# Teaching

## Dynamical Systems (2019-2020)

This will be a 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)
- Complex dynamics on the Riemann sphere (Julia sets and Fatou sets, Fractals and the Mandelbrot set—if I have time...)

## 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.