Analysis   Seminar

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

March 4th
A function which has Vanishing Mean Oscillation on the unit circle but is not essentially bounded

Abstract

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

Abstract:
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