The geometry of k-Ricci curvature and a Monge-Ampere equation

Speaker: 

Lei Ni

Institution: 

UC San Diego

Time: 

Tuesday, May 31, 2022 - 4:00pm

Location: 

ISEB 1200

A joint Geometry and Analysis seminar.

 

Abstract: The k-Ricci curvature interpolates between the Ricci curvature and holomorphic sectional curvature. For the positive case, a recent result asserts that the compact Kaehler manifolds with positive k-Ricci are  projective and rationally connected. This generalizes the previous results of Campana, Kollar-Miyaoka-Mori for the Fano case and the Heirer-Wong and Yang for holomorphic sectional curvature case. For the negative case, all compact Kaehler manifolds with negative k-Ricci admit a Kaehler-Einstein metric with negative scalar curvature. I shall explain how to get this result by solving a complex Monge-Ampere equation.

A nonlinear spectrum on closed manifolds

Speaker: 

Christos Mantoulidis

Institution: 

Rice University

Time: 

Tuesday, May 17, 2022 - 4:00pm

Host: 

Location: 

ISEB 1200

Abstract: The p-widths of a closed Riemannian manifold are a nonlinear 
analogue of the spectrum of its Laplace--Beltrami operator, which was 
defined by Gromov in the 1980s and corresponds to areas of a certain 
min-max sequence of hypersurfaces. By a recent theorem of 
Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like 
the eigenvalues do. However, even though eigenvalues are explicitly 
computable for many manifolds, there had previously not been any >= 
2-dimensional manifold for which all the p-widths are known. In recent 
joint work with Otis Chodosh, we found all p-widths on the round 
2-sphere and thus the previously unknown Liokumovich--Marques--Neves 
Weyl law constant in dimension 2.

 

Hamilton-Ivey estimates for gradient Ricci solitons

Speaker: 

Pak-Yeung Chan

Institution: 

UC San Diego

Time: 

Tuesday, May 24, 2022 - 4:00pm

Location: 

ISEB 1200

One special feature for the Ricci flow in dimension 3 is the
Hamilton-Ivey estimate. The curvature pinching estimate provides a lot of
information about the ancient solution and plays a crucial role in the
singularity formation of the flow in dimension 3. We study the pinching
estimate on 3 dimensional expanding and 4 dimensional steady gradient Ricci
solitons. A sufficient condition for a 3-dimensional expanding soliton to
have positive curvature is established. This condition is satisfied by a
large class of conical expanders. As an application, we show that any
3-dimensional gradient Ricci expander C^2 asymptotic to certain cones is
rotationally symmetric. We also prove that the norm of the curvature tensor
is bounded by the scalar curvature on 4 dimensional non Ricci flat steady
soliton singularity model and derive a quantitative lower bound of the
curvature operator for 4-dimensional steady solitons with linear scalar
curvature decay and proper potential function. This talk is based on a joint
work with Zilu Ma and Yongjia Zhang.

Geometric Flows of G2-Structures

Speaker: 

Aaron Kennon

Institution: 

UC Santa Barbara

Time: 

Tuesday, May 3, 2022 - 4:00pm to 5:00pm

Host: 

Location: 

ISEB 1200

There are several natural flows of G2-Structures that may be useful for resolving important problems in the theory of G2-Holonomy Manifolds. Here we review what is known and discuss some open problems for two such flows: The Laplacian flow and the Laplacian co-flow. We will then present some new results for each of these flows. 

Degeneration of 7-dimensional minimal hypersurfaces with bounded index

Speaker: 

Nick Edelen

Institution: 

University of Notre Dame

Time: 

Tuesday, April 26, 2022 - 4:00pm

Host: 

Location: 

ISEB 1200

A 7D minimal and locally-stable hypersurface will in general have a discrete singular set, provided it has no singularities modeled on a union of half-planes. We show in this talk that the geometry/topology/singular set of these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are bi-Lipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$ embedded minimal hypersurfaces in a fixed $8$-manifold, having index $\leq I$, area $\leq \Lambda$, and discrete singular set, divides into finitely-many diffeomorphism types.

Rigidity and instability of SU(n) symmetric spaces

Speaker: 

Thomas Murphy

Institution: 

CSU Fullerton

Time: 

Tuesday, April 12, 2022 - 4:00pm

Location: 

ISEB 1200

Focusing on the Lie group SU(n) and associated symmetric spaces,
I investigate two related topics. The main result is that the bi-invariant
Einstein metric on SU(2n+1) is isolated in the moduli space of Einstein
metrics, even though it admits infinitesimal deformations. This gives a
non-Kaehler, non-product example of this phenomenon adding to the famous
example of Koiso from the eighties.  I also explore the relationship between
the question of rigidity and instability (under the Ricci flow) of Einstein
metrics, and present results in this direction for complex Grassmannians.

Cohomology and Morse theory on symplectic manifolds

Speaker: 

David Clausen

Institution: 

UC Irvine

Time: 

Tuesday, April 19, 2022 - 4:00pm

Location: 

ISEB 1200

On symplectic manifolds, there are intrinsincally symplectic cohomologies of differential forms that are analogous to the Dolbeault cohomology on complex manifolds. These cohomologies are isomorphic to the de Rham cohomologies on odd-dimensional sphere bundles over the symplectic manifold. In this talk, I will describe how we can use this sphere bundle perspective to define a novel Morse-type theory on symplectic manifolds associated with the symplectic cohomologies. This is joint work with Xiang Tang and Li-Sheng Tseng.

Moduli spaces in algebraic geometry

Speaker: 

Kenneth Ascher

Institution: 

UC Irvine

Time: 

Tuesday, March 8, 2022 - 4:00pm

Location: 

NS2 1201

Understanding moduli spaces is one of the central questions in
algebraic geometry. This talk will survey one of the main aspects of
research in moduli theory — the compactification problem. Roughly speaking,
most naturally occurring moduli spaces are not compact and so the goal is to
come up with geometrically meaningful compactifications. We will begin by
looking at the case of algebraic curves (i.e. Riemann surfaces) and progress
to higher dimensions, where the theory is usually divided into three main
categories: general type (i.e. negatively curved), Calabi-Yau (i.e. flat),
and Fano (i.e. positively curved, where the theory is connected to the
existence of KE metrics). Time permitting, we will use this motivation to
discuss moduli spaces of K3 surfaces (simply connected compact complex
surfaces with a no-where vanishing holomorphic 2-form).

(A joint seminar with the Geometry & Topology Seminar series.)

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