Short Courses in Analysis

Maryna Viazovska: Fourier Uniqueness and Interpolation

25th October, 2023

Can we reconstruct a function by knowing only a subset of its values and a subset of the values of the function's Fourier transform?

How many values do we need to know for such a reconstruction? Can we interpolate a given subset of values? What are the possible applications of such interpolation? In this series of lectures, we will try to answer these questions.

In the first lecture, we will speak about the Cohn-Elkies linear programming bound for the sphere packing and how this bound's analysis led to the discovery of a Fourier interpolation formula. The second lecture will discuss explicit constructions of Fourier uniqueness sets and Fourier interpolation formulas. The third lecture will focus on analytic approaches to Fourier uniqueness and interpolation.

Christopher Schafhauser: On the classification of nuclear simple C*-algebras

5th August, 2021

A conjecture of George Elliott dating back to the early 1990s asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely infinite algebras, this is the famous Kirchberg-Phillips Theorem. The stably finite setting proved to be much more subtle and has been a driving force in research in C*-algebras over the last 30 years. A series of breakthroughs were made in 2015 through the classification results of Elliott, Gong, Lin, and Niu and the quasidiagonality theorem of Tikuisis, White, and Winter. Today, the classification conjecture is now a theorem under two additional regularity assumptions: Z -stability and the UCT. In my recent joint work with José Carrión, Jamie Gabe, Aaron Tikuisis, and Stuart White a much shorter and more conceptual proof of the classification theorem in the stably finite setting was provided. I hope to give an overview of the classification problem for C*-algebras and discuss some of the new techniques that led to the new proof.

Jesse Peterson: Von Neumann algebras and lattices in higher-rank groups

3rd August, 2021

An online lecture course by the University of Münster in Von Neumann algebras.

Lecture 1: We'll briskly review basic properties of semi-finite von Neumann algebras. The standard representation, completely positive maps, group von Neumann algebras, the group-measure space construction, and some characterizations of the hyperfinite II1 factor.

Lecture 2: We discuss some approximation properties that are common in "rank 1" groups: Weak amenability and biexactness.

Lecture 3: We discuss properly proximal groups as defined by Boutonnet, Ioana, and myself, and give some applications to group von Neumann algebras associated to higher-rank groups.

Lecture 4: We’ll introduce measure equivalence (ME), W*-equivalence (W*E), and von Neumann equivalence (VNE). We’ll give examples and discuss invariants.

Dave Sixsmith and Vasiliki Evdoridou: Lectures on Holomorphic Dynamics

27th October, 2020

An LMS online lecture course in holomorphic dynamics.

The series will consist of 6 one-hour lectures which will focus on the iteration of entire functions. We explore, among other things, some famous fractal Julia sets and the well-known Mandelbrot set. In particular, we will cover the following topics:

   1.  Equicontinuity, normal families, Montel's theorem, Riemann mapping theorem, the Riemann sphere.
   2.  Iteration of polynomials. Definition of the Fatou set and the Julia set for a polynomial. Examples.
   3.  The filled Julia set. Fixed and periodic points.
   4.  An introduction to the properties of the Fatou set and the Julia set.
   5.  The Mandelbrot set: its definition and properties.
   6.  Introduction to the iteration of transcendental entire functions.
   7.  Similarities and differences between polynomials and transcendental entire functions.
   8.  The escaping set: definition, properties, and its important role.
   9.  Examples of the Fatou, Julia and escaping sets for transcendental entire functions.

The lecture series is addressed to PhD students from diverse mathematical backgrounds. We shall assume a basic knowledge of complex analysis and a little topology. Some more advanced background in complex analysis will be covered in the first lecture. No knowledge of dynamics will be assumed.