Week 29.04.2024 – 05.05.2024

Vortices, turbulence, and rogue waves are typical phenomena of fluid dynamics. They can all be found, however, in simple models of lasers with optical injection. Almost 40 years ago we introduced a model of laser oscillations where, unexpectedly, conservative and dissipative dynamics coexist in the same phase space. When these laser models are extended to partial differential equations to include diffraction or dispersion, the underlying wave dynamics leads first to Turing patterns and then to regimes of defect-mediated turbulence where creation and annihilation of 2D vortices produce psychedelic spiral structures. In these regimes of spatio-temporal disorder, we observe the appearance of rogue waves corresponding to rare events, enormous peaks of light and heavily non-Gaussian probability density functions.

We consider an Erdos-Renyi random graph on n nodes where the probability of an edge being present between any two nodes is equal to a/n with a > 1. Every edge is assigned a (non-negative) weight independently at random from a general distribution. For every path between two typical vertices we introduce its hop-count (which counts the number of edges on the path) and its total weight (which adds up the weights of all edges on the path). We prove a limit theorem for the joint distribution of the appropriately scaled hop-count and general weights. This theorem, in particular, provides a limiting result for hop-count and the total weight of the shortest path between two nodes. This is a joint work with Fraser Daly and Matthias Schulte.

Configuration spaces have played an important role in mathematics and its applications. In particular, the question of how their topology changes as the cardinality of the underlying configuration changes has been studied for some fifty years and has attracted renewed attention in the last decade.

While classically additional information is associated "locally" to the points of the configuration, there are interesting examples when this additional information is "non-local". With Martin Palmer we have studied homology stability in some of these cases, including Hurwitz space and moduli spaces of asymptotic monopoles.

TBA

This term we will have a study group on the work of Dimitrov--Gao--Habegger and Kühne on uniformity in the Mordell conjecture. The first half of the schedule is meant to be an introduction to the area for non-specialists. In the second half, we will try to introduce some of the ideas from functional transcendence, dynamics and the moduli of abelian varieties which go into the proof.

The first talk will be an introduction to the history and statement of the results, with a vague hint at the methods of proof. A plan of the rest of the study group can be found here: https://sites.google.com/site/netandogra/seminars/uniform-mordell

Duffin-Schaeffer meets Littlewood and related topics

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to analytic number theory that lies in the heart of the corresponding proofs.

In this work we consider a time-periodic and random version of the ABC flow. We are concerned with two main subjects. On the one hand, we study the mixing problem of a passive tracer in the three-dimensional torus by the action of the random ABC vector field. On the other hand, we investigate the effect of the ABC flow on the growth of a magnetic field described by the kinematic dynamo equations. To deal with these questions we analyse the ABC flow as a random dynamical system and examine the ergodic properties of its associated one-point, two-point, and projective Markov chains, as well as its top Lyapunov exponent. This work settles that the random ABC vector field is an example of a space-time smooth universal exponential mixer in the three dimensions, and in addition, we obtain that it is an ideal kinematic fast dynamo. This is a joint work with Michele Coti Zelati (Imperial College London).