Week 19.11.2023 – 25.11.2023

Random permutations show up in a variety of areas in mathematics and its applications. In connection with physical applications (e.g., the lambda transition for superfluid helium), there is an interest in random spatial permutations -- that is, laws on permutations that have a 'geometric bias'. There are compelling heuristic arguments that this spatial bias has little effect on the distribution of the largest cycles of a random spatial permutation, provided that large cycles actually exist. I'll discuss a particular model of random spatial permutations (directed permutations on asymmetric tori) where these heuristics can be made precise, and large cycles can be shown to follow the expected (Poisson-Dirichlet) law.

Building on non-vanishing theorems of Kronheimer and Mrowka in instanton Floer homology, Zentner proved that if Y is a homology 3-sphere other than S^3, then its fundamental group admits a homomorphism to SL(2,C) with non-abelian image. In this talk, I’ll explain how to generalize this to any Y whose first homology is 2-torsion or 3-torsion, other than the connect sum of n copies of the three-dimensional real projective space for any n or lens spaces of order 3. This is joint work with Sudipta Ghosh and Raphael Zentner.

King's College London Mathematics School is for students aged 16-18 with an enthusiasm and aptitude for mathematics, and aims to widen participation in high-quality degrees and careers in the mathematical sciences.

The school opened in 2014 and for nearly 10 years has been evolving a pedagogy and curriculum tuned to generating confident, skilled and articulate mathematical thinkers.

In this talk I will explore those aspects of pedagogy and culture at KCLMS that universities might use to improve their teaching and learning.

I will discuss a recent progress on two classical problems. The first one comes mostly from applied mathematics and numerical analysis: find tight universal and preferably simple enclosures for zeros of Bessel functions, of their derivatives, and possibly of other special functions. The second one comes primarily from number theory: find bounds for the number of lattice points under the graph of a given function (with some restrictions on the class of functions). As an application of these results, I’ll show the validity of inequalities à la Pólya for the magnetic Aharonov--Bohm Laplacian in the disk, discuss possible generalisations, and open problems. The talk covers some joint works, mostly in progress, with N. Filonov, I. Polterovich, and D. A. Sher.