Un of Tenn | Math Department

Seminars and Colloquiums
for the week of April 23, 2012

Speaker:

Mr. Buddhi Pantha and Mr. Jeremy Auerbach, Monday
Dr. Victor Perez-Abreu, Center for Research in Mathematics, CIMAT, Mexico, Monday
Professor Conrad Plaut, Wednesday
Dr. Karl-Mikael Perfekt, Lund University, Wednesday
Mr. Frederick Byrd and Ms. Yiyang Sun, and Honors Undergraduate Students, Thursday
Prof. Mi Hee Park, Chung-Ang University, Seoul, Korea, Friday

If you are interested in giving or arranging a talk for one of our seminars or colloquiums,
please review our calendar.

If you have questions, or a date you would like to confirm, please contact Dr. Judy Day.

Monday, April 23

MATH BIOLOGY SEMINAR
TIME: 2:30 – 3:20 p.m.
ROOM: Ayres 121
SPEAKER: Buddhi Pantha and Jeremy Auerbach
TITLE: Introduction of Agent-based Models (part 5)

PROBABILITY SEMINAR
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 122
SPEAKER: Dr. Victor Perez-Abreu, Center for Research in Mathematics, CIMAT, Mexico
TITLE: Free Probability, Random Matrices and Infinite Divisibility. Part III: Random Matrices: A Bridge between Free and Classical Infinite Divisibility
ABSTRACT: The concept of free infinite divisibility of probability measures as well as some criteria, examples and characterizations will be presented. We will then consider the Bercovici-Pata bijection between free and classical infinite divisibility with emphasis on its random matrix approach.

Wednesday, April 25

TOPOLOGY SEMINAR
TIME: 2:30 - 3:20 P.M.
ROOM: Ayres Hall G004
SPEAKER: Professor Conrad Plaut
TITLE: Discrete Homotopies and the Fundamental Group - 2
ABSTRACT: Discrete homotopies were introduced by Valera Berestovskii and me as a way to define generalized fundamental groups for locally bad spaces. In this talk I will discuss a more geometric application that is joint work with Jay Wilkins: generalizing and strengthening a theorem of Gromov about generators and relators of the fundamental group in a compact Riemannian manifold. I'll give all of the background needed; the theorem is proved in the setting of compact geodesic spaces (no differential geometry needed!), and will be accessible to anyone who knows what metric spaces and the fundamental group are. Background will include the Gromov-Hausdorff metric and Gromov's precompactness theorem, which is a beautiful generalization of the basic fact that a metric space is compact if and only if it is complete and totally bounded. As an application I'll show that in any Gromov-Hausdorff precompact collection of compact metric spaces with a global bound on the number of "short loops", there are at most finitely many possible fundamental groups. This generalizes finiteness theorems of Michael Anderson, Shen-Wei, and Sormani-Wei.

ANALYSIS SEMINAR
TIME: 3:35 - 4:25 p.m.
ROOM: Ayres Hall 112
SPEAKER: Dr. Karl-Mikael Perfekt, Lund University
TITLE: Polarizability of Lipschitz domains
ABSTRACT: The polarizability of a domain is obtained by solving an electrostatic problem, a Laplace equation with two-sided boundary conditions. Recently, the polarizability of non-smooth domains has received considerable attention, and it turns out that solutions and the corresponding spectral measure have completely different behavior than in the smooth case. A standard method of constructing solutions to the PDE is provided by classical double layer potential operators. Introducing for them a suitable operator-theoretic formalism related to Krein's theory of symmetrizable operators, the differences between the smooth and the non-smooth cases will be explained through the use of spectral theory and related Hilbert space techniques. We identify when the PDE can be solved, and importantly, when it does not have finite energy solutions. The structure of spectral measures is investigated, and measures are shown to converge weak-star when smooth domains are deformed into a non-smooth domain. Furthermore, a highly efficient numerical scheme that computes the polarizability of a cube in 3D has been developed. Joint work with Johan Helsing.

Thursday, April 26

JUNIOR COLLOQUIUM - HONORS SEMINAR
SPEAKERS: Mr. Frederick Byrd and Ms. Yiyang Sun, and Honors Undergraduate Students
TIME: 3:35 p.m.
ROOM: Ayres Hall 405
TITLE: Applications of Ultraproducts in Ring Theory
SPEAKER: Mr. Frederick Byrd
ABSTRACT: The concept of an ultraproduct leads to many interesting results in ring theory. Using this construction, it is possible to create new rings from old ones, which may have entirely different properties than the original rings. Though the component rings of an ultraproduct may have many restrictive characteristics, their ultraproduct typically will not have these characteristics unless it is a field. This can be used to generate counterexamples to many conjectures about ring theory while still retaining the familiarity of well-known rings. Of particular interest are the use of ultraproducts in factorization problems. In this thesis we attempt to use an ultrapower to find an example of an element with non-unique irreducible factorizations whose lengths are unbounded, as well as examine and review many interesting properties of ultrapower rings.

TITLE: Mathematical Analysis of Poker Games
SPEAKER: Mr. Yiyang Sun
ABSTRACT: Poker is an intriguing game which has attracted many scholars' attention throughout History. The study of two-person zero-sum poker models with independent uniform(0,1) hands goes back to Borel and von Neumann. In this presentation we will take a look at three different mathematical poker game models. First we will start with von Neumann's simplified two player poker model where each player antes the same amount to form the initial amount to start the game, then we will change up the betting rules to make it a blind bet poker model. Finally, we will throw in a extra variable to capture the other uncertainties in a poker game.

Pizza will be available at 3:15 p.m.

Friday, April 27

COLLOQUIUM
TIME: 3:35 – 4:25 pm
ROOM: Ayres Hall 405
SPEAKER: Prof. Mi Hee Park, Chung-Ang University, Seoul, Korea
TITLE: Dimension theory of mixed polynomial and power series rings
ABSTRACT: We consider the mixed polynomial and power series ring extensions, or, for short, mixed extensions. A mixed extension over a ring R in variables x1; … ; xn is denoted by R[x1]] …[xn]], where each [xi]] is fixed as either [xi] or [[xi]]. One extreme is the polynomial extension R[x1; …; xn] and the other extreme is the power series extension R[[x1; … ; xn]]. The motivation for our work comes from the following question raised by Robert Gilmer and Jim Coykendall: Let R be a commutative ring with identity. When dim(R[[x]]) < ?, is dim(R[[x]]) ? 2(dimR) + 1? We answer the question in the negative by computing the Krull dimension of mixed extensions over a PrÄufer domain. We also discuss similarities and di®erences among several mixed extensions over fields. Especially, given a field extension k ó K, we recall the algebraicity of the extension k [x1]] …[xn]] ,! K[x1]] …[xn]] and then compute the dimension of its generic fiber ring. Also, we characterize the field extensions k ó K such that the prime spectra Spec(K[x1]] …[xn]]) and Spec(k[x1]] … [xn]]) are homeomorphic.

Refreshments will be available in Ayres 401 at 3:15 p.m.

Past notices:

March 19, 2012 - spring break