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.


      Fall 2018

      The topic for this Fall is Einstein metrics and special holonomy. Below is a summary of our goals for the semester.

      • Understand what are some of the obstructions to the existence of Einstein metrics in low dimensions.
      • Understand what is holonomy, Berger's classification of irreducible honolomy representations, and why metrics of special holonomy are Einstein.
      • Understand various constructions of Einstein metrics with special holonomy
      • Understand the convergence and moduli theory of Einstein metrics.
      • If time permits, discuss a few constructions of Einstein with generic holonomy. In particular, how some of them arise as limits (in an appropriate sense) of metrics of special holonomy.
      • If time permits, use the existence of Einstein metrics to derive information about the topology of the underlying manifolds.

      Time and location: We meet weekly on Mondays, from 4:15 pm to 5:15 pm, in 5-127.

      The schedule for our seminar is below. Abstracts for talks will be linked in the titles as they become available.

      Date   Title Speaker References
      8/27  Organizational meeting

      9/3   Labour day.

      9/10   Introduction to Einstein manifolds. Lisandra Hernandez [Be87]
      9/17   Holonomy representations and Einstein metrics. Marlon Gomes [Be87]
      9/24   Kähler-Einstein metrics: examples and obstructions. Michael Albanese [Jo00]
      10/1   The Calabi conjecture (and Kähler-Ricci flow). Jae Ho Cho [Jo00]
      10/8   Fall break!

      10/15   Hyper-Kähler geometry. John Sheridan [Bv99], [Hi91], [Hu97], [Hu11]
      10/22   Quaternion-Kähler geometry. Marlon Gomes [BG08], [HKLR87],
      10/29   G_2 geometry: introduction. Jordan Rainone [Jo00], [Jo07]
      11/5   Joyce's construction of compact G_2 manifolds: part I . Marlon Gomes [Jo00], [Jo07]
      11/12   Joyce's construction of compact G_2 manifolds: part II. Jean-François Arbour [Jo00], [Jo07]
      11/19   Spin(7) geometry Matthew Lam [Jo00], [Jo07]
      11/26   Moduli and convergence theory. Zhongshan An [LW99]
      12/3   Aspects of Ricci-flat, ALE 4-manifolds. Demetre Kazaras [LV16]

      List of possible extra topics : 3-manifolds, Sasaki-Einstein metrics, applications to 4-manifold topology.

      References

      • [Be87] : Arthur L. Besse, Einstein manifolds. Springer, Berlin, 1987.
      • [Bv99] : Arnaud Beuville, Riemannian Holonomy and Algebraic Geometry , arXiv:math/9902110v1.
      • [BG08] : Charles Boyer and Krzysztof Galicki, Sasakian Geometry. Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
      • [GL88] : Krzysztof Galicki and H. Blaine Lawson Jr., Quaternionic reduction and quaternionic orbifolds , Math. Ann. 282 (1988), no. 1, pp. 1-21.
      • [HKLR87] : Nigel J. Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček, HyperKähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, pp. 535-589.
      • [Hi91] : Nigel J. Hitchin, Hyperkähler manifolds, Séminaire Bourbaki (1991-1992) 34 (1991), pp. 137-166.
      • [Hu97] : Daniel Huybrechts, Compact Hyperkaehler Manifolds: Basic results, arXiv:alg-geom/9705025v1.
      • [Hu11] : Daniel Huybrecths, Hyperkähler manifolds and sheaves , Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), v. 2, (2011), pp. 450-460.
      • [Jo00] : Dominic Joyce, Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
      • [Jo07] : Dominic Joyce, Riemannian holonomy groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics, v.12, Oxford University Press, Oxford, 2007.
      • [LW99] : Claude LeBrun and McKenzie Wang (editors), Essays on Einstein Manifolds , Surveys in Differential Geometry, v.6, International Press, Boston, 2001.
      • [LV16] : Michael T. Lock and Jeff A. Viaclovsky, Quotient singularities, eta invariants, and self-dual metrics., Geometry and Topology 20 (2016), pp. 1773-1806.