Graduate Level Material

Upper undergraduate, graduate or post-doc level material

• Analysis 2
• Measure Theory
• The Elements of Finite Elements
• Elliptic Systems
• History of Foundations of Mathematics
• Axiom of Choice
• Mathematical Economics
• Probability Basics

In the following I try to motivate the results and the proofs as much as possible.

Analysis 2

Third Year Honours Level course. Topology, Axiom of Choice and equivalents, Measure Theory, Lebesgue Integral, Hilbert spaces, Lebesgue’s differentiation theorem.

Measure Theory

A series of 5 lectures at a workshop for graduate students. Lebesgue and more general measures, Fubini’s Theorem, Sobolev spaces.

The Elements of Finite Elements

A brief introduction, assuming some background in Sobolev spaces. Convergence estimates, inverse estimates, generalisations.

Elliptic Systems

A series of 5 lectures at a workshop for graduate students. Elliptic systems model vector-valued quantities in an equilibrium situation. Examples are a vector-field describing the molecular orientation of a liquid crystal, and the displacement of an elastic body under an external force. Solutions of elliptic systems typically have singularities, unlike the situation for scalar-valued elliptic problems.

History of Foundations of Mathematics

(A 2024 MSI colloquium.) From 1870 to current. Some of the material is more technical than I could give in the colloquium. But at least the history and the individuals involved will be interesting! There is an extensive annotated bibliography.

Axiom of Choice

Proofs of the equivalence of 5 standard versions of the Axiom of Choice.

Mathematical Economics

Basic terminology, Edgeworth box, core of an economy, Walras equilibrium.

Probability Basics

Measure theoretic approach to probability. Borel-Cantelli lemma, weak and strong versions of the law of large numbers, renewal theorem, continuous-time jump Markov processes [CJM].