# 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.

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