Friday February 02, 2018 4:00 PM  6:00 PM P131 Math Tower
 Robert Abramovic, Stony Brook University
How to Solve the Dirac Equation on an Asymptotically Flat Riemannian Manifold with Nonnegative Scalar CurvatureI 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 P131 Math Tower
 Robert Abramovic, Stony Brook University
An Introduction to the ADM EnergyBeginning 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 P131 Math Tower
 Marlon De Oliveira Gomes, Stony Brook University
Scalar curvature as a moment mapIn 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 KempfNess theorem in the finitedimensional 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 (Kstability) in the case of polarized algebraic varietes, and recent developments in the subject.

Friday March 02, 2018 4:00 PM  6:00 PM P131 Math Tower
 Shaosai Huang, Stony Brook University
Currents in metric spacesWe will discuss the paper of AmbrosioKirchheim 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 P131 Math Tower
 JeanFrançois Arbour, Stony Brook University
Ricci flow with surgery on 4manifolds 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 4manifolds 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 4manifolds with PIC, to get to Hamilton's classification.

