## Dynamical Systems (2019-2020)

This will be a broad introduction to dynamical systems, intended for upper-level undergraduates and beginning graduate students.

Topics covered in Dynamical Systems I are tentatively planned to include:

• Topological dynamics (transitivity, attractors, chaos, structural stability)
• Ergodic theory (Poincaré recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures)
• Low-dimensional dynamics.

Lecture notes are available here.

## Communication in Mathematics (Autumn 2019)

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.

This year's course is significantly expanded from last year (which is why it is now worth 2 credit points instead of 1!)

Lecture notes are available here.

## Differential Geometry (2018-2019)

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

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

##### Differential Geometry I (Lectures 1-27):
• 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 (Lectures 28-53):
• 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.

## 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.