## 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: Α rigorous derivation of a ternary Boltzmann equation for a classical system of particles.

Abstract: In this paper we present a rigorous derivation of a ternary Boltzmann equation describing the motion of a classical system of particles with three particle instantaneous interactions. The equation serves as a kinetic model for a dense gas in non-equilibrium, and is for the first time derived from laws of instantaneous three particle interactions, preserving momentum and energy.

Thursday, October 18, 17:05

Marco Méndez (University of Chicago/Princeton University)

Title: The Allen–Cahn equation and the theory of minimal surfaces.

Abstract: The Allen–Cahn equation behaves as a desingularization of the area functional. This allows for a purely PDE approach to the construction of minimal hypersurfaces in closed Riemannian manifolds. After presenting an overview of the subject, I will discuss recent results regarding a Weyl Law and its consequences for the density of minimal hypersurfaces in generic metrics. This is joint work with P. Gaspar.

* 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: Extremal metrics for the min-max width.

Abstract: We present our study on the min-max width of Riemannian three-dimensional spheres. This is a natural geometric invariant which is closely related to critical values of the area functional acting on closed surfaces, and can be interpreted as the first eigenvalue of a non-linear spectrum of a Riemannian metric, as suggested by Gromov. We will focus first on optimal bounds for the above invariant involving their volumes in a fixed conformal class. If time permits, we will discuss some general properties of extremal metrics for the min-max width. This is all part of a joint work with Lucas Ambrozio.

Thursday, November 1, 16:05

Richard Schoen (UC Irvine/Princeton University)

Title: Scalar Curvature and Minimal Hypersurface Singularities.

Abstract: We will describe the minimal hypersurface approach to scalar curvature problems and explain the difficulty that singularities present in high dimensions. We will then describe our approach to handling them for certain questions such as the positive mass theorem.

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

Thursday, November 29, 16:05

Abstract: In a metric space, discrete homotopy theory means replacing curves and homotopies by discrete analogs called $\varepsilon$-chains and $\varepsilon$-homotopies. One can then imitate the construction of the universal covering space to produce covering spaces, called $\varepsilon$-covers, with deck groups $\pi_{\varepsilon}(X)$ that are a kind of fundamental group at the scale of $\varepsilon>0$. This kind of construction, and an equivalent one by Sormani-Wei, have various mathematical applications related to geometric analysis such as finiteness theorems for the fundamental group and the ability to define spectra related to the length spectrum (lengths of closed geodesics) in spaces such as fractals with resistance metrics in which all non-constant curves have infinite length. Various new questions are opened up concerning the relationship between these spectra and the Laplace spectrum for Riemannian manifolds. This will be purely an introductory talk, fully understandable to anyone who knows what metric spaces and covering spaces are. If there is interest I can go more deeply into the topics in future talks, and discuss open questions that might be of interest to graduate students and other faculty. Some of work is joint with Valera Berestovskii and my former student Jay Wilkins.