Schedule

Speaker: Richard Schoen (UC Irvine)

Title: How minimal hypersurface and MOTS singularities affect relativity theorems 

Abstract: Some theorems in relativity work directly only in low dimensions because of the possibility of singularities in area minimizing hypersurfaces and MOTS. These include both Riemannian and spacetime positive mass theorems as well as the Riemannian Penrose inequality. In some cases the singularities are relatively easy to work around and in others they present significant problems to overcome. This will be a general lecture which will attempt to give perspective and more clarity to this issue. 

Lunch Break 12:00 - 1:30

Speaker: Guofang Wei (UC Santa Barbara) 

Title: Singular Weyl's law with Ricci curvature bounded below 

Abstract: The classical Weyl’s law describes the asymptotic behavior of eigenvalues of the Laplace Beltrami operator in terms of the geometry of the underlying space. Namely, the growth order is given by (half of) the dimension and the limit by the volume. The study has a long history and is important in mathematics and physics. In a recent joint work with X. Dai, S. Honda and J. Pan, we find two surprising types of Weyl’s laws for some compact Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm as some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for Ricci limit spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed by J. Pan and G. Wei (GAFA 2022).

Speaker: Xiaodong Cao (Cornell University)

Title: A short Survey on Einstein 4-manifolds and Ricci Solitons 

Abstract: In this talk, we will give some update on Einstein 4-manifolds with various positive curvatures; we will also survey some recent development on 4-dimensional Ricci solitons.

Speaker: Ailana Fraser (University of British Columbia) 

Title: Higher codimension minimal surfaces in Riemannian geometry 

Abstract: A fundamental question in Riemannian geometry is to understand the relationships between the curvature and the geometry and topology of Riemannian manifolds. The classical theorem of Bonnet-Myers gives an upper bound on the length of any stable geodesic in terms of a lower positive bound on the Ricci curvature. In this talk we will discuss Bonnet-Myers type theorems for stable minimal surfaces in manifolds with positive isotropic curvature. We will also discuss related Bernstein questions.

End of the day

Speaker: Peter Petersen (UCLA)

Title: A Universal Principle for Manifolds

Abstract: We will offer an easy to check principle that can be used to prove several results: The full Whitney Embedding Theorem and for de Rham cohomology: Homotopy Invariance, The de Rham Isomorphism Theorem, Poincaré Duality, Künneth Theorem etc. The principle bypasses the need for a “good” cover and the usual Leray-Hirsch spectral sequence arguments.

Speaker: Matthew Gursky (University of Notre Dame)

Title: Some rigidity results for 4-d (AH) Einstein metrics 

Abstract: I will begin by reviewing some older rigidity results for Einstein metrics on closed manifolds, then describe some results in the asymptotically hyperbolic setting that can be seen as their natural extensions. This is joint work with Stephen McKeown and Aaron Tyrrel.

Speaker: Lu Wang (Yale University)

Title: A mean curvature flow approach to density of minimal cones 

Abstract: Minimal cones are models for singularities in minimal submanifolds, as well as stationary solutions to the mean curvature flow. In this talk, I will explain how to utilize mean curvature flow to yield near optimal estimates on density of topologically nontrivial minimal cones. This is joint with Jacob Bernstein.

End of Conference