GEOMETRIC ANALYSIS SEMINAR

The seminar will be held weekly on Monday afternoon at 3:30pm in room B219 (Van Vleck Hall.)

Scheduled speakers

May 12: Andre Neves (Princeton). Translating solitons to Lagrangian mean curvature flow.

April 14: Marshall Slemrod (UW Madison). Transonic Flow and Supersonic Differential Geometry

Abstract: This talk will discuss recent results by G Q Chen, M Slemrod, and D Wang on the proofs of weak solutions for (i) 2 dimensional steady irrotational flow over an airfoil and (ii) the initial value problem for isometric immersion of a 2 dimensional Riemannian manifold with negative Gauss curvature in three dimensional Euclidean space. Surprising perhaps(?) the two problems are very similar and the method of compensated compactness applies in both cases.

December 10: Spyros Alexakis (Princeton). The decomposition of global conformal invariants: On a conjecture of Deser and Schwimmer.

Global conformal invariants are integrals of local geometric quantities which remain invariant under conformal changes of the underlying metric. I will present (parts of) my recent proof of a conjecture of Deser and Schwimmer, which states that any such global invariant can be decomposed into standard “building blocks” of three types. I will also discuss some of the impications of the above conjecture for conformal geometry and for the AdS-CFT correspondence.

December 3: Aobing Li (Math department, UW Madison). A fully nonlinear Yamabe Problem.

The Yamabe problem is to look for a metric with constant scalar curvature in each conformal class of metrics. The equivalent equation is a semilinear equation. In our talk, we will generalize it to a problem and the equivalent equation is a fully nonlinear elliptic eqaution. We will consider the generalized problem on compact manifolds with/without boundaries.

November 12: Jeffrey Streets (Math department, Princeton). Ricci Yang-Mills Flow

I will introduce a geometric evolution equation coupling the Ricci flow for a Riemannian metric and the Yang-Mills flow for a connection on a principal bundle in a natural, nontrivial way. This equation is introduced in the hope that by choosing a nice enough principal bundle, the existence properties will be nicer than that of Ricci flow alone. I will discuss many basic analytic properties of this equation, and describe two convergence results.

October 22: Jingyi Chen (Maths department, U British Columbia ), Stable minimal disks in manifolds with nonnegative isotropic curvature.

Abstract: Isotropic curvature arises naturally in the second variation of surface area. Using minimal surface techniques, Micallef-Moore showed that a simply connected manifold of dimension greater than 3 with positive isotropic curvautre is homeomorphic to a sphere. We will discuss a free boundary problem for disk type minimal surfaces with boundary in a compact hypersurface of a manifold with nonnegative isotropic curvature.

October 15: Meijun Zhu (Oklahoma State U), Local sharp embedding inequalities.

Abstract: In this talk I shall introduce the concept of local sharp embedding inequalities. The applications of such inequalities on manifolds will be described. In particular I will show that the local inequalities directly yield global sharp inequalities on Sⁿ (sharp Sobolev inequality for n>4; Onofri inequality for n=2). I shall also talk about certain calculus inequality (an analogue to Hardy inequality), which is used to prove the local sharp inequality for n=2.

October 8: Dan Knopf (UT Austin), Convergence and stability of locally R^N-invariant solutions of Ricci flow.