Colloquium

Research seminars

VIGRE Working Groups

Clifford lectures

Speaker, Institution

"TBA"

Abstract:

TBA

Karl H. Hofmann, Technische Universität Darmstadt and Tulane University

"Bourbaki at T and T:  A historical reflection on the Bourbaki impact around 1955-1965 by sampling the University of  Tübingen and Tulane University "

Abstract:

The Bourbaki movement, which started in France in the nineteen thirties and forties, had considerable influence on the style and the direction of mathematics and mathematicians in the second half of the last century.  It appears to be interesting to note the difference in the reception of Bourbaki program in various countries. Since I know German universities, notably Tübingen (which was rather prominent in mathematics after WWII), and since I have known Tulane for some time, I venture to offer some observations on the historical plane by looking back at both Tübingen and Tulane around the middle of the last century. Obviously, this is meant to be a non-technical presentation which imposes no prerequisites on the audience.

Speaker, Institution

"Title"

Tim Elston, University of North Carolina

"Mathematical modeling of intracellular signaling pathways"

Abstract:

Cells most respond to a constantly changing environment. Information about their surroundings is transmitted through signaling pathways that receive external cues and initiate the appropriate response. The complexity of these signaling pathways has led to use of mathematical and computational approaches to understand how cells receive and process information. The first part of this talk provides an introduction to intracellular signal transduction and some of the key biological questions in this field. In particular, the idea of pathway specificity is introduced. Pathway specificity refers to the cell's ability to generate appropriate responses to distinct external cues even though the underlying signaling pathways share common components. Next results demonstrating how mathematical modeling has been used to elucidate the mechanisms responsible for pathway specificity are presented. Finally, a developmental decision that is mediated by the yeast mating response pathway is discussed. Depending on the concentration of pheromone yeast either undergo chemotrophic growth toward a mating partner or initiate a mating response. Mathematical modeling is used to provide insight into how information about the pheromone concentration is encoded and transmitted through this pathway to ensure the appropriate transcriptional program is followed.

Meijun Zhu, University of Oklahoma

"Sharp Sobolev inequalities: from historic and geometric view points"

I shall review the history of the study of sharp Sobolev inequalities on Rⁿ (thus Sⁿ) (back to the early work of Hardy and Littlewood in 1929, including the story of the Bliss's Lemma), and describe their relation to the Yamabe problem and sharp Sobolev inequalities on manifolds. These motivates us to obtain the local sharp inequalities, which yield, among other results, a much simpler proof of the Onofri inequality.

I shall also explain that these sharp inequalities on Sⁿ often encoded in certain geometric flow problems. In fact, the steady state metrics of those flows are usually described by the sharp inequalities. Our current project on the study of low dimensional geometric flows will be described.

Timothy Alvin Thornton, University of California, Berkeley

"Case-Control Association Testing with Incomplete Genealogy"

Abstract:

We consider the problem of testing for association between a complex trait and a genetic marker in a case-control design in which some individuals are related. Using related individuals in case-control studies has compelling advantages. When related individuals are included in a study, correlations among relatives must be taken into account to ensure validity of the test. We first give an overview of proposed methods when the genealogy of individuals in a study is completely specified. We then consider the case when the genealogy is incomplete and present a new approach to this problem.

Speaker, Institution

"Title"

John Mayer , UAB

"Outside Looking In! Characterizing an Indecomposable Plane Continuum From Its Complement"

Abstract:
t is a long-standing open question whether the Julia set of some rational function is an indecomposable continuum. By definition, a continuum is iff it is not the union of two of its proper subcontinua. There are several ways of recognizing intrinsically that a continuum X is indecomposable. For instance, X is indecomposable if and only if every proper subcontinuum of X is nowhere dense in X. We provide a condition for testing whether the Julia set of a rational function is an indecomposable continuum using data from its complement. In the complex dynamics context, that means we are investigating the (possibly topologically and dynamically complicated) Julia set from the point of view of the (always topologically and dynamically simple) Fatou set.

To state our theorem we need to define some terms.  A generalized crosscut of a complementary domain U is an open arc  such that . Let U be a plane domain and A a generalized crosscut of U.  We call each of the two components of U\A a crosscut neighborhood.  If V is a crosscut neighborhood determined by generalized crosscut A, we call the continuum a shadow of A.  A sequence  of (not necessarily distinct) complementary domains of a continuum X satisfies the double-pass condition iff, for any sequence of generalized crosscuts  of , there is a sequence of shadows  of  such that. We prove the following

Characterization Theorem: A plane continuum X is indecomposable iff X has a sequence  of Xcomplementary domains satisfying the double-pass condition.

Recently, Clinton Curry has extended the Characterization Theorem (with an appropriately modified definition of generalized surface crosscut) to continua in compact surfaces.

