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10:50 am Tuesday, April 13, 2010 Algebraic Geometry Seminar: Canonical subgroups for abelian varieties
Joe Rabinoff (Harvard) in HB 453- An elliptic curve over the integer ring of a p-adic field whose special fiber is ordinary has a canonical line contained in its p-torsion. This fact has many arithmetic applications: for instance, it shows that there is a canonical partially-defined section of the natural map of modular curves X_0(Np) -> X_0(N). Lubin was the first to notice that elliptic curves with "not too supersingular" reduction also contain a canonical order-p subgroup. I'll begin the talk by giving an overview of Lubin and Katz's theory of the canonical subgroup of an elliptic curve. I'll then explain one approach to defining the canonical subgroup of a general abelian variety (and even p-divisible group), and state a very general existence result.
Submitted by evanmb@gmail.com |

1:00 pm Tuesday, April 13, 2010 SPECIAL ERGODIC THEORY SEMINAR FOR GRADUATE STUDENTS : 'Dynamical properties of arbitrary beta-shifts and related systems via weak specification properties'Dan Thompson (PennState) in HB 453- The beta-transformation f(x) = beta x (mod 1) has been widely studied since its introduction by Renyi in 1957. The sustained interest in the study of the beta-transformation arises from its connection with number theory and its special role as a model example of a one-dimensional expanding dynamical system which admits discontinuities. The natural coding space for the beta-transformation is the beta-shift. Many results which are well known for shifts of finite type are challenging to generalise to beta-shifts, and a variety of basic questions about beta-shifts remain open. We discuss a topological dynamical property called almost specification which applies to every beta-shift. This property was introduced recently by Pfister and Sullivan and has proved to be a useful tool in studying the beta-shift. I will describe how this property works, why the beta-shift has it, and what it's good for.
Submitted by hardt@rice.edu |

4:00 pm Tuesday, April 13, 2010 Bochner Lecture III: Optimal transport and applications
Allessio Figalli (University of Texas Austin and Ecole Polytechnique) in HB 227The optimal transport problem consists in finding the "most efficient" way of moving a distribution of mass from one place to another. This problem, formulated by Monge in 1781, has been solved only recently: in 1987 Brenier showed existence and uniqueness of optimal transports on Euclidean spaces when the cost for moving a unit mass from a point x to a point y is |x-y|^2. This result has been the starting point for many developments in different directions: for instance, optimal transport has revealed to be a powerful tool for proving functional and geometric inequalities (like isoperimetric or Sobolev inequalities) or for studying the asymptotic of evolution equations (like heat or porous medium equations). More recently, optimal transport has found applications in Riemannian geometry: first of all, after the solution of the optimal transport problem on Riemannian manifolds (McCann 2001), people started to realize that optimal transport could be used to study Ricci curvature bounds. Moreover, in these last 2-3 years, it was discovered that optimal transport techniques could even be used to study the geometry of the cut-locus of the manifold. In these lectures, which are supposed to be accessible to a broad audience, I will start from the very beginning, explaining the strategy to solve the optimal transport problem both in Euclidean spaces and on Riemannian manifolds. Then, to show some of the possible applications, I'll apply optimal transport techniques to prove functional inequalities. Finally, if time permits, I'll explain the link between optimal transport on Riemannian manifolds, Monge-Ampere equations, and the geometry of the cut-locus of the manifold. Submitted by mathweb@rice.edu |