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MIT 18.966 Geometry of Manifolds II

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Abstract

18.966 Geometry of Manifolds II

Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a continuation of 18.965 and focuses more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. Prior exposure to calculus on manifolds, as in 18.952, recommended.

  • 1992
  • 1998
  • 2004 (Alan Edelman)
  • 2005 (Tomasz Mrowka)
  • 2006 (Victor Guillemin)
  • 2007 (Denis Auroux)
  • 2008 (Denis Auroux)
  • 2009 (Tomasz Mrowka)
  • 2010 (Tomasz Mrowka)
  • 2011 (Peter Ozsváth)
  • 2012 (Colding)
  • 2013
    • 18.966 — Spring 2013 (Guth)
      • The theme of the class is the connection between analysis on the one hand and the topology of manifolds on the other hand.  There are three main topics.
      • The first is transversality, covering Sard’s theorem and its applications in topology. The applications include degrees of maps, linking numbers, and the Hopf invariant.
      • The second topic is vector bundles, connections, and characteristic classes.  We will study the Euler class, and Chern and Pontryagin classes.  One of the main results we will study is the Gauss-Bonnet-Chern theorem.
      • The third topic is Morse theory, connecting the critical points of a function to the topology of the manifold.  We will begin with Morse theory on finite dimensional manifolds, and then study Morse theory on the space of loops on a manifold, building up to a proof of the Bott periodicity theorem on the homotopy groups of the unitary group.
      • Texts: In the first unit, we will use the book Topology from the Differentiable Viewpoint, by John Milnor. In the third unit, we will use the book Morse Theory, also by John Milnor. For the second unit, we will use in-class lectures, and I may post some references on the webpage.
  • 2014
    • 18.966 — Spring 2014 (Murphy)
      • Contact geometry is a topic at the intersection of many fields, closely related to classical dynamics, complex analysis, and geometric topology. It is best thought of as the odd-dimensional cousin to symplectic geometry. The beginning of the class will cover the basic theory, and afterwards we will specialize to the 3-dimensional case, where a number of additional tools strengthen our knowledge of that dimension (particularly Giroux flexibility).
      • Contact isotopies. Gray stability, Darboux, and Legendrian neighborhood theorems. Connections between contact and symplectic geometry. Geometry of hypersurfaces.
      • Tight contact structures and Bennequin’s inequality. Applications of Bishop disk fillings. Giroux flexibility. Eliashberg’s proof of Cerf’s theorem. Connect sum decompositions of tight contact manifolds. Classification of overtwisted contact structures. Legendrian knots.
      • As time allows confoliation theory and/or topics from high dimensional contact geometry will be covered.
      • Text Book: No official text, but Geiges’ book is an excellent resource that covers most of the topics discussed. There will also be class notes.
    • https://math.mit.edu/classes/18.966/2014SP/notes.pdf
  • 2015 (Colding)
  • 2016 (Paul Seidel)
  • 2017 (Minicozzi)
  • 2018
  • 2019
  • 2020 (Minicozzi)
  • 2021 (Colding)
  • 2022
    • 18.966 — Spring 2022 (Colding)
    • https://math.mit.edu/classes/18.966/2022SP/
    • Form and shape can be described by differential equations. Many of these equations originate in various branches of science and engineering. They are fundamental and, in a sense, canonical. The fact that they make sense geometrically means that they are relevant everywhere and have fundamental properties that appear over and over again in many settings. Understanding them requires simultaneous insight into analysis and geometry and the interplay between these.
    • In this class we will discuss a number of different ideas and estimates that have a wide range of applications to many fields including geometry, analysis, probability and applied mathematics. Common for them all is that they originate in geometry.
    • Topics will include (but not restricted to):
      • Continues and discrete Laplacian.
      • Drift Laplacian and weighted inequalities.
      • Gradient estimates.
      • Harnack inequalities.
      • Sharp gradient estimate and monotonicity.
      • L2 eigenfunctions.
      • Ornstein-Uhlenbeck operator and Hermite polynomials.
      • Li-Yau differential Harnack inequality.
      • Hamilton’s matrix maximum principle.
      • Perelman’s monotonicity.
  • 2023
  • 2024
  • Not offered in Spring 2025.
  • 2026 (Colding)

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