2026-2027 Advances in Mathematical Analysis: metric embeddings and geometry of fractals (with Nicolas Angelini), 24 hours.
We aim to cover:
Isometric and biLipschitz embedding into Hilbert space, Schoenberg's theorem.
Significative examples: Laakso diamond space, expanders, Hamming cube, Heisenberg group.
Snowflakes, doubling spaces, Assouad's embedding theorem.
Embedding of finite metric spaces into Hilbert spaces (Bourgain's theorem and Linial-London-Rabinovich's result).
Cheeger's constant, Poincaré inequality on graphs, coarse embeddability, proof of non-coarse embeddability of expanders into Hilbert spaces.
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2024-2025 Term I: MA947 - Graduate real Analysis. All the information can be found at the university webpage.
The course will follow the lecture notes written by David Bate that you can find either on the moodle or here.
The course will be organized as follows:
Chapter 1: Measures. (Lectures 1, 2, 3, 4).
Chapter 2: Integration (Lectures 5, 6).
Chapter 3: Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Lebesgue decomposition theorem (Lectures 7, 8, 9).
Chapter 4: Riesz's representation theorem, lattices of functions and Daniell integral (Lectures 10, 11, 12, 13, 14).
Chapter 5: Fubini's theorem. (Lectures 15, 16, 17).
Chapter 6: Covering theorems, Hardy-Littlewood maximal function and Lebesgue differentiation (Lectures 18, 19, 20, 21).
Chapter 7: Lipschitz functions. AC and BV in the real line. Rademacher Theorem (Lectures 22, 23, 24, 25).
Chapter 9: Rectifiable sets and approximate tangent planes (Lectures 26, 27, 28, 29).
Chapter 10: Purely unrectifiable sets (Lecture 30).
Some interesting material about real analysis and geometric measure theory useful for the course is:
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Spring 2023: Teaching assistant for the course of Complex Analysis II (held by Behnam Esmayli).
This is the solution to the exercise sheets we provided during the first week: Exercise sheet (solved).
For all the other weeks, we will solve the exercises given by Behnam during classes, that you may find here.