The Nonlinear Analysis Seminar aims to investigate nonlinear problems arising in differential geometry, analysis, and partial differential equations. A seminar talk may cover original research or report on an interesting paper. All meetings of the Nonlinear Analysis Seminar will be held on Tuesdays from 12:45-1:45 PM in the Room 5417, the CUNY Graduate Center, unless otherwise stated.Faculty OrganizerDr. Stephen Preston(Brooklyn College and the Graduate Center, CUNY) email: Student OrganizerJae Min Lee(CUNY Graduate Center) email: jlee10@gradcenter.cuny.edu website: http://opencuny.org/jaeminlee/ Spring 2017 ScheduleFebruary 7, 2017 Stephen Preston(Brooklyn College and Graduate Center, CUNY) "Metrics on Shape Spaces" Abstract I will describe several approaches to the Riemannian geometry of the space of shapes: that is, unparametrized closed curves in the plane. One method is to work on the space of all parametrized curves, then quotient out by the reparametrization group (circle diffeomorphisms). Another is to parametrize all curves by arc length. I will describe a few choices of Riemannian metrics and what their geodesic equations look like, as well as summarizing what is known about the attainability problem (whether one can join two shapes by a minimizing geodesic). Finally I will present some recent work due to myself and Barbara Tumpach on the elastic metric on the space of unit-speed curves, with some explicit formulas and numerical results for the gradient descent method in this case. The talk will be accessible to graduate students familiar with differential geometry and differential equations. February 14, 2017 No seminar February 21, 2017 Jae Min Lee(The Graduate Center, CUNY) "Local Well-posedness of the Camassa-Holm equation on the real line" Abstract I will establish the local well-posedness of the Camassa-Holm equation on the real line. That is, we will show the local existence and uniqueness of the solution, and the continuous dependence of the solution on the initial data. The main idea is to use the Lagrangian flow to rewrite the PDE as a smooth ODE on a Banach Space. We will also use the topological group properties of the group of diffeomorphisms to prove the smoothness of the equation and continuous dependence. Some backgrounds in Real and Functional Analysis will be useful, but I will make this talk accessible to most graduate students. February 28, 2017 No seminar March 14, 2017 Jakob Møller-Andersen(Florida State University) "Riemannian Geometries on Spaces of Curves" Abstract In this talk we give an introduction to infinite dimensional Riemannian Geometry on spaces of curves. We'll illustrate some of the properties and problems unique to the infinite dimensional setting and contrast these to the finite dimensional theory. A shape space of geometrically equivalent curves is defined, and we show how one can equip this space with a family of Riemannian metrics through the use of Riemannian submersions and analyze the resulting geodesic equations. Finally we illustrate how this theory can be used in applications. |