# Colloquia

## Dr. Stefan Mueller 11/21/2013

Sign conditions for injectivity and
surjectivity of generalized
polynomial maps

Dr. Stefan Mueller

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.

Date, Location:
2013-21-11

## 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.

Date, Location:
2013-11-19

## 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.

Date, Location:
2013-10-29

## 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

Abstract

Date, Location:
2013-10-10

## 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

Date, Location:
2013-10-07

Math in the City – A model for a project
based Learning Experience

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.

Date, Location:
2013-09-18

## 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

Date, Location:
2013-09-06

## 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.

Date, Location:
2013-08-22

## Dr. Zach Etienne 5/3/2013

Solving the Einstein-Maxwell Equations to Model the Most Energetic Outbursts in the Universe

Dr. Zach Etienne

Date: 5/3/2013
Time: 4:00PM-5:00PM,
Place: 315 Armstrong Hall
*Refreshments served at 3:30p in Armstrong Hall, Room 310

Abstract: Black hole--neutron star (BH-NS) and black hole--black hole (BH-BH) binary systems are among the most promising candidates for gravitational waves (GWs) detectable by Advanced LIGO and NANOGrav. In addition, spectacular and observable electromagnetic (EM) radiation may accompany these GWs. However, the knowledge we gain from such "multi-messenger" systems will be limited by our theoretical understanding of them. To improve this understanding, we have developed state-of-the art numerical relativity codes designed to model these systems. The codes solve a strongly hyperbolic formulation of Einstein field equations and a conservative formulation of the GRMHD equations for the magnetized plasma that preserves the critically-important divergenceless condition for the magnetic field. Our simulations were the first to demonstrate that BH-NS binary mergers not only produce GWs detectable by Advanced LIGO, but also under the right conditions may provide the engine for launching a gamma-ray burst. Our BH-BH-disk simulations, in which the binary is surrounded by a tenuous, magnetized accretion disk, were the first to show that when the binary is widely separated, a jet of EM energy from the binary forms, with the luminosity peaking at merger. We intend to consider binaries at even larger separations in the future to determine what EM signatures might be observed when the emitted GWs fall in the NANOGrav band. I will discuss the new algorithms we developed to perform these simulations, their mathematical underpinnings, and our latest results.

Date, Location:
2013-05-03

## Professor Brian Alspach 5/14/2013

The Coxeter Group
Project: A Progress Report

Professor Brian Alspach

Date: 5/14/2013
Time: 10:30AM-11:30AM,
Place: 315 Armstrong Hall

Abstract: If one forms a Cayley graph on the symmetric group using only
transpositions for the connection set, then the graph is bipartite. In
2006, Araki proved that if such a Cayley graph is connected, then for any
two vertices u and v in different parts, there is a Hamilton path whose
terminal vertices are u and v. Of course, the symmetric group is a
Coxeter group. The speaker has embarked on a project of extending Araki's
Theorem to Cayley graphs on other families of Coxeter groups. This talks
is a summary of what has been achieved so far.

Date, Location:
2013-04-30