VIGRE Seminar Fall 2008 - Spring 2009

 

Location and Time: Tuesdays from 2:00p - 3:00p in Boyd Graduate Studies Research Center ("Boyd") room 304

Email the organizers at: arapmv@math.uga.edu, bagci@math.uga.edu, jwong@math.uga.edu

• Tuesday August 26th, Tin Kong, University of Georgia
  • Title of talk: The role of Mathematics in Climate Change
  • Abstract: This is a report of the Climate Change workshop I attended in MSRI this summer. A brief introduction to Climate Change/Global Warming will be presented, follow by the mathematical idea involved in Climate Change research.

• Tuesday September 2nd, Benjamin Jones, University of Georgia
  • Title of talk: Schubert Varieties
  • Abstract: The purpose of this talk is to give some basic ideas and details about Grassmannian varieties, Schubert varieties, and their importance in representation theory and combinatorics.

• Tuesday September 9th, Lev Konstantinovskiy, University of Georgia
  • Title of talk: What is a stack?
  • Abstract: Take a paper disc and fold it once horizontally, then vertically. You will obtain a quotient space, topologically a disc. However the the vertex at the rectangular corner of your resulting shape will be quite thick, hard, sharp and dangerous What happened is that we just took the quotient of D2 under the action of Z_2 X Z_2 and the vertex has the whole group as its stabilizer. There is nothing special about this vertex in the topological quotient space. However using the stack structure we will see this vertex as something special, since the stack structure stores stabilizers. Stacks naturally arise in algebraic geometry when one studies a space parametrizing objects that have automorphisms, for example algebraic curves of a given genus.

• Tuesday September 16th, Jingyue Wang, University of Georgia
  • Title of talk: Variational Models for Image Restoration
  • Abstract: Image restoration deals with the removal of noise from digital images. I'll give an introduction to variational models for image restoration with an emphasis on the Rudin-Osher-Fatemi model, its computation and some fast algorithms.

• Tuesday September 23rd, Monica Vazirani, University of California, Davis
  • Title of talk: Cores, Shi arrangements, and Catalan numbers
  • Abstract: Catalan numbers are known to count many mathematical objects. (See Richard Stanley's ``Enumerative Combinatorics" or http://math.mit.edu/~rstan/ec/catalan.pdf and http://math.mit.edu/~rstan/ec/catadd.pdf for a list of over 150 different combinatorial interpretations.) Some of the more well-known include triangulations of an $n+2$-gon or ways of closing up $n$ pairs of parentheses. In particular, the $n$-th Catalan number counts dominant regions in the Shi arrangement (of type $A_{n-1}$) and partitions that are both $n$-cores and $n+1$-cores. This fits into a more general framework, considering the $m$-Shi arrangement and partitions that are both $n$-cores and $mn+1$-cores. In joint work with Susanna Fishel, we give a bijective proof of this result, (given necessary definitions along the way) using the techniques of J. Anderson.

• Tuesday September 30th, Pete Clark, University of Georgia
  • Title of talk: Things to do with a conditionally convergent series
  • Abstract: I wish to pass along to you a conjecture concerning almost sure limit sets of series with random signs. (Nevertheless the statement is quite elementary -- I had undergraduates at McGill working on this general area.) The conjectural result would complete an analogy between theorems about conditionally convergent series: Riemann rearrangement : Levy-Steinitz :: Rademacher-Paley-Zygmund : ??? The first of these is the familiar statement about rearranging any conditionally convergent series so as to converge to any real number. Much of the talk will be devoted to the other two theorems. By the end of the talk, someone should be able to guess the conjecture I have in mind!.

• Tuesday October 7th, Frank Lubeck, RWTH Aachen University
  • Title of talk: Computing with groups on a computer
  • Abstract: In this talk I will mention some ways to represent (mostly finite) groups on a computer: by a finite presentation, by generating permutations, by generating matrices, as "black box" groups. I will show what can be computed with these groups, this depends on the representation. Some questions are for example: What is the order of the group? What can we say about the structure of the group? Can we compute subgroups? Can we decide if a given permutation or matrix is contained in our group? The talk will include some demonstrations using the computer algebra system GAP (www.gap-system.org).

