Colloquia 2011-2012

Current Talks:

Abstract: We discuss several concepts, techniques, and tools that are useful in the study of low dimensional dynamics and focus on a few well-known results to highlight those background techniques. We particularly emphasize the role that symbolic dynamics plays in understanding dynamical systems generated by unimodal maps. We then discuss adding machine maps and highlight several recent results in the field. We conclude by showing how a better understanding of adding machine maps can lead to more generalized results in the family of unimodal maps.

Past Talks:

Abstract: The starting point for our considerations is the notion of graded Hopf algebra, particularly those graded over the nonnegative integers. When the latter are replaced by finite sets, one arrives at the notion of Hopf monoid in Joyal's category of species. The goal of this talk is to go one step further, replacing finite sets by finite real hyperplane arrangements. Geometric considerations allow us to define a generalized notion of "Hopf algebra" in this setting. The key ingredient in this construction is furnished by the projection maps of Tits. The case of finite sets (Hopf monoids in species) is recovered by restricting to braid arrangements.

This is joint work in progress with Swapneel Mahajan.

Abstract: Rehabilitation is an interdisciplinary field of study with the primary aim of enhancing health, function and quality of life among persons who have or who may be at risk of developing, acute injuries or long-term conditions. Rehabilitation research also incorporates the disciplines of athletic training, exercise sciences, occupational therapy, physical therapy, as well as other fields such as public health and engineering. The research of rehabilitation spans the entire life course, from infancy to older adulthood, and addresses a wide variety of patient populations. I will present some case studies that show how mathematics can potentially provide research solutions to understand human locomotion with robot devices and brain activity.

Abstract: Most of the rings one encounters as basic examples have what's known as the "Invariant Basis Number" property, namely, for every pair of positive integers m and n, if the free left R-modules _RR^m and _RRⁿ are isomorphic, then m=n. (For instance, the IBN property of fields is used to show that the dimension of a vector space over a field is well defined.) In seminal work completed in the early 1960's, Bill Leavitt produced a specific, universal collection of algebras which fail to have IBN. While it's fair to say that these algebras were initially viewed as mere pathologies, it's just as fair to say that these now-so-called Leavitt algebras currently play a central, fundamental role in numerous lines of research in both algebra and analysis.

More generally, from any directed graph E and any field K one can build the Leavitt path algebra L_K(E). In particular, the Leavitt algebras arise in this more general context as the algebras corresponding to the graphs consisting of a single vertex. I'll give an overview of some of the work on Leavitt path algebras which has occurred in their first seven years of existence, as well as mention some of the future directions and open questions in the subject. There should be something for everyone in this presentation, including and especially algebraists, analysts, and graph theorists. We'll also present a basic number theoretic result which provides the foundation of one of the recent main results in Leavitt path algebras. The talk will be aimed at a general audience.

Abstract: Many commentators contend that Hume made some obvious mathematical errors in the section "Of scepticism with regard to reason" of his A Treatise of Human Nature. In my talk I will argue that a proper understanding of Hume’s historical context exonerates Hume of any serious mathematical errors and help to provide a plausible understanding of the nature of his argument in his Treatise. More specifically, I shall point to a different way of understanding Hume on probability that avoids an anachronistic Bayesian reading of Hume’s sceptical argument. In addition, I shall contend that a non-Bayesian approach helps to alleviate the puzzlement about some other seemingly odd mathematical moves in Hume’s argument and allows us to reconstruct Hume’s reasoning in a plausible way.

Abstract: We charge a metal body with electrons. The distribution of electrons minimizing the energy is called the equilibrium measure. Gauss was the first to begin the analysis of the electrostatic equilibrium problem with external fields. He wrote: "The determination of the distribution of the mass lies, in most cases, beyond the powers of present day analysis."

We will study the equilibrium measure for logarithmic and Riesz energy. Results from the UCUR project of Chelsey David will also be discussed (poster is on the 4th floor). Finally, we show how ping-pong and fractal-like dogs relate to potential theory.

Abstract: A smooth projective variety X is called "regular" if there exists an algebraic action of the invertible upper triangular 2x2 matrices on X such that the unipotent radical has exactly one fixed point. A remarkable approach for studying the cohomology algebra of a regular variety is developed by Akyildiz and Carrell. Among the important examples of regular varieties are homogenous spaces. In particular, when applied to flag varieties, the Akyildiz-Carrell method yields remarkable results in representation theory.

The variety of complete quadrics, which is used by Schubert in his famous computation of the number of space quadrics tangent to 9 quadrics in general position, is a particular ``wonderful compactification'' of the space of non-singular quadric hypersurfaces in n dimensional complex projective space.

In this talk, we first give an overview of the work of Akyildiz and Carrell, then characterize the regular subvarieties of the variety of complete quadrics.

Abstract: In this talk, I will survey the development of fundamental ideas, results and will formulate a few open problems in coloring theory ranging from graph coloring to mixed hypergraph coloring. This talk will be suitable for graduate and upper level undergraduate students.

Department of Mathematics and Statistics
ILB 325
University of South Alabama
307 North University Blvd
Mobile, AL 36688-0002
phone: (251) 460-6264 (voice), (251) 460-7969 (fax)
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current page:
http://www.southalabama.edu/mathstat/colloquia/index.shtml