Geometry and Topology Seminar
|September 15||Jiyuan Han (University of Wisconsin-Madison)||"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"||Local|
|September 22||Sigurd Angenent (UW-Madison)||"Topology of closed geodesics on surfaces and curve shortening"||Local|
|September 29||Ke Zhu (Minnesota State University)||"Isometric Embedding via Heat Kernel"||Bing Wang|
|October 6||Shaosai Huang (Stony Brook)||TBA||Bing Wang|
|October 20||Shengwen Wang (Johns Hopkins)||TBA||Lu Wang|
|October 27||Marco Mendez-Guaraco (Chicago)||TBA||Lu Wang|
|November 17||Ovidiu Munteanu (University of Connecticut)||TBA||Bing Wang|
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with Jeff Viaclovsky.
"Topology of closed geodesics on surfaces and curve shortening"
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.
"Isometric Embedding via Heat Kernel"
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.
Archive of past Geometry seminars