Geometry of Lagrangian submanifolds related to isoparametric hypersurfaces

Speaker: 

Yoshihiro Ohnita

Institution: 

Dept of Math, Osaka City University, Japan

Time: 

Tuesday, March 11, 2014 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In this talk I shall provide a survey of my recent works and their environs on differential geometry of Lagrangian submanifolds in specific K\"ahler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and so on. I shall emphasis on the relationship between certain minimal Lagrangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres. This talk is mainly based on my joint work with Associate Professor Hui Ma (Tsinghua University, Beijing).

Polar actions on symmetric spaces

Speaker: 

Jurgen Berndt

Institution: 

Dept of Math, King's College London

Time: 

Tuesday, February 18, 2014 - 4:00pm to 5:00pm

Location: 

RH 306

An isometric action of a connected Lie group on a Riemannian manifold is called polar if there exists a connected closed submanifold that meets each orbit of the action and intersects it orthogonally. Dadok established in 1985 a remarkable, and mysterious, relation between polar actions on Euclidean spaces and Riemannian symmetric spaces. Soon afterwards an attempt was made to classify polar actions on symmetric spaces. For irreducible symmetric spaces of compact type the final step of the classification has just been achieved by Kollross and Lytchak. In the talk I want to focus on symmetric spaces of noncompact type. For actions of reductive groups one can use the concept of duality between symmetric spaces of compact type and of noncompact type. However, new examples and phenomena arise from the geometry induced by actions of parabolic subgroups, for which there is no analogon in the compact case. I plan to discuss the main difficulties one encounters here and some partial solutions.

Finite subgroups of symplectic Cremona group

Speaker: 

Weiwei Wu

Institution: 

Michigan State University

Time: 

Tuesday, January 28, 2014 - 4:00pm

Location: 

RH 306

Finite subgroup of Cremona group is a classical topic in algebraic geometry since the 19th century.  In this talk we explain an extension of this problem to the symplectic category.  In particular, we will explain the symplectic counterparts of two classical theorems.  The first one due to Noether, says a plane Cremona map is decomposed into a sequence of quadratic transformations, which is generalized to the symplectic category on the homological level.  The second one is due to Castelnuovo and Kantor, which says a minimal G-surface either has a conic bundle structure or is a Del Pezzo surface.  The latter theorem lies the ground of classifications of finite Cremona subgroups due to Dolgachev and Iskovskikh.  This is an ongoing program joint with Weimin Chen and Tian-Jun Li

There are finitely many surgeries in Perelman's Ricci flow

Speaker: 

Richard Bamler

Institution: 

Stanford University

Time: 

Tuesday, March 18, 2014 - 4:00pm to 5:00pm

Location: 

RH 306

Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré
and Geometrization Conjectures, some of its basic properties are still unknown. For
example it has been an open question whether the surgeries eventually stop to occur
(i.e. whether there are finitely many surgeries) and whether the full geometric
decomposition of the underlying manifold is exhibited by the flow as times goes to infinity.

In this talk I will show that the number of surgeries is indeed finite and that the
curvature is globally bounded by C t^{-1} for large t. Using this curvature
bound it is possible to give a more precise picture of the long-time behavior of the
flow.

Ricci Curvature and the manifold learning problem

Speaker: 

Antonio Ache

Institution: 

Princeton University

Time: 

Tuesday, January 21, 2014 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

In the first half of this talk we will review several notions of coarse or weak
Ricci Curvature on metric measure spaces which include the work of Yann
Ollivier. The discussion of the notion of coarse Ricci curvature will serve as
motivation for developing a method to estimate the Ricci curvature of a an
embedded submaifold of Euclidean space from a point cloud which has applications
to the Manifold Learning Problem. Our method is based on combining the notion of
``Carre du Champ" introduced by Bakry-Emery with a result of Belkin and Niyogi
which shows that it is possible to recover the rough laplacian of embedded
submanifolds of the Euclidean space from point clouds. This is joint work with
Micah Warren.

Partial C^0-estimates

Speaker: 

Gang Tian

Institution: 

Princeton University, Beijing University

Time: 

Thursday, December 5, 2013 - 4:00pm to 5:00pm

Host: 

Location: 

NatSci 1201

It has been a challenging problem to studying the existence of Kahler-Einstein metrics on Fano manifolds. A Fano manifold is a compact Kahler manifold with positive first Chern class. There are obstructions to the existence of Kahler-Einstein metrics on Fano manifolds. In these lectures, I will report on recent progresses on the study of Kahler-Einstein metrics on Fano manifolds. The first lecture will be a general one. I will discuss approaches to studying the existence problem. I will discuss the difficulties and tools in these approaches and results we have for studying them. In the second lecture, I will discuss the partial C^0-estimate which plays a crucial role in recent progresses on the existence of Kahler-Einstein metrics. I will show main technical aspects of proving such an estimate.

Obstructions in rational cohomology to positive curvature and symmetry

Speaker: 

Lee Kennard

Institution: 

UC Santa Barbara

Time: 

Tuesday, January 14, 2014 - 4:00pm to 5:00pm

Location: 

RH 306

In large dimensions, the only known compact, simply connected Riemannian manifolds with positive sectional curvature are spheres and projective spaces. The natural metrics on these spaces have large isometry groups, so it is natural to consider highly symmetric metrics when searching for new examples. On the other hand, there are many topological obstructions to a manifold admitting a positively curved metric with large symmetry. I will discuss a new obstruction in this setting. This is joint work with Manuel Amann (KIT).

Sharp estimate on the first positive eigenvalue of Kohn Laplacian and rigidity theorem

Speaker: 

Son Ngoc Duong

Institution: 

UC Irvine

Time: 

Tuesday, November 19, 2013 - 4:00pm

Location: 

RH 306

In this talk, I will present the sharp estimate for the first positive eigenvalue
of the the Kohn Laplacian and an Obata (1962) type theorem on the characterization
of the (CR) sphere for closed Pseudo-Hermitian Manifolds.

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