Teaching
Below is the list of courses Will taught, together with all the lecture notes we gathered so far.
If you have any of Will's notes that are not linked below, we would be grateful if you shared them with us.
Beweise und Grundstrukturen (Frühling 2021)
Axiomatische Mengenlehre und mathematische Logik bilden die Fundamente, auf denen unser Fach aufgebaut ist.
Der Kurs beginnt mit einer Einführung in die Zermelo-Fraenkel-Mengenlehre. Nebenbei werden wir beweisen, dass Zahlen (!) existieren - zuerst die natürlichen Zahlen, dann die reellen Zahlen und schliesslich andere „grosse” Kardinalzahlen. Wir diskutieren die Implikationen des Auswahlaxioms und der berühmten Kontinuumshypothese.
Sobald die grundlegenden Strukturen fest etabliert sind, gehen wir zur „Kunst des Beweises” über. Das Ziel ist es, Ihnen zu helfen, Beweise zu verstehen und zu konstruieren, und zu lernen, klare und prägnante Mathematik zu schreiben.
Ein wahres Ensemble von Themen aus der Kombinatorik, Algebra und Zahlentheorie (wenn es die Zeit erlaubt) wird vorgestellt - diese Themen sind so gewählt, dass sie gute Beispiele zur Veranschaulichung einer Reihe grundlegender Beweismethoden liefern und fundamentale Ideen vorstellen, die Teil des Standard-Toolkits eines jeden Mathematikers sind. Als besonderes Highlight werden wir eine Auswahl der grössten klassischen Beweise aller Zeiten sehen.
Differential Geometry (2020-2021)
Lecture notes for the 2018-2019 version of the course are available here.
Lecture notes for the 2020-2021 version of the course are available here.
This course is a general introduction to Differential Geometry, intended for upper-level undergraduates and beginning graduate students.
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)
Lecture notes are available here.
The web pages containing the notes on the \(\LaTeX\) part of the course can be downloaded as a zip file here.
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.
Dynamical Systems (2019-2020)
Lecture notes are available here.
This course was broad introduction to dynamical systems, intended for upper-level undergraduates and beginning graduate students.
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, 2021)
Lecture notes for the 2018 version of the course are available here.
This course was a general introduction to Algebraic Topology, intended for upper-level undergraduates and beginning graduate students.
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.