A free boundary problem in pseudoconvex domains

Speaker: 

Chi Fai Chau

Institution: 

UC, Irvine

Time: 

Tuesday, November 28, 2023 - 4:00pm

Host: 

Location: 

ISEB 1200

A domain with C^2 boundary in complex space is called pseudoconvex if it has a C^2 defining function with positive complex hessian on its boundary. Pseudoconvexity is a generalization of convexity. It can be realised as a domain with geometric condition on the boundary and its topology can be studied by Morse theory. In this talk, we will discuss the Morse index theorem for free boundary minimal disks for partial energy in strictly pseudoconvex domain and the relation between holomorphicity and stability of the free boundary minimal disk. We will also give an example to illustrate the necessity of strict pseudoconvexity in our index estimate.

On complete Calabi-Yau manifolds asymptotic to cones

Speaker: 

Junsheng Zhang

Institution: 

UC Berkeley

Time: 

Tuesday, November 14, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

ISEB 1200

We proved a ``no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate  to the asymptotic cone for such manifolds. Joint work with Song Sun.

 

The parabolic U(1)-Higgs equations and codimension-two mean curvature flows

Speaker: 

Davide Parise

Institution: 

UC San Diego

Time: 

Tuesday, October 3, 2023 - 4:00pm

Location: 

ISEB 1200

Mean curvature flow is the negative gradient flow of the area
functional, and it has attracted a lot of interest in the past few years. In
this talk, we will discuss a PDE-based, gauge theoretic, construction of
codimension-two mean curvature flows based on the Yang-Mills-Higgs
functionals, a natural family of energies associated to sections and metric
connections of Hermitian line bundles. The underlying idea is to approximate
the flow by the solution of a parabolic system of equations and study the
corresponding singular limit of these solutions as the scaling parameter
goes to zero. This is based on joint work with A. Pigati and D. Stern.

The transformation theorem for type-changing semi-Riemannian manifolds

Speaker: 

Louann Rieger

Institution: 

University of Zurich

Time: 

Tuesday, October 17, 2023 - 4:00pm

Host: 

Location: 

ISEB 1200

In 1983 Hartle and Hawking put forth that signature type-change may be conceptually interesting, leading to the so-called no-boundary proposal for the initial conditions for the universe, which has no beginning because there is no singularity or boundary to the spacetime. But there is an origin of time. In mathematical terms, we are dealing with signature type-changing manifolds where a positive definite Riemannian region is smoothly joined to a Lorentzian region at the surface of transition where time begins.

We utilize a transformation prescription to transform an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Then we prove the transformation theorem saying that locally the metric \tilde{g} associated with a signature-type changing manifold (M, \tilde{g}) is equivalent to the metric obtained from a Lorentzian metric g via the aforementioned transformation prescription. By augmenting the assumption by certain constraints, mutatis mutandis, the global version of the transformation theorem can be proven as well.

The transformation theorem provides a useful tool to quickly determine whether a singular signature type-changing manifold under consideration belongs to the class of transverse type changing semi-Riemannian manifolds.

Harmonic maps in general relativity

Speaker: 

Sumio Yamada

Institution: 

Gakushuin University

Time: 

Tuesday, May 2, 2023 - 4:00pm

Host: 

Location: 

ISEB 1200

Herman Weyl in 1916 described the Schwarzschild metric
by a single harmonic function with a pole.  Since then, the Einstein
equation with time symmetry can be regarded as an elliptic variational
problem, and I will report on the recent progress in this direction,
including a series of collaborative work with Gilbert Weinstein and Marcus
Khuri.  We will introduce spactimes of dimension four and five with some
rotational symmetries, and discuss the difference in 4 and 5, and some new
geometric consequences..

On the moduli spaces of ALH*-gravitational instantons

Speaker: 

Yu-Shen Lin

Institution: 

Boston University

Time: 

Monday, June 5, 2023 - 3:30pm

Location: 

RH 340N

Gravitational instantons are defined as non-compact hyperKahler
4-manifolds with L^2 curvature decay. They are all bubbling limits of K3
surfaces and thus serve as stepping stones for understanding the K3 metrics.
In this talk, we will focus on a special kind of them called
ALH*-gravitational instantons. We will explain the Torelli theorem, describe
their moduli spaces and some partial compactifications of the moduli spaces.
This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi,
X. Zhu and S. Soundararajan.

Special date/time and joint with Geometry and Topology Seminar.

The Curvature Operator of the Second Kind

Speaker: 

Xiaolong Li

Institution: 

Wichita State University

Time: 

Tuesday, May 23, 2023 - 4:00pm

Location: 

ISEB 1200

The Riemann curvature tensor on a Riemannian manifold induces two
kinds of curvature operators: the first kind acting on two-forms and the
second kind acting on (traceless) symmetric two-tensors. The curvature
operator of the second kind recently attracted a lot of attention due to the
resolution of Nishikawa's conjecture by X.Cao-Gursky-Tran and myself. In
this talk, I will survey some recent works on the curvature operator of the
second kind on Riemannian and Kahler manifolds and also mention some
interesting open problems. The newest result, joint with Harry Fluck at
Cornell University, is an investigation of the curvature operator of the
second kind in dimension three and its Ricci flow invariance.

Quivers, stacks, and mirror symmetry

Speaker: 

Siu-Cheong Lau

Institution: 

Boston University

Time: 

Tuesday, June 6, 2023 - 4:00pm

Location: 

ISEB 1200

In this talk, we will start by introducing quiver representations
and some of their applications.  Then we will review noncommutative crepant
resolutions of singularities of Van den Bergh.  We will find that the notion
of quiver stacks will be useful in unifying geometric and quiver
resolutions.  Finally, we will explain our motivation and construction of
these quiver stacks from a symplectic mirror point of view.

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