Co-Authors: Clinton P. Curry (University of Alabama at Birmingham) and E. D. Tymchatyn (University of Saskatchewan)

Sinai Robins , Temple University

"A solid angle theory for real polytopes"

We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. This theory captures a new measure of volume, which is a kind of discrete volume of polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation parameters.

One of the main results is an extension of Macdonald's solid angle quasipolynomial for rational polytopes to a real analytic function of the dilation parameter, for any convex polytope whose vertices have arbitrary real coordinates. Some of this work is joint with my student David DeSario. I'll present some computer graphics to illustrate the ideas more clearly.

Marc Chamberland , Grinnell College

"The Computer's Role in Mathematical Discovery and Proof"

The use of computer packages has brought us to a point where the computer can be used for many tasks: discover new mathematical patterns and relationships, create impressive graphics to expose mathematical structure, falsify conjectures, confirm analytically derived results, and perhaps most impressively for the purist, suggest approaches for formal proofs. This is the thrust of experimental mathematics. This talk will give some examples to discover or prove results concerning geometry, integrals, binomial sums, dynamics and infinite series.

Javier Rojo, Rice University, Department of Statistics, Rice University.

"Tail inference for probability distribution functions"

Abstract:

After presenting a review of some concepts of tail-ordering, tail-heaviness, and tail categorization of probability distributions, methodology for testing for medium-tailed distributions against either small- or long-tailed distributions is presented and its operating characteristics examined.

Nick Ercolani, University of Arizona

"A Variational Theory for Defects in Patterns"

Joceline Lega, University of Arizona

"Molecular dynamics simulations of live particles"

Abstract:

I will show results of molecular dynamics simulations of hard spheres with non-classical collision rules. In particular, I will focus on how local interactions at the microscopic level between these particles can lead to large-scale coherent dynamics at the mesoscopic level.

This work is inspired by collective behaviors, in the form of vortices and jets, recently observed in bacterial colonies. I will start with a brief summary of basic experimental facts and review a hydrodynamic model developed in collaboration with Thierry Passot (Observatoire de la Cote d'Azur, Nice, France). I will then motivate the need for a complementary approach that includes microscopic considerations, and describe the principal computational issues that arise in molecular dynamics simulations, as well as the standard ways to address them.

Finally, I will discuss how classical collision rules that conserve energy and momentum may be modified to describe ensembles of live particles, and will show results of numerical simulations in which such rules have been implemented. Randomness, included in the form of random reorientation of the direction of motion of the particles, plays an important role in the type of collective behaviors that are observed.

Rafael Irizarry, Johns Hopkins

"Statistics for the Genomics Revolution: Some Examples"

The completion of the human genome project has revolutionized biology and science in general. The information gained from this project along with powerful new technologies has led to de so-called Genomics Revolution and has transformed biology to be a much more quantitative science. Mathematical models have been extremely important in physics, chemistry, and, to some extent, biology. However most biological processes of interest are far too complex for these models to be useful. The impact of Statistics on the Genomics Revolution has been through the development of powerful data analysis techniques, many of which rely on empirically motivated models. In this talk I will give some examples of useful tools developed to aid in the analysis of data produced by one of the most widely used technologies: microarrays.

Thanksgiving Holiday

Stilian A Stoev, University of Michigan

"Max-stable processes: representations, ergodic properties and
some statistical applications"

Max-stable stochastic processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this talk, various representations of max-stable processes will be discussed. Then, in terms of these "spectral" representations, necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes will be presented.The large classes of moving maxima and mixed moving maxima processes are shown to be mixing. Other examples of ergodic doubly stochastic processes and non-ergodic processes will be given. The developed ergodicity and mixing conditions involve a certain measure of dependence. We will address the statistical problem of estimating this measure of dependence.

Aaron Folgeson , University of Utah

"Computational Modeling of Blood Clotting"

Intravascular hemostasis and thrombosis occur under flow and this can profoundly influence the progress of clot formation. This talk will focus on two different aspects of our efforts to model and probe the interactions of flow and clotting. One involves the biochemistry of the coagulation enzyme network and how the behavior of this system is affected by flow-mediated platelet deposition on an injury and by flow-mediated transport of the enzymes and their precursors. The other involves a continuum model that describes platelet thrombosis initiated by a ruptured atherosclerotic plaque in a coronary-artery-sized vessel. This model includes full treatment of the fluid dynamics, and the aggregation of platelets in response to the plaque rupture and further chemical signals. Among the behaviors seen with this model are the growth of wall-adherent platelet thrombi to occlude the vessel and stop the flow, and the transient growth and subsequent embolization of thrombi leaving behind a passivated injured surface.