find ``missing" information and how to account for it. Students also improve their communication skills (both oral and written), they learn how to better work in groups, how to meet deadlines, and many other soft skills that will help them on the job market and in the workplace.

## Professor Anthony Evans 9/6/2013

A class of orthogonal

latin square graphs

Professor Anthony Evans

Date: 9/6/2013

Time: 3:45PM-4:45PM

Place: 315 Armstrong Hall

An \emph{orthogonal latin square graph} is a graph whose

vertices are latin squares of the same order, adjacency being synonymous

with orthogonality. We are interested in orthogonal latin square graphs in

which each square is orthogonal to the Cayley table $M$ of a group $G$ and

is obtained from $M$ by permuting columns. These permutations, regarded as

permutations of $G$, are \emph{orthomorphisms} of $G$ and the graphs so

obtained are \emph{orthomorphism graphs}.

We will discuss results and problems in the study of orthomorphism graphs

## Professor Anant Godbole 8/22/2013

A Survey of Old and New Results on Universal Cycles

Date: 8/22/13 Thursday Friday

Time: 4:30PM-5:30PM

Place: Armstrong 315

Professor Anant Godbole

Department of Mathematics and Statistics

East Tennessee State University

Abstract:

A fundamental result in graph theory is that a connected graph with even

vertex degrees is Eulerian. For digraphs, we can state the result as

follows: A digraph, with at most one weakly connected component, for with

the indegree $i(v)$ of any vertex $v$ equals its outdegree $o(v)$, is

Eulerian. This result led to deBruijn proving the result that bears his

name, about universal cycles of words of arbitrary length from an

arbitrary alphabet; for example, the cyclic string 11101000 contains each

binary three-letter word exactly once as a substring. Chung, Diaconis and

Graham's landmark paper extended the notion of universal cycles to

subsets, partitions, and permutations. This talk will focus on further

progress, much of which has been made by students at my REU program. I

will present results on universal and overlap cycles of restricted words,

graphs, hypergraphs, matroids, venn diagrams, posets, subsets, words of

fixed weight, complementary classes, and juggling patterns.

## Pages