• Tuesday October 14th, Maxim Arap, University of Georgia
  • Title of talk: On a conjecture about lines in smooth hypersurfaces.
  • Abstract: This seminar will introduce basic definitions and objects of study in algebraic geometry in order to state an elementary conjecture of O. Debarre and (independently) J. de Jong about lines in smooth hypersurfaces. The proof (due to O. Debarre) of this conjecture in the case of hypersurfaces of degree d < 6 will be presented. This presentation will be elementary and will assume no previous background in algebraic geometry.

• Tuesday October 21st, Alexander Kleshchev , University of Oregon
  • Title of talk: Representation Theory of Symmetric Groups
  • Abstract: This is an expository talk accessible to all graduate students. The talk will be on representation theory of symmetric groups. We will review some old and new topics such as hook formula, branching rules, and deep connections with Lie theory.

• Tuesday October 28th, Jeremy Wong, University of Georgia
  • Title of talk: Filling Manifolds
  • Abstract: We will survey and discuss mostly open aspects of the following question: When does a Riemannian manifold bound a manifold with prescribed geometry?

• Tuesday November 4th, Lirong Yu, University of Georgia
  • Title of talk: Black-Scholes Option pricing formula -- a simple discussion
  • Abstract: Most of the models and techniques employed by today's financial analysts are rooted in a model developed by Fischer Black and Myro Scholes in 1973. I want to discuss this model and its application, the famous Black-Scholes option pricing formula.

• Tuesday November 11th, Jim Solazzo, Coastal Carolina University
  • Title of talk: Frames, Graphs and Equiangular Lines
  • Abstract: In this talk I will give a brief introduction to frame theory. I will also discuss how some well known results in the graph theory community answer certain questions in frame theory. The only prerequisite for this talk is linear algebra.
    

• Tuesday November 18th, No Meeting

 

• Tuesday November 25th Thanksgiving Break No Meeting.

• Tuesday December 2nd, Sandra Ritz, University of Southern California
  • Title of talk: The Burau Representation of the Braid Group
  • Abstract: An introduction and some history of the braid group $B_n$, including different definitions and applications. This will be followed by a topological description of the Burau representation, and a summary of some of its properties.

• Tuesday January 13th, 2009, Jonathan Kujawa, University of Oklahoma
  • Title of talk: Burnt Pancakes, DNA, and the Symmetric Group: or, how Bill Gates became a billionaire.
  • Abstract: The tremendous amounts of data coming from breakthroughs in research on DNA raises many new, interesting, and important questions in discrete mathematics (algebra, combinatorics, etc.). We will give a nontechnical introduction to one such problem and point out some of the interesting mathematical questions it raises.

• Tuesday January 20th, 2009, David Krumm, UGA
  • Title of talk: Computing the Hilbert Function of a Complete Intersection
  • Abstract: I will address the question of when the Hilbert function and the Hilbert polynomial of an algebraic variety start agreeing. In the process of answering this question, I will give explicit formulas for both the Hilbert function and the Hilbert polynomial that are valid for varieties which are complete intersections, and I'll discuss computational applications of these formulas.

• Tuesday January 27th, 2009, David Swinarski, UGA
  • Title of talk: Automorphisms of Riemann surfaces
  • Abstract: I will discuss my favorite results on automorphisms of Riemann surfaces/algebraic curves. Some old, some new, some well-known, some obscure. We'll do lots of examples. I may also sketch how these questions arose in my research.

• Tuesday February 3rd, 2009, Jie Yu, UGA
  • Title of talk: A regime-switching model and applications in derivative pricing
  • Abstract: I will start with fundamental knowledge in the study of financial derivatives and some popular results. Then I will introduce a regime-switching model and its applications in derivative pricing, examples will be given for application with real data.