## University of Tennessee - Geometric analysis seminar

### Fall 2018

Thursday, September 6, 16:05

Mariana Smit Vega Garcia (University of Washington)

Title: Recent developments in the thin obstacle problem.

Abstract: The study of the classical obstacle problem began in the 60's with the pioneering works of G. Stampacchia, H. Lewy and J. L. Lions. During the past five decades it has led to beautiful and deep developments in calculus of variations and geometric partial differential equations. One of its crowning achievements has been the development, due to L. Caffarelli, of the theory of free boundaries. Nowadays the obstacle problem continues to offer many challenges and its study is as active as ever. In particular, over the past years there has been some interesting progress the thin obstacle problem, also called Signorini problem. In this talk I will overview the thin obstacle problem for a divergence form elliptic operator, and describe a few methods used to tackle two fundamental questions: what is the optimal regularity of the solution, and what can be said about the free boundary, in particular the regular and singular sets. The proofs are based on Almgren, Weiss and Monneau type monotonicity formulas. This is joint work with Nicola Garofalo and Arshak Petrosyan.

Thursday, September 20, 16:05

Peter McGrath (UPenn)

Abstract: Let B^3 be the unit ball in R^3 and consider the family of surfaces contained in B^3 with boundary on the unit sphere S^2. The critical points of the area functional amongst this class are called Free Boundary Minimal Surfaces. The latter surfaces are physically realized by soap films in equilibrium and have been the subject of intense study. In the 1980s, it was proved that flat equatorial disks are the only free boundary minimal surfaces with the topology of a disk. It is conjectured that a surface called the critical catenoid is the unique (up to ambient rotations) embedded free boundary minimal annulus. I will discuss some recent progress towards resolving this conjecture. I will also discuss some sharp bounds (Joint work with Brian Freidin) for the areas of free boundary minimal surfaces in positively curved geodesic balls which extend works of Fraser-Schoen and Brendle in the Euclidean setting.

Thursday, September 27, 16:05

Julien Paupert (Arizona State University)

Title: Presentations for cusped arithmetic hyperbolic lattices

Abstract: We present a general method to compute a presentation for any cusped hyperbolic lattice $\Gamma$, applying a classical result of Macbeath to a suitable $\Gamma$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ for $d=1,3,7$ and the quaternionic lattice ${\rm PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$. This is joint work with Alice Mark.

* Julien will also give a Colloquium talk on Friday, September 28.

Thursday, October 11, 16:05

Ioakeim Ampatzoglou (University of Texas at Austin)

Title: TBA.

Abstract: TBA.

Thursday, October 18, 17:05

Marco Méndez (University of Chicago)

Title: TBA.

Abstract: TBA.

* Marco will also give a Jr. Colloquium talk prior to the seminar, at 15:40.

Thursday, October 25, 16:05

Rafael Montezuma (Princeton University)

Title: TBA.

Abstract: TBA.

Thursday, November 1, 16:05

Richard Schoen (UC Irvine)

Title: TBA.

Abstract: TBA.

* Rick will also give a Colloquium talk on Friday, November 2.

Spring 2018.
Fall 2017.