## 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)

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.