Friday September 08, 2017 4:00 PM  6:00 PM P131 Math Tower
 Christina Sormani, Stony Brook University
Introduction to GromovHausdorff Convergence

Friday September 15, 2017 4:00 PM  6:00 PM P131 Math Tower
 Christina Sormani, Stony Brook University
Intrinsic Flat Convergence: Definition, Examples, and Theorems

Friday September 22, 2017 4:00 PM  6:00 PM P131 Math Tower
 Yuanqi Wang, Stony Brook University
Stable reflexive sheaves on projective spacesWe 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 P131 Math Tower
 JeanFrançois Arbour, Stony Brook University
Relative index formula for elliptic operators on manifolds with cylindrical endsThis 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 P131 Math Tower
 JeanFrançois Arbour, Stony Brook University
Relative index formula for elliptic operators on manifolds with cylindrical endsThis 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 coclosed.

Friday October 13, 2017 4:00 PM  6:00 PM P131 Math Tower
 Demetre Kazaras, Stony Brook University
Positive Scalar Curvature and Minimal HypersurfacesA 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 P131 Math Tower
 Marlon De Oliveira Gomes, Stony Brook University
Uniqueness of KählerEinstein metrics up to automorphismsThe Uniformization Theorem of PoincaréKoebe states that on every compact Riemann surfaces there are preferred Riemannian metrics, compatible with their complex structures and characterized, up to conformal rescaling, as metrics of constant Gaussian curvature.
KählerEinstein 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 KählerEinstein metrics of negative and zero Einstein constants. Then I will discuss a theorem of Bando and Mabuchi that states that KählerEinstein 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 P131 Math Tower
 Jordan Rainone, Stony Brook University
Original Proof of the Positive Mass TheoremThe Positive Mass Theorem is originally stated as follows: "An isolated gravitational system with nonnegative local matter/energy density will have nonnegative total mass (measured gravitationally at spacial infinity) & and if the total mass is equal to 0 then our system is the flat Minkowski spacetime." 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 3manifold with nonnegative scalar curvature has nonnegative 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 P131 Math Tower
 Yu Li, Stony Brook University
On the Heat Kernel under the Ricci FlowWe will discuss some estimates of heat kernel under the Ricci flow, Perelman's entropy, LiYauHarnack inequality and some other applications to noncompact manifolds.

Friday December 01, 2017 4:00 PM  6:00 PM P131 Math Tower
 Christina Sormani, Stony Brook University
Almost Rigidity of the Positive Mass TheoremThe Rigidity of the Positive Mass Theorem of SchoenYau 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 LanHsuan
Huang and Dan Lee, and with Iva Stavrov proving special cases of this Almost Rigidity conjecture. The problem remains open in full generality.

