Location: Engineering Lecture Hall 100


Saturday, January 27, 2018
10:00AM -11:00AM Registration
11:00AM -12:00PM Tristan Rivière A variational approach to minimal surfaces in arbitrary codimensions
12:00PM Group Picture next to Engineering Tower
12:10PM -1:30PM Lunch at Phoenix Food Court
1:30PM -2:30PM Chikako Mese Harmonic maps into CAT(1) spaces
2:30PM -3:00PM Break
3:00PM -4:00PM Song Sun Degenerations of Calabi-Yau metrics
4:00PM -4:30PM Break
4:30PM -5:30PM Shing-Tung Yau Invariant metrics on negatively pinched complete Kähler manifolds
6:30PM -9:00PM Banquet Dinner at Capital Seafood Irvine Spectrum
Sunday, January 28, 2018
9:00AM -10:00AM Vladimir Markovic The analytic shapes of Teichmüller spaces
10:00AM -10:20AM Break
10:20AM -11:20AM Shouhei Honda Spectral properties on metric measure spaces with Ricci bounds from below
11:20AM -11:30AM Break
11:30AM -12:30PM Peter Topping Ricci flow and Ricci limit spaces

  Titles and Abstracts:

Shouhei Honda (Tohoku University), Spectral properties on metric measure spaces with Ricci bounds from below.

In 2000, Cheeger-Colding proved that under measured Gromov-Hausdorff (mGH) convergence with Ricci bounds from below, called the Ricci limit setting, the eigenvalues of the Laplacian behave continuously. This result is now called the Spectral Convergence (SC), and it was conjectured by Fukaya in 1987 who introduced the notion of mGH-convergence with the same conclusion under bounded sectional curvature. The SC can be rewritten equivalently by a behavior on $H^{1, 2}$-Sobolev functions, and it is now extended to a more general setting, called RCD-setting, by Gigli-Mondino-Savare in 2015.

In this talk, I will explain two generalizations of this SC in the RCD-setting. The first one is on nonlinear PDEs, including $p$-Laplacian for all $1 \le p \le \infty$. The SC above corresponds to the case when $p=2$. In particular, we will prove the continuity of Cheeger's isoperimetric constants via $1$-Laplacian and BV-functions. The second one is on the local setting. Although the local SC is not satisfied in general, however, it holds in the "generic" case. This generic property provides an affirmative answer to a question by Petrunin in 2003 in a stronger form. If there is sufficient time, I will explain Weyl's law and further developments related to spectral properties on such spaces. These are new even in the Ricci limit setting. This talk is based on joint works with L.Ambrosio, L.Ambrosio-D.Tewodrose, and L.Ambrosio-J.Portegies.


Vladimir Markovic (Caltech), The analytic shapes of Teichmüller spaces.

Teichmüller spaces carry a lot of structure. As complex manifolds they biholomorphically embed as bounded domains in $C^n$. We discuss geometric properties of such embeddings and the role of Teichmüller dynamics in this theory.


Chikako Mese (Johns Hopkins University)
, Harmonic maps into CAT(1) spaces.

The pioneering work of Gromov-Schoen and Korevaar-Schoen established the theory of harmonic maps into NPC spaces, i.e. complete metric space of non-positive curvature. In this talk, we will discuss harmonic map theory when we relax the curvature assumption and consider metric spaces with curvature bounded from above.


Tristan Rivière (ETH Zürich), A variational approach to minimal surfaces in arbitrary codimensions.

We shall present a new variational approach to the construction of minimal 2-dimensional surfaces in arbitrary closed manifold. From the analysis side, the main strategy consists in applying Palais deformation theory to viscous approximations of the area and to pass to the limit as the viscous parameter tend to zero. We shall review the difficulties for implementing this a-priori simple scheme. This will bring us in particular to the study of parametrized stationary varifolds for which we shall present the complete regularity theory. We shall also address the problem of controlling the index of the limiting minimal surfaces. In the second part of the talk, we will present the more geometric side of this variational approach that we call "minmax hierarchies'' and we will present its implementation on $S^3$.


Song Sun (UC Berkeley), Degenerations of Calabi-Yau metrics.

Yau's solution to the Calabi conjecture gives Ricci-flat metrics on compact Kahler manifolds with vanishing first Chern class. These metrics often occur in natural families and it is an interesting question to understand singularity formation of these metrics, and the relation with algebraic geometry. I will discuss some recent results around this.


Peter Topping (Warwick University), Ricci flow and Ricci limit spaces.

Ricci flow theory has been developing rapidly over the last couple of years, with the ability to handle Ricci flows with unbounded curvature finally becoming a reality. This is vastly expanding the range of potential applications. I will describe some recent work in this direction with Miles Simon that shows the right way to pose the 3D Ricci flow in this setting in order to obtain applications. Amongst these applications is a proof that 3D Ricci limit spaces are locally bi-Hölder homeomorphic to smooth manifolds, which solves more than an old conjecture of Anderson-Cheeger-Colding-Tian in this dimension.


Shing-Tung Yau (Harvard University), Invariant metrics on negatively pinched complete Kähler manifolds.