Seminar Schedule
September 10 - September 14, 2007
All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.
MONDAY, September 10, 2007
Algebra
2:30pm, Room 410
Speaker: Benjamin Jones, University of Georgia
Title of talk: Singular Chern Classes of Schubert Varieties: Part 2
Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies
Topology
3:30pm, Room 303
Speaker: Ken Baker (Georgia Tech)
Title of talk: Creating small Seifert fibered spaces by Dehn surgery on knots
Abstract: Generically, a small Seifert fibered space may be viewed as a thrice-punctured sphere cross S1 with three solid tori attached along the boundary tori. Sometimes an embedded solid torus may be excised from S3 and then reattached along the resulting boundary torus in a different manner to produce one of these small Seifert fibered spaces; i.e. Dehn surgery on a knot in S3 sometimes produces a small Seifert fibered space. The classification of such knots and Dehn surgeries remains open. We'll talk about the context for this problem, what's known, and some of our current on-going joint research with Cameron Gordon and John Luecke.
TUESDAY, September 11, 2007
VIGRE - Graduate Student Seminar
2:00pm, Room 304
Speaker: Alex Rice, University of Georgia
Title of talk: Density and Substance: An Investigation into the Size of Integer Subsets
Abstract: What is the probability that an integer is prime? How about square? Squarefree? What portion of the integers can be written as the sum of three squares? What must be true of an integer subset in order for its reciprocal series to converge? How can we use these ideas from number theory to catch multinational corporations cooking their books and cheating on their taxes? These questions and more will be answered as we investigate two methods for measuring the size of a subset of positive integers, search for relationships between the two measures, and prove several intriguing results that both confirm and betray our intuition. The majority of the talk will be very accessible to anyone who is familiar with limits, infinite series, and set theory. A smaller portion of the talk will feature some more involved number theory, such as the three squares theorem and the Mobius function, but I will certainly give definitions and explanations of such things. Everything is pretty easy to wrap your head around.
FRG Analysis and Additive Combinatorics Working Group
3:30pm, Room 410
Speaker: Neil Lyall, University of Georgia
Title of talk: Sarkozy's Theorem
Abstract: This is the first meeting of an informal seminar series, specifically aimed at graduate students. It is our intent to give expository lectures on a number of related results from analytic number theory, combinatorics and harmonic analysis (depending on the participants'/speakers' interests).
At the first meeting we shall use the Hardy-Littlewood circle method to prove Sarkozy's theorem: If A is a subset of positive density in the integers, then there exists two elements a and a' in A such that a-a' is a perfect square. The argument we plan to present (a modification of one due to Ben Green), although elementary, in fact gives better quantitative bounds than those which were originally obtained by Sarkozy (but alas fall way short of the best bounds that are currently known). On the positive side, this argument is relatively simple and can be easily extended to prove a more general (and new) result on the existence of certain polynomial configurations in difference sets (or sumsets). If time permits we may briefly discuss these generalizations.
WEDNESDAY, September 12, 2007
Algebraic Geometry
2:30pm, Room 410
Speaker: Peter Clark, University of Georgia
Title of talk: Algebraic points on algebraic strata
Abstract: It is a common situation in algebraic geometry for a desired property to hold at a "very general point" of some parameter space. We recall what this means and why it is a satisfactory statement over the complex numbers but completely vacuous over a countable field like the field of all algebraic numbers. It is natural to wonder whether certain geometric objects which have been proven to exist over C can in fact be defined over some number field. In fact we can prove this in the case of endomorphism algebras of abelian varieties, using a theorem due (independently) to R. Rumely and L. Moret-Bailly.
Faculty and Graduate Student Social
3:00pm, Room 409
Coffee, Cookies, Tea
Arithmetic Geometry/Number Theory
3:30pm, Room 304
Speaker: Peter Clark, University of Georgia
Title of talk: Existence of abelian varieties with prescribed endomorphism algebra
Abstract: This is a continuation of the 2:30 pm algebraic geometry seminar talk: namely, we seek to show that the classification of endomorphism algebras of g-dimensional abelian varieties over Q-bar is the same as it is over the complex numbers. Because this is the arithmetic geometry seminar, we will also discuss the corresponding problem over Q, which is significantly (indeed, so far as I know, prohibitively) more difficult. If possible, we may end early and solicit remarks on the density theorem of Rumely and Moret-Bailly.
Mathematical Physics
3:45pm, Room 302
Speakers: Cal Burgoyne, Emily Pritchett, and Robert Varley, University of Georgia
Title of talk: Continuation of the introduction to statistical mechanics, including entropy and Helmholtz free energy
THURSDAY, September 13, 2007
VIGRE – Algebraic Geometry
3:30pm, Room 323
Applied Math
2:00pm, Room 302
Speaker: Dong Hoon Shin, University of Georgia
Title of talk: Stochastic Optimization Algorithms for Pricing
Abstract: This work provides a Markov-modulated stochastic approximation based approach for pricing American put options under a regime switching geometric Brownian motionmarket model. The solutions of pricing American options may be characterized by certain threshold values. Here, a class of Markov-modulated stochastic approximation (SA) algorithms is developed to determine the optimal threshold levels. For option pricing in a finite horizon, a SA procedure is carried out for a fixed time T. As T varies, the optimal threshold values obtained via SA trace out a curve, called the threshold frontier. Numerical experiments are reported to demonstrate the effectiveness of the approach. Our approach provides a viable computational tool and has advantage in terms of the reduced computational complexity compared with the variational or quasivariationa l inequality methods for optimal stopping.
VIGRE – Tropical Geometry
2:00pm, Room 304
VIGRE – Circle Packing
3:30pm, Room 222
VIGRE-Number Theory
2:30pm, Room 326
FRIDAY, September 14, 2007
VIGRE-Algebra
1:30pm, Room 302
Geometry
2:30pm, Room 410
Speaker: Yang Liu, University of Georgia
Title of talk: An introduction to integral geometry (following J. Milnor)
Abstract: Starting with the Crofton's formula, which is the "origin" of integral geometry, I will explain the proof of Milnor's tube formula and intrinsic volume formula of product of convex sets. Then I will go to explain the Gauss-Bonnet theorem for polyhedra, following Milnor's proof by defining Steiner's spheric measure and angle.