# The construction of the splitting maps

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For a geodesic ball with non-negative Ricci curvature and almost

maximal volume, we give the existence proof of splitting map without

compactness argument. There are two technical new points, the first one is

the way of finding n-directional points by induction and stratified Gou-Gu

Theorem, the second one is the error estimates of projections. The content

of the talk is technical, but we will explain the basic geometric intuition

behind the technical proof. This is a joint work with Jie Zhou.

# Ancient mean curvature flows and their applications to topology

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The mean curvature flow is an evolution of hypersurfaces satisfying a geometric heat equation. The mean curvature flow in general develops singularities, and the topology of the hypersurface is changed through singularities. To study the topological change, we consider ancient flows which can be obtained by the blow-up of the flow at singularities. In this talk, we will discuss how to use the classification result of ancient flows for singularities analysis and topology.

# Unitary Representations of 3-manifold Groups and the Atiyah-Floer Conjecture

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A useful tool to study a 3-manifold is the space of the

representations of its fundamental group, a.k.a. the 3-manifold group, into

a Lie group. Any 3-manifold can be decomposed as the union of two

handlebodies. Thus, representations of the 3-manifold group into a Lie group

can be obtained by intersecting representation varieties of the two

handlebodies. Casson utilized this observation to define his celebrated

invariant. Later Taubes introduced an alternative approach to define Casson

invariant using more geometric objects. By building on Taubes' work, Floer

refined Casson invariant into a graded vector space whose Euler

characteristic is twice the Casson invariant. The Atiyah-Floer conjecture

states that Casson's original approach can be also used to define a graded

vector space and the resulting invariant of 3-manifolds is isomorphic to

Floer's theory. In this talk, after giving some background, I will give an

exposition of what is known about the Atiyah-Floer conjecture and discuss

some recent progress, which is based on a joint work with Kenji Fukaya and

Maksim Lipyanskyi. I will only assume a basic background in algebraic

topology and geometry.

# Weak solutions for general complex Hessian equations

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Joint with Analysis seminar.

Pluripotential solutions to the complex Monge-Ampere equations are very useful in modern complex geometry. It is thus very natural to seek analogues for other Hessian type complex equations that appear in geometric analysis. In the talk I will discuss several approaches towards such a theory and their limitations. Then I will focus on a newly coined notion of a $L^p$-viscosity solutions and discuss uniform estimates for a large class of Hessian equations. This is a joint work with S. Abja and G. Olive.

# Hermitian geometry and curvature positivity

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A general theme in differential/complex geometry is that curvature positivity conditions imposes strong geometric and topological constraint on the underlying manifold. In this talk, I will discuss a new curvature positivity condition in Hermitian geometry and prove a liouville type theorem for (1, 1)-forms for manifolds satisfying the positivity condition. I will discuss various interactions between this curvature condition and other notions in Hermitian geometry. Lastly, I will discuss some examples and potential applications.

# PDEs on Non-Kahler Calabi-Yau Manifolds

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We will discuss the non-Kahler Calabi-Yau geometry introduced by string theorists C. Hull and A. Strominger. We propose to study these spaces via a parabolic PDE which is a nonlinear flow of non-Kahler metrics. This talk will survey works with T. Collins, T. Fei, D.H. Phong, S.-T. Yau, and X.-W. Zhang.

# 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.