Geometric Analysis Learning Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Instructions for subscribing to Stony Brook Math Department Calendars

Friday
February 02, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Robert Abramovic, Stony Brook University
How to Solve the Dirac Equation on an Asymptotically Flat Riemannian Manifold with Nonnegative Scalar Curvature

I will show how the Dirac equation can be solved on an asymptotically flat Riemann manifold with nonnegative scalar curvature. First, I will define asymptotic flatness, spinor bundles, the spin connection, and introduce the Dirac Operator. Then, I will outline how to obtain a nontrivial solution to the Dirac equation on this manifold by defining an appropriate Hilbert space for the Dirac operator to act on, applying the Riesz Representation Theorem, and using the fact the L^2 kernel of the Dirac operator is trivial. After I finish the outline, I will prove all the steps in detail and demonstrate why the solution is nontrivial. Finally, I will place the work in context with other results about harmonic spinors. I will be solving the Dirac Equation on an Asymptotically Flat Spin Manifold.


Friday
February 16, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Robert Abramovic, Stony Brook University
An Introduction to the ADM Energy

Beginning with the Newtonian approximation, I will introduce a notion of mass and use it to prove the "conformally flat" positive mass theorem in three dimensions. Motivated by the Einstein constraint equations, I will define the general notion of mass for a general metric on R^n minus a ball and explore its relationship with scalar curvature. For an asymptotically flat metric, this mass is precisely the ADM energy. I will conclude by showing that the ADM energy of the Euclidean metric on R^n is zero and that the ADM energy of the spatial part of the Schwarzchild metric returns the mass parameter present in its definition. If there is time, I will introduce one or two positive mass theorems (without proof).


Friday
February 23, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Marlon De Oliveira Gomes, Stony Brook University
Scalar curvature as a moment map

In this talk, I will begin by briefly describing what was know about the existence problem for cscK metric prior to Donaldson's work. I will then move on to discuss the Kempf-Ness theorem in the finite-dimensional setting, and how the concepts involved can be understood in the cscK case, by describing the formulation of scalar curvature as a moment map. Time permitting, I will also describe the (conjectural) associated algebraic notion of stablity (K-stability) in the case of polarized algebraic varietes, and recent developments in the subject.


Friday
March 02, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Shaosai Huang, Stony Brook University
Currents in metric spaces

We will discuss the paper of Ambrosio-Kirchheim which lays the foundation of the theory of currents in metric spaces. More specifically, we will introduce the concept of currents and they basic properties, and we will also prove the compactness and slicing theorems. Time permitting, we will discuss the closure and boundary rectifiability theorems.


Friday
March 09, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Jean-François Arbour, Stony Brook University
Ricci flow with surgery on 4-manifolds with positive isotropic curvature.

Positive Isotropic Curvature (PIC) is a condition on the curvature tensor of a Riemannian manifold introduced in 88 by Micaleff and Moore. They used it to give a proof of the sharp pointwise (1/4)-pinched sphere theorem. In a seminal paper in 97, Hamilton introduced Ricci flow with surgery and obtained a classification result for closed 4-manifolds admitting a metric with PIC. In this talk, I will follow B.L. Chen and X. Zhu's 2006 paper which is an exposition of Perelman's work on Ricci flow with surgery in the context of closed 4-manifolds with PIC, to get to Hamilton's classification.


Friday
April 06, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Jae Ho Cho, Stony Brook University
3-Manifolds with positive Ricci curvature

In this talk, I will introduce the paper of Richard Hamilton which said that for any initial condition given by 3 dimensional compact manifold with positive Ricci curvature, the normalized Ricci flow converges exponentially to the constant positive sectional curvature. (So we can say every such manifold which is simply connected is diffeomorphic to a round 3-sphere.) To begin with, I will introduce the Ricci flow and try to go through details as many as possible. Time permitting, I will also introduce some of variations of this problem.


Friday
April 13, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Demetre Kazaras, Stony Brook University
Riemannian metrics of constant mass

I will survey some results of L. Habermann on constructing and studying certain canonical metrics in a given Yamabe-positive conformal class. These metrics have the special property of having constant “mass”. The “mass” of manifold M at point p is obtained by turning M-{p} into an asymptotically-flat space and measuring its ADM-mass in the General-Relativity sense. There are also some applications to the moduli space of conformal classes on a given manifold.


Friday
April 20, 2018

4:00 PM - 6:00 PM
P-131 Math Tower
Fangyu Zou, Stony Brook University
Existence of C^{1,1} geodesics in the space of Kahler metrics

The space of Kahler metrics has a Riemannian structure by endowing with a L^2 metric. The geodesic equation can be transformed into a degenerated complex Monge-Ampere equation of Dirichlet type which can be solved by continuity method. In this talk, I will present Prof. Chen's work on the existence of C^{1,1} geodesics following his 2000 pape "The space of Kahler metrics".


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