# Colloquia

## Professor Jie Ma 4/16/2014

The maximum number of proper

colorings in graphs with fixed

numbers of vertices and edges

Jie Ma

Date: 4/16/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract:

We study an old problem of Linial and Wilf to find the graphs

with n vertices and m edges which maximize the number of proper q-colorings

of their vertices. In a breakthrough paper, Loh, Pikhurko and

Sudakov asymptotically reduced the problem to an optimization problem. We

prove the following result which tells us how the optimal solution must

look like: for any instance, each solution of the optimization

problem corresponds to either a complete multipartite graph or a graph

obtained from a complete multipartite graph by removing certain edges. We

then apply this result to general instances, including a conjecture of

Lazebnik from 1989 which asserts that for any q>=s>= 2, the Turan graph

T_s(n) has the maximum number of q-colorings among all graphs with the same

number of vertices and edges. We disprove this conjecture by providing

infinity many counterexamples in the interval s+7 <= q <= O(s^{3/2}). On

the positive side, we show that when q= \Omega(s^2) the Turan graph indeed

achieves the maximum number of q-colorings. Joint work with Humberto Naves.

## Professor Martin Feinberg 4/7/2014

An Introduction to Chemical

Reaction Network Theory

Professor Martin Feinberg

Date: 4/7/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract: Here

## Professor John Tyson 3/6/2014

Irreversible Transitions,Bistability and Checkpoints in the Eukaryotic

Cell Cycle

Professor John Tyson

Date: 3/6/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract: Here

## Professor Zach Etienne 2/28/2014

Throwing in the Kitchen Sink: Adding Mixed Type PDEs to Better Solve Einstein's Equations

Professor Zach Etienne

Date: 2/28/2014

Time: 2:30PM-3:30PM

Place: 315 Armstrong Hall

Abstract:

With the first direct observations of gravitational waves (GWs) only a few years away, an exciting new window on the Universe is about to be opened. But our interpretation of these observations will be limited by our understanding of how information about the sources generating these waves is encoded in the waves themselves. The parameter space of likely sources is large, and filling the space of corresponding theoretical GWs will require a large number of computationally expensive numerical relativity simulations.

Numerical relativity (NR) solves Einstein's equations of general relativity on supercomputers, and with the computational challenge of generating thoeretical GWs comes an arguably even greater mathematical one: finding an optimal formulation of Einstein's equations for NR. These formulations generally decompose the intrinsically 4D machine of Einstein's equations into a set of time evolution and constraint equations---similar to Maxwell's equations of electromagnetism. Once data on the initial 3D spatial hypersurface are specified, the time evolution equations are evaluated, gradually building the four-dimensional spacetime one 3D hypersurface at a time. We are free to choose coordinates however we like in this 4D manifold, which are generally specified on each 3D hypersurface via a set of coordinate gauge evolution equations.

Robust coordinate gauge evolution equations are very hard to come by and are critically important to the stability and accuracy of NR simulations. Most of the NR community continues to use the highly-robust---though nearly decade-old---"moving-puncture gauge conditions" for such simulations. We present dramatic improvements to this hyperbolic PDE gauge condition, which include the addition of parabolic and elliptic terms. The net result is a reduction of numerical errors by an order of magnitude without added computational expense.

## Professor David Offner 2/26/2014

Polychromatic colorings

of the hypercube

Professor David Offner

Date: 2/26/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract: Here

## 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.

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