Jan 14th: Xin Fu, UC Irvine
Codimension four regularity of generalized Einstein structures
Abstract:
We establish codimension 4 regularity of noncollapsed sequences of metrics with bounds on natural generalizations of the Ricci tensor. We obtain a priori L2 curvature estimates on such spaces, with diffeomorphism finiteness results and rigidity theorems as corollaries.
This is Joint with Aaron Naber, Jeffrey Streets.
Zoom Recording Passcode: !b.M7K+t
Jan 21th: Po-Ning Chen, UC Riverside
Conserved quantities in general relativity.
Abstract:
One of the fundamental problems in general relativity is to define conserved quantities such as energy and angular momentum. The equivalence principle implies that there is no density for gravitational energy. Hence, there have been considerable difficulties finding acceptable definitions of these concepts in general Relativity since Einstein's time.
In this talk, I will give an overview of the recent development on defining conserved quantities such as energy and angular momentum in general relativity.
Zoom Recording Passcode: Yq*Zj73g
Jan 28th: Yiyue Zhang, UC Irvine
Title: Proof of positive mass theorems using harmonic level sets.
Abstract:
I will introduce the harmonic level set method developed by Stern in 2019. This technique has been used to prove the positive mass theorems in various settings, for example, the Riemannian case, the spacetime case, the hyperbolic case, and the positive mass theorem with charge. I will focus on the hyperbolic PMT and the charged PMT. We give a lower bound for the mass in the asymptotically hyperbolic setting. We also prove some rigidity results as corollaries.
This is joint work with Bray, Hirsch, Kazaras, and Khuri.
Zoom Recording Passcode: 9tw92.u^
Feb 4th: Po-Ning Chen, UC Riverside (In-person)
Conserved quantities in general relativity II
Abstract:
One of the fundamental problems in general relativity is to define conserved quantities such as energy and angular momentum. The equivalence principle implies that there is no density for gravitational energy. Hence, there have been considerable difficulties finding acceptable definitions of these concepts in general Relativity since Einstein's time.
In this talk, I will give an overview of the recent development on defining conserved quantities such as energy and angular momentum in general relativity.
Feb 11th: Yuchin Sun, UC Santa Cruz (Online)
Morse index bound of minimal two torus
Abstract:
Min-max construction of minimal spheres using harmonic replacement is introduced by Colding and Minicozzi and generalized by Zhou to conformal harmonic torus. This min-max construction gives existence for minimal surfaces of arbitrary codimension which are not area minimizing. The difference between spheres and tori is that tori has varying conformal structures. We construct deformation with respect to the varying conformal structures and prove that the Morse index of the min-max conformal harmonic torus is bounded by one.
Zoom Recording Passcode: X!4oj&$A
Feb 18th: Michael McNulty, UC Riverside (In-person)
On the Stability of Self-Similar Blow-Up for Nonlinear Wave Equations
Abstract:
Of fundamental importance to the study of nonlinear wave equations is the well-posedness of the associated Cauchy problem. Self-similar solutions provide examples that can be initially smooth yet fail to be continuously differentiable after a finite amount of time. After reviewing what is known for a variety of such equations, we will introduce the strong-field Skyrme model. This model is a particular limiting case of the geometric field theory called the Skyrme model. We will present recent progress toward establishing the stability of an explicit self-similar solution of the strong-field Skyrme model’s equation of motion.
Feb 25th: Russell Phelan, UC Riverside (Online)
Bounding Betti Number Growth for G-Manifolds whose Quotients Tile Space Forms
Abstract:
We will look at non-negatively curved simply-connected manifolds that admit cohomogeneity 2 isometric Lie group actions. Motivated by the Bott Conjecture, (that all non-negatively curved simply-connected manifolds are rationally elliptic) we examine manifolds M such that the orbit space admits a metric of constant positive (resp. zero) curvature such that it tiles S^2 (resp. R^2). In this case, knowledge of the accumulation of conjugate points in constant curvature enables a Morse theoretic argument that shows elliptic behavior for a principal orbit in M. An unpublished theorem of Halperin lets us leverage this behavior to show this class of manifolds is rationally elliptic.
Zoom Recording Passcode: R^1X5Wzx
Mar 4th: Siyuan Lu, McMaster University (Online)
Rigidity of Riemannian Penrose inequality with corners and its implications
Abstract:
Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show
that suitable singular metrics attaining the optimal value in the Riemannian Penrose
inequality is necessarily smooth in properly specified coordinates. If applied to
hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result
gives the rigidity of isometric hypersurfaces with the same mean curvature.
This is a joint work with Pengzi Miao.
Mar 11th: Jonathan Zhu , Princeton University (Online)
TBA