Seminars & Events for Joint Princeton Rutgers Geometric PDEs Seminar

October 16, 2015
3:00pm - 4:00pm
Embedded Willmore tori in three-manifolds with small area constraint

While there are lots of contributions on Willmore surfaces in the three-dimensional Euclidean space, the literature on curved manifolds is still relatively limited. One of the main aspects of the Willmore problem is the loss of compactness under conformal transformations. We construct embedded Willmore tori in manifolds with a small area constraint by analysing how the Willmore energy under the action of the Möbius group is affected by the curvature of the ambient manifold. The loss of compactness is then taken care of using minimisation arguments or Morse theory. 

Speaker: Andrea Malchiodi, Scuola Normale Superiore-Pisa
Location:
Fine Hall 314
October 16, 2015
4:15pm - 5:15pm
The two membranes problem

We will consider the two membranes obstacle problem for two different operators, possibly non-local. In the case when the two operators have different orders, we discuss how to obtain $C^{1,\gamma}$ regularity of the
solutions. In particular, for two fractional Laplacians of different orders, one obtains optimal regularity and a characterization of the boundary of the coincidence set. This is a joint work with L. Caffarellii and O. Savin. 

Speaker: Daniela Da Silva, Columbia University
Location:
Fine Hall 314
March 28, 2016
1:32pm - 1:32pm
Hypersurfaces of low entropy

The entropy is a natural geometric quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the complexity of a hypersurface of Euclidean space. It is closely related to the mean curvature flow. On the one hand, the entropy controls what types of singularities the flow develops. On the other, the flow may be used to study the entropy. In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy can't be too complicated. 

Speaker: Jacob Bernstein, Johns Hopkins University
Location:
Rutgers - Hill Center, Room 705
April 5, 2016
4:00pm - 5:00pm
Moser-Trudinger type inequalities, mean field equations and Onsager vortices

In this talk, I will present a recent work confirming the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. The proof is based on a new and powerful lower bound of total mass for mean field equations. Other applications of the lower bound include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on tori and the sphere, etc. The resolution of several interesting problems in these areas will be presented. The work is jointly done with Amir Moradifam from UC Riverside. 

Speaker: Changfeng Gui, University of Connecticut
Location:
Rutgers - Hill Center, Room 705
April 5, 2016
5:15pm - 6:15pm
Hypersurfaces of low entropy

The entropy is a natural geometric quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the complexity of a hypersurface of Euclidean space. It is closely related to the mean curvature flow. On the one hand, the entropy controls what types of singularities the flow develops. On the other, the flow may be used to study the entropy. In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy can't be too complicated. 

Speaker: Jacob Bernstein , Johns Hopkins University
Location:
Rutgers - Hill Center, Room 705