Geometric Analysis Learning Seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Friday
September 08, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Christina Sormani, Stony Brook University
Introduction to Gromov-Hausdorff Convergence


Friday
September 15, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Christina Sormani, Stony Brook University
Intrinsic Flat Convergence: Definition, Examples, and Theorems


Friday
September 22, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Yuanqi Wang, Stony Brook University
Stable reflexive sheaves on projective spaces

We will discuss a rank 2 reflexive sheaf on P3,
and the construction of rank 2 stable bundles on P2 with
c_{1}=0, c_{2}=2.


Friday
September 29, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Jean-François Arbour, Stony Brook University
Relative index formula for elliptic operators on manifolds with cylindrical ends

This is the first of two talks on R. Lockhart and R. McOwen's 1984 paper "Elliptic Differential Operators on Noncompact Manifolds". The paper is concerned with elliptic operators acting on weighted Sobolev spaces on manifolds with cylindrical ends. The main result is that there is a discrete set of "critical weights" outside of which the acting operator is Fredholm. Moreover, a formula is given describing the jump in Fredholm index when a critical weight is crossed. Their results found many important applications later on for gluing problems arising in geometry.


Friday
October 06, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Jean-François Arbour, Stony Brook University
Relative index formula for elliptic operators on manifolds with cylindrical ends

This is the second of two talks on R. Lockhart and R. McOwen's 1984 paper "Elliptic Differential Operators on Noncompact Manifolds". The paper is concerned with elliptic operators acting on weighted Sobolev spaces on manifolds with cylindrical ends. In this talk, I will discuss the proof of the formula describing the jump in Fredholm index when a critical weight is crossed. Following the authors, I will then show how to use this formula to understand whether or not certain L2 harmonic forms on manifolds with isolated conical singularities are also closed and co-closed.


Friday
October 13, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Demetre Kazaras, Stony Brook University
Positive Scalar Curvature and Minimal Hypersurfaces

A celebrated result of Schoen and Yau from ’79 states that any stable minimal hypersurface in a manifold of positive scalar curvature (psc) is Yamabe positive (i.e. there is a psc metric in the same conformal class as the restriction metric). When used alongside regularity results from geometric measure theory, this observation is a major tool used to study psc metrics on manifolds of dimension 8 and below (above dimension 8, the regularity results fail dramatically). In this talk, I will describe the ’79 paper and discuss a recent preprint of Schoen and Yau where a method is produced which applies to all dimensions.


Friday
October 20, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Marlon De Oliveira Gomes, Stony Brook University
Uniqueness of Kähler-Einstein metrics up to automorphisms

The Uniformization Theorem of Poincaré-Koebe states that on every compact Riemann surfaces there are preferred Riemannian metrics, compatible with their complex structures and characterized, up to conformal re-scaling, as metrics of constant Gaussian curvature.
Kähler-Einstein metrics, on compact complex manifolds may be viewed as generalized versions of canonical metrics in higher dimensions, characterized as Kähler metrics of constant Ricci curvature.

In this talk I will focus on the issue of uniqueness of such canonical metrics. I will begin by reviewing the situation of Kähler-Einstein metrics of negative and zero Einstein constants. Then I will discuss a theorem of Bando and Mabuchi that states that Kähler-Einstein metrics are unique in their Kähler classes, up to the action of the automorphism group of the manifold, thus settling the positive case.


Friday
November 03, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Jordan Rainone, Stony Brook University
Original Proof of the Positive Mass Theorem

The Positive Mass Theorem is originally stated as follows: "An isolated gravitational system with non-negative local matter/energy density will have non-negative total mass (measured gravitationally at spacial infinity) & and if the total mass is equal to 0 then our system is the flat Minkowski space-time." In their paper Schoen and Yau show that resolving the following much easier to understand (for mathematicians) theorem is equivalent to resolving the above: "An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature has non-negative ADM mass, and if the mass is 0 then our manifold is the flat Euclidean space."

In my talk I will spend some time explaining how these two statements are related and why we should believe/want them to be true. The larger part of my talk will focus on explaining the original proof as presented in Schoen and Yau's 1979 paper. The proof is very geometric (relying on the construction of a minimal surface) and doesn't involve much analysis.


Friday
November 17, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Yu Li, Stony Brook University
On the Heat Kernel under the Ricci Flow

We will discuss some estimates of heat kernel under the Ricci flow, Perelman's entropy, Li-Yau-Harnack inequality and some other applications to noncompact manifolds.


Friday
December 01, 2017

4:00 PM - 6:00 PM
P-131 Math Tower
Christina Sormani, Stony Brook University
Almost Rigidity of the Positive Mass Theorem

The Rigidity of the Positive Mass Theorem of Schoen-Yau states that a manifold with nonnegative scalar curvature that is asymptotically flat and has ADM mass = 0 is isometric to Euclidean space. The Almost Rigidity Conjecture states that if a sequence of manifolds with nonnegative scalar curvature that are asymptotically flat have ADM mass decreasing to 0 then the sequence converges in the pointed intrinsic flat sense to Euclidean space. I will present joint papers with Dan Lee, with Lan-Hsuan
Huang and Dan Lee, and with Iva Stavrov proving special cases of this Almost Rigidity conjecture. The problem remains open in full generality.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars