Friday February 01, 2019 4:00 PM  6:00 PM P131 Math Tower
 TBA, Stony Brook University
Organizational meeting + TBA

Friday February 08, 2019 4:00 PM  6:00 PM P131 Math Tower
 Lisandra Vazquez, Stony Brook University
Isometric Embeddings and the Implicit Function TheoremThe main goal of this talk is to present the proof of the Nash Embedding Theorem which states that every Riemannian manifold is realizable as a submanifold of some Euclidean space by a smooth isometric embedding. The problem of isometric embeddings has been around since the introduction of the concept of an abstract manifold. Yet as conceptually fundamental as Nash’s result is, it is in the proof that Nash gives us one of the most important techniques in geometric analysis, known today as the NashMoser iteration or hard implicit function theorem. Thus we hope to give a brief history of the problem of isometric embeddings, from the local results of Cartan and Janet to the theorem of Nash and prepare the stage for the proof of Nash’s theorem by detailing the NashMoser iteration and its applications to solving nonlinear PDE’s.
References:
 Nash, J. “The Imbedding problem for Riemannian Manifolds”, Annals Math., Vol 63, No. 1, 1956
 Han, Q., Hong, JX. “Isometric Embedding of Riemannian Manifolds in Euclidean Spaces”
 Krantz, S. , Parks, H. , “The Implicit Function Theorem”

Friday February 22, 2019 4:00 PM  6:00 PM P131 Math Tower
 Demetre Kazaras, Stony Brook University
Bray's proof of the Penrose Conjecture

Friday March 01, 2019 4:00 PM  6:00 PM P131 Math Tower
 Vardan Oganesyan, Stony Brook University
Second variation and stability of minimal Lagrangian submanifolds in Kahler manifoldsWe say that an immersed minimal submanifold is stable if the second variation of the volume is nonnegative under all deformations. We will prove two Theorems.
1) Suppose that M is a Kahler manifold with nonpositive first Chern class. Then any minimal Lagrangian submanifold is stable.
2) Assume that M is KahlerEinstein with Einstein constant k. Then an immersed Lagrangian L is stable under Hamiltonian deformations if and only if the first eigenvalue of the Laplacian of L is greater than or equal to k.
These theorems were proved by Oh and Le in 1990.

Friday March 08, 2019 4:00 PM  6:00 PM P131 Math Tower
 Saman Habibi Esfahani, Stony Brook University
On Existence, Regularity and Singularity of Harmonic MapsWe study an existence and regularity theory for Harmonic maps between Riemanian manifolds. We review an existence theorem by Eells and Sampson, for compact target manifolds with nonpositive curvature. Then we see from a theorem by Schoen and Uhlenbeck that a bounded energy minimizing map is regular except for a closed set S of Hausdorff dimension at most n3.
References:
A regularity theory for harmonic maps by Schoen and Uhlenbeck
Theorems on Regularity and Singularity of Energy minimizing maps by Leon Simon

Friday March 15, 2019 4:00 PM  6:00 PM P131 Math Tower
 Jingchen Hu, Shanghai Tech University
Existence of Einstein–Hermitian metrics in stable holomorphic vector bundlesIn algebraic geometry, there is a notion of a (semi)stable bundle over a projective variety. On the other hand, in differential geometry, there is a notion of Einstein–Hermitian vector bundles over the complex manifold. It turns out these ideas coincide.
In this talk, show the existence of EinsteinHermitian metrics on stable holomorphic vector bundle over a Kahler manifold.
Reference: K.Uhlenbeck, S.T.Yau On the existence of hermitian‐yang‐mills connections in stable vector bundles

