# Colloquia

## 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.

## 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.

## 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.

## Dr. Gesham Magombedze 4/29/2013

Mathematical modeling of host-pathogen immunological interactions and

molecular biological systems

Dr. Gesham Magombedze

Date: 4/29/2013

Time: 3:30PM-4:30PM,

Place: 315 Armstrong Hall

*Refreshments served at 3:00p in Armstrong Hall, Room 310

Abstract: Mycobacterial infections, such as Mycobacterium avium subspecies paratuberculosis which causes Johne's disease in cattle and other ruminants and Mycobacterium tuberculosis, which is the etiological agent of tuberculosis in humans, are characterized by a persistent and slow infection progression, that can be rapid under certain conditions. Mycobacterial pathogens have the ability to adapt to the changing intracellular environment in response to a dynamic immune response. The underlying mechanisms of how the bacteria can persist despite of the host mounting a robust immune response are not clearly understood. In this talk I will illustrate how mathematical models can be used to: (i) address questions arising from biological systems and (ii) provide insights in understanding some of the bacterial mechanisms associated with its persistence using a mathematical framework that integrates gene expression data and biochemical systems theory. I will present a mathematical immunological model that describes the cattle immune response mechanisms associated with mycobacterial infection progression.

## Dr. Arvind Baskaran 4/26/2013

Kinetic Density Functional Theory : A mesoscale approach to understanding the effect of flow on freezing transition

Dr. Arvind Baskaran

Date: 4/26/2013

Time: 3:30PM-4:30PM,

Place: 315 Armstrong Hall

*Refreshments served at 3:00p in Armstrong Hall, Room 310

Abstract: A clear understanding of freezing and melting is of great importance in many applications starting from growth of nano-crystals in solutions for solar cells to making the perfect ice cream. This work aims to understand how the flow of the liquid phase affects the freezing process (solid-liquid phase transition).This talk will outline the problem with the perspective of an interdisciplinary mathematician.The mathematical challenges involved in developing a solvable predictive model and developing efficient methods of solving the so developed model will be discussed.Some results demonstrating the predictive ability of the so developed approach will be discussed.

This work was done in collaboration with John Lowengrub and Aparna Baskaran

## Dr. Carlos Palenzuela 4/23/2013

Revealing the dark side of the

Universe with gravitational waves

Dr. Carlos Palenzuela

Date: 4/23/2013

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

*Refreshments served at 3p in Armstrong Hall, Room 310

Abstract:In the next few years a new generation of gravitational wave detectors will allow us to "listen" some of the most energetic events in the universe; the coalescence of binary compact objects such as black holes and/or neutron stars. If the magnetic field around these objects is sufficiently strong, the binary may produce an electromagnetic burst which we may be able to "see", especially if it is in the form of a collimated jet. A new era of multi-messenger astronomy, involving detections of EM, GW and possible neutrino signals, will provide more insight into the physical processes involved in the collisions. After giving an overview on gravitational waves and the underlying equations, I will describe the dynamics of compact binary mergers, focusing on the gravitational waveforms and the possible EM counterparts from these systems.

## Dr. Casian Pantea 4/22/2013

Geometry of interaction networks: dynamics from structure

Dr. Casian Pantea

Date: 4/22/2013

Time: 3:30PM-4:30PM,

Place: 315 Armstrong Hall

*Refreshments served at 3:00p in Armstrong Hall, Room 310

Abstract: The behavior of biological interaction networks (such as networks of biochemical reactions, infectious diseases within a population, or species in an ecosystem) is commonly modeled using intricate systems of differential equations. These models usually contain a large number of parameters whose values are rarely known in practice. However, even when no information regarding parameter values is available, wide classes of interaction networks can be shown to have surprisingly stable behavior, induced by the topology of the network alone. In this talk I will consider some of the problems and results of chemical reaction network theory, a body of work which attacks the question: "What behaviours of an interaction network are a function of its structure, and are robust to different choices of parameters?" In particular, I will discuss recent results relating the topology of a network with the possibility for Hopf bifurcations in the corresponding ODE system.

## Professor Yue Zhao 4/19/2013

On Edge Chromatic

Critical Graphs

Date: 4/19/2013

Time: 3:30PM-4:30PM,

Place: 315 Armstrong Hall

Professor Yue Zhao

Abstract: Around 1965, Vizing proposed four conjectures about edge

chromatic critical graphs. Since then, many people have been working on

these conjectures. But these four conjectures remain open. In this talk, we will present some results on these four conjectures.

## Dr. Scott Noble 4/18/2013

Computing PDEs to Discover the Unexpected: Predicting Light Signatures of Black Holes

Dr. Scott Noble

Date: 4/18/2013

Time: 1:00PM-2:00PM,

Place: 320 Armstrong Hall

*Refreshments served at 12:30 in Armstrong Hall, Room 310

Abstract: Even though binary black hole (BBH) systems are expected to come in a wide range of masses, only the mergers of supermassive black holes at the centers of galaxies are expected to live in gas-rich environments. The presence of matter opens up the possibility that gravitational aspects of the binary's interaction can be transmitted electromagnetically to distant observers via dissipation of gas motion. Matching theoretical predictions to observations of systems before and after merger has the potential to improve our estimates of merger rates, and tell us about the spin and mass distributions of supermassive black holes. Seeing the light from the precise moment of merger---if such a robust signature exists---presents us with additional information such as more evidence that black holes merge, how material behaves in the strong-field dynamical regime of gravity, and a new and independent class of redshift-distance measurements if found with accompanying gravitational radiation. All of these exciting possibilities require realistic predictions for how magnetized gas responds to a BBH evolution. Therefore, realistic, accurate magnetohydrodynamics simulations using Einstein's theory of General Relativity must be performed. Such calculations require the numerical solution of the partial differential equations that describe the gravity and matter dynamics. The accuracy of the solution demands using state-of-the-art computational techniques and massive supercomputing resources. In this talk, I will survey what we know about accreting single black hole systems, to gain an understanding of what we may expect through a simpler, better known problem. Then, I will provide a theoretical introduction to the topic and highlight key aspect of the numerical methods we employ. The results from our first steps on this new campaign will be presented, including the prediction of a nontrivial electromagnetic period signal from an orbiting binary black hole. We will show how this periodic signal could be used to determine properties of the orbit. I will then conclude with a few ideas we have for future work on this endeavor.

Dr. Noble is currently an Associate Research Scientist at the Center for Computational Relativity and Gravitation (CCRG) at the Rochester Institute of Technology and is a candidate for a position in the Department of Mathematics.

## Professor James Sellers 4/17/2013

On Euler’s Theorem Relating Odd-Part and Distinct-Part Partitions

Professor James Sellers

Date: 4/17/2013

Time: 4:00PM-5:000PM,

Place: 315 Armstrong Hall

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