Differential Geometry (2020-2021)

This course is a general introduction to Differential Geometry, intended for upper-level undergraduates and beginning graduate students.

Lecture Notes for the 2018-2019 version of the course are available as a single PDF for ETH/UZH students here.

The 2020-2021 version of the course will fairly similar, at least to begin with. Updated lecture notes for this year's course will appear gradually here.

Differential Geometry I:
  • Smooth manifolds, submanifolds, vector fields,
  • Lie groups, homogeneous spaces,
  • Vector bundles, tensor fields, differential forms,
  • Integration on manifolds and the de Rham Theorem,
  • Principal bundles.
Differential Geometry II:
  • Connections on vector bundles, parallel transport, covariant derivatives.
  • Curvature and holonomy on vector bundles, Chern-Weil theory.
  • Connections and curvature on principal bundles.
  • Geodesics and sprays, sectional curvature, Ricci curvature.
  • The metric structure of a Riemannian manifold,
  • Curvature vs. Topology.

Communication in Mathematics (Autumn 2020)

This course teaches fundamental communication skills in mathematics.

Topics covered include:

  • how to write a thesis (more generally, a mathematics paper),
  • elementary $\LaTeX$ skills and language conventions,
  • how to write a personal statement for Masters and PhD applications.

There are no formal mathematical prerequisites.  

Lecture notes are available here.

Dynamical Systems (2019-2020)

This course was broad introduction to dynamical systems, intended for upper-level undergraduates and beginning graduate students.

Lecture notes are available here.

Dynamical Systems I (Lectures 1-28):
  • Topological dynamics (transitivity, attractors, chaos, structural stability)
  • Low-dimensional dynamics (the Sharkovsky Theorem, rotation numbers)
  • Ergodic theory (Poincaré recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures)
Dynamical Systems II (Lectures 29-50):
  • Local hyperbolic dynamics (the Grobman-Hartman Theorem, the local Stable Manifold Theorem).
  • Global hyperbolic dynamics on manifolds (the Shadowing Theorem, the Lambda Lemma, transverse homoclinic points and chaos, Omega Stability Theorem)

Algebraic Topology (2017-2018)

This course was a general introduction to Algebraic Topology, intended for upper-level undergraduates and beginning graduate students.

I produced full lecture notes, which can be found here.

Algebraic Topology I (Lectures 1-23):
  • Basic homological algebra and category theory,
  • The fundamental group,
  • Singular homology,
  • Cell complexes and cellular homology,
  • The Eilenberg-Steenrod axioms.
Algebraic Topology II (Lectures 24-45):
  • Universal coefficients,
  • The Eilenberg-Zilber Theorem and the Künneth Formula,
  • The cohomology ring,
  • Fibre bundles, the Leray-Hirsch Theorem, and the Gysin sequence,
  • Topological manifolds and Poincaré duality,
  • Higher homotopy groups and fibrations.