# Geometry & Analysis Seminar Fall 2016

McHenry Library Room 4130

For further information please contact Professor Longzhi Lin or call 831-459-2400

**September 29, 2016**

**Thursday,**October 6, 2016

**"Weakly Horospherically Convex Surfaces in Huperboic 3-Space"****Jingyong Zhu, University of Science and Technology of China**

A global correspondence between the immersed weakly horospherically convex hypersurfaces and a class of the conformal metrics on the domains of a round sphere was established by Vicent Bonini, Jose Espinar and Jie Qing in 2012. Some important results in their paper were proved when the dimension of hypersurfaces is great than or equal to 3. For example, the embeddedness of those hypersurfaces along the normal flow, the consistency of the boundary of surfaces at the infinity with that of Gauss map image, the symmetry of hypersurfaces and the Bernstein theorem. All the proofs of these results need the same analytic proposition about the asymptotic behavior of the conformal factor, which is open in 2-dimension. In this talk, I want to introduce you my recent work about these issues in 2-dimension. This is a joint work with Vincent Bonini and Professor Jie Qing.

**Thursday,**October 13, 2016

**"Two dimensional water waves in holomorphic coordinates"****Mihaela Ifrim, UC Berkeley**

We consider this problem expressed in position-velocity potential holomorphic coordinates. We will explain the set up of the problem(s) and try to present the advantages of choosing such a framework. Viewing this problem(s) as a quasilinear dispersive equation, we develop new methods whcih will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. The talk will try to be self contained.

**Thursday, October 20, 2016****Given a compact Riemannian manifold $(M,g)$ with boundary, the geodesic ray transform is the mapping which takes a function on $M$ to its integrals over the maximally extended geodesics of $(M,g)$. We are interested primarily in two questions: whether this transform is injective, and whether there is a stability estimate between appropriate Sobolev spaces for its inversion. It is well known that for so-called “simple manifolds”, which in particular do not have caustics, the transform is injective, and there is a stability estimate. On the other hand, in the two dimensional case it has been proven that as soon as there are caustics no stability estimate between any Sobolev spaces is possible. This is the case even though there are two dimensional examples which have caustics, but for which the transform is injective. The question motivating this talk is whether the same phenomenon happens in three dimensions. The talk will examine recent results on the stability of the inversion of the geodesic ray transform in the presence of caustics in three dimensions, contrasting them with what is known on the injectivity.**

Sean Holman, Unviersity of Manchester**"Stability of the geodesic ray transform in the presence of caustics"**Sean Holman, Unviersity of Manchester

**Thursday,**October 27, 2016**We will begin with two situations in which the Schrodinger-Newton system of equations arise: as the non relativistic limit of the Einstein Klein Gordon equation in the study of scalar field dark matter (SFDM), and as a description of quantum collapse due to gravitational interaction. After this, we will discuss the character of the solutions to these equations, mainly in showing the existence of a discrete family of bound eigenstates in the case of spherical symmetry.**

Robert Hingtgen, UC Santa Cruz**The Schrodinger-Newton system**Robert Hingtgen, UC Santa Cruz

Robert Hingtgen, UC Santa Cruz

**Thursday, November 3**, 2016**"The Schrodinger-Newton system" (continued)**Robert Hingtgen, UC Santa Cruz

In the continuation of the last talk, we will discuss the analytical methods used in assessing the character of solutions to the Schrodinger-Newton system. We will then make a small aside in discussing the solutions of the Schrodinger equation in the case of the Hydrogen atom before returning to the SN system and viewing it as the Schrodinger equation with a perturbation from the Coulomb potential.

**Thursday,**November 10, 2016*"The geodesic X-ray transform with a GL(n,C) connection on simple surfaces"***Francois Monard, UC Santa Cruz**

The geodesic X-ray transform with a GL(n,C) connection on simple surfaces

We will discuss recent results regarding the injectivity and inversion of geodesic X-ray transforms with connections, defined on certain Riemannian surfaces with boundary. Such a problem arises for instance as the linearization of the inverse problem of reconstructing a connection from knowledge of its parallel transport along geodesics (in short, its "scattering data"). It also arises in medical imaging applications, in the context of attenuated x-ray transforms where the attenuation term depends linearly on the tangent vector.

While prior litterature tackled the case of injectivity for unitary connections, the present case tackles inversion for general, non-unitary ones. The starting point is the derivation of Fredholm inversion formulas, obtained by studying certain transport equations on the unit tangent bundle. The error operators involved are then explicit enough, that further properties can be inferred on the injectivity of such equations (and as a result, of the transform itself) for almost all connections, including a significant drop in regularity requirements.

Numerical examples will be presented at the end.

This is joint work with Gabriel Paternain (Cambridge).

**Thursday,**November 17, 2016

*"Stable shock formation for solutions to the multidimensional compressible Euler equations in the presence of non-zero vorticity"*

Jonathan Luk, Stanford UniversityJonathan Luk, Stanford University

It is well-known since the foundational work of Riemann that plane symmetric solutions to the compressible Euler equations may form shocks in finite time. For a class of simple plane symmetric solutions, we prove that the phenomenon of shock-formation is stable under perturbations of the initial data that break the plane symmetry with potentially non-vanishing vorticity. In particular, this is the first constructive shock-formation result for which the vorticity is allowed to be non-vanishing at the shock. We show in particular that the vorticity remains bounded all the way up to the shock, and that the dynamics are well-described by the irrotational compressible Euler equations. This is a joint work with J. Speck (MIT), which is partly an extension of an earlier joint work with J. Speck (MIT), G. Holzegel (Imperial) and W. Wong (Michigan State).

Thanksgiving Holiday!!

**Thursday,**November 24, 2016***NO SEMINAR***Thanksgiving Holiday!!

Xin Zhou, UC Santa Barbara

**Friday, December 2**, 2016 *change of day* 4:00p.m.**Min-max minimal hypersurfaces with free boundary**Xin Zhou, UC Santa Barbara

This is a joint work with Martin Li. Minimal surfaces with free boundary are natural ctirical points of the area functional in compact smoth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with gree boundary in any compact smooth Euclidean domain. The minimal surface with gree boundary were constructed using the celebrated min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions. At the end, I will introduce a few conjetures that can be approaced based on our work.

Wei Yuan, Sun-Yat Sen University, China

**Friday, December 2**, 2016 *change of day* 3:00p.m.**"Volume comparison with respect to scalar curvature"**

Wei Yuan, Sun-Yat Sen University, China

In this talk, we will investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison hold for small geodesic balls of metrics near V-static metrics, which are exactly those metrics we can expect such comparison hold. As for global results, we give volume comparison for metrics near Einstein metrics with certain restrictions. As an application, we recover a volume comparison result of compact hyperbolic manifolds due to Besson-Courtois-Gallot, which provides a partial answer to a conjecture of Schoen on volume of hyperbolic manifolds.