# On categorical dynamical systems

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One can regard a self-map on a space as a dynamical system, and study its

long-term behavior under larger iterations. In this talk, we will introduce

the categorical version of such dynamical systems, and establish invariants

that one can associate to endofunctors of categories from the dynamical

perspective. We will also explain some results on categorical dynamical

systems that are related to holomorphic dynamics, symplectic dynamics,

Teichmuller theory, and Poincare rotation numbers. No background in

dynamical systems is assumed for this talk.

# Generalized soap bubbles and the topology of manifolds with positive scalar curvature

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Abstract: It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will present some recent progress on this question, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov), as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. Applications include a Schoen-Yau Liouville theorem for all locally conformally flat manifolds. The proofs of these results are based on analyzing generalized soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. This talk is based on joint work with O. Chodosh.

# On the Kahler-Einstein metric near isolated log-canonical singularity

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We construct K ̈ahler-Einstein metric with negative curvature near an isolated log canonical singularity by solving Monge-Amp`ere equation with Dirichlet boundary. We continue to consider the geometry of the Kahler-Einstein metric we constructed. In particular, in complex dimension 2, we show that all complete local K ̈ahler-Einstein metrics near isolated singularity are asymptotic the same as the model metric constructed by Kobayashi and Nakamura.

# Semi-classical isotropic functions and applications

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Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semi-classical analysis. In this talk I will describe how to associate spaces of semi-classical oscillatory functions to isotropic submanifolds of phase space, and sketch their symbol calculus. As a special case we obtain the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin. I will also discuss a couple applications of the theory. This is based on joint works with Victor Guillemin and Alejandro Uribe.

# Semi-classical isotropic functions and applications

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Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semi-classical analysis. In this talk I will describe how to associate spaces of semi-classical oscillatory functions to isotropic submanifolds of phase space, and sketch their symbol calculus. As a special case we obtain the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin. I will also discuss a couple applications of the theory. This is based on joint works with Victor Guillemin and Alejandro Uribe.

# SEMINAR CANCELLED - Classification Results for Expanding and Shrinking gradient Kahler-Ricci solitons.

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A complete Kahler metric g on a Kahler manifold M is a "gradient Kahler-Ricci soliton" if there exists a smooth real-valued function f:M-->R with \nabla f holomorphic such that Ric(g)-Hess(f)+\lambda g=0 for \lambda a real number. I will present some classification results for such manifolds. This is joint work with Alix Deruelle (Université Paris-Sud) and Song Sun (UC Berkeley).

# Isoperimetry and volume preserving stability in real projective spaces

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In this talk I will address the problem of classifying volume

preserving stable constant mean curvature hypersurfaces in Riemannian

manifolds. I will present recent classification in the real projective space

of any dimension and, consequently, the solution of the isoperimetric

problem.

# Scalar curvature and harmonic functions

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We'll discuss a new technique for relating scalar curvature bounds to the global structure of 3-dimensional manifolds, exploiting a relationship between the scalar curvature and the topology of level sets of harmonic functions. We will describe several geometric applications in both the compact and asymptotically flat settings, including a simple and effective new proof (joint with Bray, Kazaras, and Khuri) of the three-dimensional Riemannian positive mass theorem.

# Epsilon-regularity for minimal surfaces near quadratic cones

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Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen