The construction of the splitting maps

Speaker: 

Guoyi Xu

Institution: 

Tsinghua University

Time: 

Tuesday, March 9, 2021 - 4:00pm

Location: 

Zoom

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

Speaker: 

Kyeongsu Choi

Institution: 

Korea Institute for Advanced Study (KIAS)

Time: 

Tuesday, February 9, 2021 - 4:00pm to 5:00pm

Location: 

Zoom

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

Speaker: 

Aliakbar Daemi

Institution: 

Washington University in St. Louis

Time: 

Tuesday, February 23, 2021 - 4:00pm

Location: 

Zoom

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

Speaker: 

Slawomir Dinew

Institution: 

Jagiellonian University

Time: 

Tuesday, January 12, 2021 - 3:00pm to 4:00pm

Location: 

Zoom

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

Speaker: 

Freid Tong

Institution: 

Columbia University

Time: 

Tuesday, December 8, 2020 - 4:00pm to 5:00pm

Location: 

Zoom

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

Speaker: 

Sebastien Picard

Institution: 

University of British Columbia (UBC)

Time: 

Tuesday, December 1, 2020 - 4:00pm to 5:00pm

Location: 

ZOOM

 

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

Speaker: 

Yu-Wei Fan

Institution: 

UC Berkeley

Time: 

Tuesday, October 13, 2020 - 4:00pm

Location: 

Zoom

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

Speaker: 

Chao Li

Institution: 

Princeton University

Time: 

Tuesday, October 27, 2020 - 4:00pm

Host: 

Location: 

Zoom

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

Speaker: 

Xin Fu

Institution: 

UC Irvine

Time: 

Tuesday, October 6, 2020 - 4:00pm to 5:00pm

Host: 

Location: 

Zoom - https://uci.zoom.us/j/91999884028

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.

 

Link:  https://uci.zoom.us/j/91999884028

Pages

Subscribe to RSS - Differential Geometry