We show that the set of integers n for which n | (2n)!/(n!)2 has a positive asymptotic density, answering a question of Pomerance. The proof uses a mix of ideas from harmonic analysis and the anatomy of integers, and is joint work with Sergei Konyagin.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
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.
An LMS online lecture course in harmonic analysis.
An LMS online lecture course in functional analysis.
How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis! If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.
The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations.
We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.
We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture").
New results will be presented on damped subcritical Klein-Gordon equations, joint with Nicolas Burq and Genvieve Raugel.
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