johanna.mcbryde@icms.org.uk      0131 650 9816

Manifolds

Theme: Structure and symmetry
Theme head: Professor Ivan Cheltsov

Lectures [Thursdays, 13:00-15:00]

  • Lecture 1, 10.01.2019: Submanifolds of Euclidean space, the Implicit Function Theorem and Sard's Theorem, abstract manifolds.
  • Lecture 2, 17.01.2019: Tangent vectors and the tangent bundle, vector bundless.
  • Lecture 3, 24.01.2019: Vector fields and flows, Lie derivative, the Frobenius Theorem.
  • Lecture 4, 31.01.2019: Differential forms, de Rham complex and Poincaré Lemma.
  • Lecture 5, 07.02.2019: Integration of differential forms, Stokes' Theorem and Poincaré duality.
  • Lecture 6, 14.02.2019: Riemannian metrics, connections, the Levi-Civita connection.
  • Lecture 7, 21.02.2019: Geodesics, the exponential map.
  • Lecture 8, 28.02.2019: Curvature and integrability, Riemannian curvature.
  • Lecture 9, 07.03.2019: The second Fundamental Form, Gauss Formula and the Theorema Egregium.
  • Lecture 10, 14.03.2019: Euler characteristic, the Gauss-Bonnet Theorem for surfaces.

Assessment

  • Assignment 1 is due on Friday of Week 8, March 1st, 2019
  • Assignment 2 is due on Friday April 19th, 2019

Prerequisites

Working knowledge of metric spaces, linear algebra, group theory, vector calculus, topological spaces, the fundamental group, homology. 

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