Analysis   Seminar    

Unless specified, all seminars are Wednesday 4-5pm at 250 Math Building.                                             

March 4th            
Adam Orenstein,  SUNY at Buffalo
                      A function which has Vanishing Mean Oscillation on the unit circle but is not essentially bounded


April 1st             Pengfei Zhang,  University of Houston
                      Homoclinic points for generic convex billiards

We proved that for a C^r-generic convex billiard (r=2,3...), every hyperbolic periodic point of the induced billiard map admits some homoclinic points. In 
                      particular, C^r open and dense convex billiards have transverse homoclinic intersections and positive topological entropy. This provides a positive answer to a
                      conjecture of V. Donnay.          

April 15th           
Jingjing Huang,  University of Toronto
                      Rational points near hypersurfaces: with applications to the Dimension Growth Conjecture and metric diophantine approximation

                      Abstract: The distribution of rational points on algebraic varieties is a central problem in number theory. An even more general problem is to investigate rational
                      points near manifolds, where the algebraic condition is replaced with the non-vanishing curvature condition. In this talk, we will establish a sharp bound for the
                      number of rational points of a given height and within a given distance to a hypersurface. This has surprising applications to counting rational points lying on the
                      manifold; indeed setting the distance to zero, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally stated for projective varieties)
                      in this general setup. In the second half of the talk, we will focus on metric diophantine approximation on manifolds. A long standing conjecture in this area is the
                      Generalized Baker-Schmidt Problem. As another consequence of the main counting result above, we settle this problem for all hypersurfaces with non-vanishing Gaussian
                      curvatures. Our main innovation in the proof of the counting result is a bootstrap method that relies on the synthesis of Poisson summation, projective duality and
                      the method of stationary phase. 

April 22nd           
Zhizhang Xie,  Texas A&M University
                      Higher signatures on Witt spaces

                      Abstract: The signature is a fundamental homotopy invariant for topological manifolds. However, for spaces with singularities, this usual notion of signature ceases
                      to exist, since, in general, spaces with singularities fail the usual Poincar duality. A generalized Poincar duality theorem for spaces with singularities was proven
                      by Goresky and MacPherson using intersection homology. The classical signature was then extended to Witt spaces by Siegel using this generalized Poincar duality.
                      Witt spaces are a natural class of spaces with singularities. For example, all complex algebraic varieties are Witt spaces. In this talk, I will describe a
                      combinatorial approach to higher signatures of Witt spaces, using methods of noncommutative geometry. The talk is based on joint work with Nigel Higson.

Past Analysis Seminar