Thursday, April 29, 2010 - 19:00 , Location: Skiles 202 , Michael Lacey , Georgia Tech , Organizer:
Club Math Presents The Mathematics of Futurama, by Dr. Michael Lacey.
Athens/Atlanta Number Theory Seminar - Lecture 2 - Some applications of potential theory to number theoretical problems on analytic curvesTuesday, April 13, 2010 - 17:15 , Location: Skiles 269 , Antoine Chambert-Loir , Universite de Rennes/Institute for Advanced Study , Organizer: Matt Baker
Athens/Atlanta Number Theory Seminar - Lecture 1 - Degree three cohomology of function fields of surfacesTuesday, April 13, 2010 - 16:00 , Location: Skiles 269 , Venapally Suresh , University of Hyderabad / Emory University , Organizer: Matt Baker
Let k be a global field or a local field. Class field theory says that every central division algebra over k is cyclic. Let l be a prime not equal to the characteristic of k. If k contains a primitive l-th root of unity, then this leads to the fact that every element in H^2(k, µ_l ) is a symbol. A natural question is a higher dimensional analogue of this result: Let F be a function field in one variable over k which contains a primitive l-th root of unity. Is every element in H^3(F, µ_l ) a symbol? In this talk we answer this question in affirmative for k a p-adic field or a global field of positive characteristic. The main tool is a certain local global principle for elements of H^3(F, µ_l ) in terms of symbols in H^2(F µ_l ). We also show that this local-global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields.
Monday, April 12, 2010 - 17:00 , Location: Klaus 1116W , Richard Schoen , Stanford University , Organizer: John McCuan
In 1854 Riemann extended Gauss' ideas on curved geometries from two dimensional surfaces to higher dimensions. Since that time mathematicians have tried to understand the structure of geometric spaces based on their curvature properties. It turns out that basic questions remain unanswered in this direction. In this lecture we will give a history of such questions for spaces with positive curvature, and describe the progress that has been made as well as some outstanding conjectures which remain to be settled.
Monday, April 12, 2010 - 08:00 , Location: Skiles 269 , Southeast Geometry Seminar , School of Mathematics, Georgia Tech , Organizer: John McCuan
The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham; The Georgia Institute of Technology; Emory University; The University of Tennessee Knoxville. The presentations will include topics on geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. See the Schedule for times and abstracts of talks.
Wednesday, April 7, 2010 - 16:30 , Location: Skiles 255 , Allan Sly , Microsoft Research, Redmond, WA , Organizer: Prasad Tetali
Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of papers verified this on $Z^2$ except at $\beta=\beta_c$ where the behavior remained unknown. In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains. Based on joint work with Eyal Lubetzky.
Tuesday, April 6, 2010 - 11:00 , Location: ISyE Executive Classroom , Adrian Lewis , School of Operations Research and Information, Cornell University , Organizer: Annette Rohrs
Concrete optimization problems, while often nonsmooth, are not pathologically so. The class of "semi-algebraic" sets and functions - those arising from polynomial inequalities - nicely exemplifies nonsmoothness in practice. Semi-algebraic sets (and their generalizations) are common, easy to recognize, and richly structured, supporting powerful variational properties. In particular I will discuss a generic property of such sets - partial smoothness - and its relationship with a proximal algorithm for nonsmooth composite minimization, a versatile model for practical optimization.
Monday, April 5, 2010 - 15:00 , Location: Skiles 255 , Guy Degla , Institute of Mathematics and Physical Sciences, Benin , Organizer: Wilfrid Gangbo
The purpose of this talk is to highlight some versions of the Krein-Rutman theorem which have been widely and deeply applied in many fields (e.g., Mathematical Analysis, Geometric Analysis, Physical Sciences, Transport theory and Information Sciences). These versions are motivated by optimization theory, perturbation theory, bifurcation theory, etc. and give rise to some simple but useful comparison methods, in ordered Banach spaces, such as the Dodds-Fremlin theorem and the De Pagter theorem.
Wednesday, March 17, 2010 - 13:30 , Location: ISyE Executive Classroom , Merrick Furst , College of Computing, Georgia Tech , Organizer:
Santosh Vempala and I have been exploring an intriguing new approach to convex optimization. Intuition about first-order interior point methods tells us that a main impediment to quickly finding an inside track to optimal is that a convex body's boundary can get in one's way in so many directions from so many places. If the surface of a convex body is made to be perfectly reflecting then from every interior vantage point it essentially disappears. Wondering about what this might mean for designing a new type of first-order interior point method, a preliminary analysis offers a surprising and suggestive result. Scale a convex body a sufficient amount in the direction of optimization. Mirror its surface and look directly upwards from anywhere. Then, in the distance, you will see a point that is as close as desired to optimal. We wouldn't recommend a direct implementation, since it doesn't work in practice. However, by trial and error we have developed a new algorithm for convex optimization, which we are calling Reflex. Reflex alternates greedy random reflecting steps, that can get stuck in narrow reflecting corridors, with simply-biased random reflecting steps that escape. We have early experimental experience using a first implementation of Reflex, implemented in Matlab, solving LP's (can be faster than Matlab's linprog), SDP's (dense with several thousand variables), quadratic cone problems, and some standard NETLIB problems.
Sunday, March 14, 2010 - 13:59 , Location: Skiles Courtyard , N/A , GT , Organizer:
Come celebrate pi day with math club! Pot-luck, so bring food! Math club will be providing the pies, so we ask that everyone else try to bring more substantial food. ;)Bring any games and such you want as well.