Student Differential Geometry seminar

      Mathematics Department
      Stony Brook University

      About the seminar

      Our goal is to bring together students in the Differential Geometry group Stony Brook to discuss topics in the interface between Geometric Analysis, Complex Geometry and affine areas. The seminar is jointly organized by  Marlon de Oliveira Gomes and Lisandra Hernandez Vazquez.


      Spring 2019

      The topic for this Spring is Topics in Scalar Curvature. The Scalar Curvature of a Riemannian manifold is a function, which assigns to a point in the manifold a number measuring the second-order variation (up to a dimenisional constant) of the volumes of geodesics balls centered at the point, relative to the volume of balls of corresponding radii in Euclidean space. It can be computed by tensorial means as the trace of the Ricci tensor.

      A classical problem in Riemannian geometry is that of prescribing curvature conditions. In a nutshell, the goal of this seminar is to study this problem for the notion of scalar curvature. We shall begin by summarizing results from the 70s and 80s, including:

      • The Yamabe (constant scalar curvature) problem.
      • Metrics with scalar curvature with a prescribed sign.
      • Obstructions to positiveness.
      • New metrics of positive scalar curvature from old.
      • The positive mass theorem.
      We plan on moving on to topics of current interest once we are done with the basics. The choice of these topics will be made with the audience's interest in mind.

      Time and location: We meet weekly on Mondays, from 5:30 pm to 6:30 pm, in 5-127.

      Below is a preliminary schedule for our seminar. Abstracts for talks will be linked in the titles as they become available.

      Date   Title Speaker References
      1/28  Organizational meeting

      2/4   No meeting.

      2/11   Scalar Curvature, Conformal Geometry, and the Yamabe Problem. Marlon Gomes [LP87], [Sc17]
      2/18   Metrics of Negative Scalar Curvature. Jordan Rainone [KW75], [Sc17]
      2/25   Metrics of Non-negative Scalar Curvature and Incompressible Minimal Surfaces . Jae Ho Cho [SY79a]
      3/4   The Gromov-Lawson Construction of Positive Scalar Curvature Metrics, I. Jiasheng Teh [GL80]
      3/11   The Gromov-Lawson Construction of Positive Scalar Curvature Metrics, II. Michael Albanese [GL80]
      3/18   Spring Break!

      3/25   The Positive Mass Theorem, I. Zhongshan An [SY79c], [SY81]
      4/1   The Positive Mass Theorem, II. Owen Mireles Briones [SY79c], [SY81]

      References

      • [KW74] : Jerry L. Kazdan and F. W. Warner, Curvature Functions for Compact 2-manifolds, Ann. of Math. 99 (1974), pp. 14-47.
      • [KW75] : Jerry L. Kazdan and F. W. Warner, Scalar Curvature and Conformal Deformation of Riemannian Structure, J. Diff. Geometry 10 (1975), pp. 113-134.
      • [SY79a] : Richard Schoen and Shing-Tung Yau, Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-negative Scalar Curvature, Ann. of Math. 110 (1979), no. 1, pp. 127-142.
      • [SC79b] : Richard Schoen and Shing-Tung Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math 28 (1979), pp. 159-183.
      • [SC79c] : Richard Schoen and Shing-Tung Yau, On the Proof of the Positive Mass Theorem in General Relaivity, Comm. Math. Phys. 65 (1979), no. 1, pp. 45-76.
      • [GL80] : Mikhail Gromov and H. Blaine Lawson Jr., The Classification of Simply-connected manifolds of Positive Scalar Curvature. , Ann. of Math. 111 (1980), pp. 423-434.
      • [SC81] : Richard Schoen and Shing-Tung Yau, Proof of the Positive Mass Theorem. II , Comm. Math. Phys. 79 , (1981), no. 2, pp. 231-260.
      • [LP87] : John M. Lee and Thomas H. Parker, The Yamabe Problem, Bull. Amer. Math. Soc. 17 , (1987), no. 1, pp. 37-91.
      • [St92] : Stephen Stoltz, Simply-connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), pp. 511-540.
      • [Sc17] : Richard Schoen, Topics in Scalar Curvature, Spring 2017. Notes by Chao Li, available here.