Tag - Complex analysis

László Székelyhidi: Morrey’s conjecture

14th February, 2022

Morrey’s conjecture arose from a rather innocent-looking question in 1952: is there a local condition characterizing 'ellipticity' in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first progress was made with V. Sverak’s ingenious counterexample. Nevertheless, the case pertaining to planar maps remains open despite much progress, and has fascinated many through its interesting connections to complex analysis, geometric function theory, harmonic analysis, probability and martingales, differential inclusions and the geometry of matrix space. In the talk, I will give an overview of some of these connections and some of the recent progress.

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.