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

One of the main problems in modern Kahler geometry is to determine when does a Kahler manifold admit a constant scalar curvature Kahler metric, in a given Kahler class. This problem, whose roots date back to Poincare and Kobe's uniformization theorem, has attracted a great deal of attention specially after Calabi's celebrated conjectures in the 1954 ICM, related to the Kahler-Einstein case. Obstructions to the existence of such metrics were found by Matsushima, Lichnerowicz and Futaki, all having to do with the algebra of the automorphism group of the underlying complex structure. By the 90s, a widely circulated conjecture was that as long as the automorphism group was discrete (in which case the know obstructions would vanish), cscK metrics should exist. Tian showed in '96 that this folklore conjecture was false, giving further indication that a conjecture of Yau - relating existence of such metrics to an algebraic notion of stability - should be true.

In a seemingly unrelated story, Atiyah and Bott had introduced a way to look into the problem of classifying flat unitary connections on a vector bundle over a Riemann surface in terms of symplectic reduction. Their work inspired Donaldson to give a new proof of a theorem of Narasimhan and Seshadri, relating stable bundles on a Riemann surface to unitary connections with constant curvature. This was later generalized to a higher-dimensional setting, by the Hitchin-Kobayashi correspondence. This set of ideas formed a successful application of a result by of Kempf and Ness (relating Geometric Invariant Theory to symplectic geometry) in an infinite-dimensional setting. Donaldson, having been an active participant in the solution the Hitchin-Kobayashi correspondence, realized that the cscK problem can also be understood as a formal application of Kempf-Ness. The key point is that the scalar curvature in Hermitian geometry can be interpreted as a moment map for the action of a suitable infinite-dimensional Lie group (a fact independently proved earlier by Fujiki). This observation not only sheds light on the known obstructions from before, but also indicates a way to formalize Yau's conjecture.

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.


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