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.

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

I will be using scattering amplitudes, instead of the Lagrangian of General Relativity (GR), to compute classical observables in GR. In the first part of the seminar I will consider the elastic scattering of two massive particles, describing two black holes, and I will show how to compute the eikonal up to two-loop order, corresponding to third Post-Minkowskian (3PM) order, that contains all the classical information. From it I will compute the first observable that is the classical deflection angle. In the second part of the seminar I will consider inelastic processes with the emission of soft gravitons. In this case the eikonal becomes an operator containing the creation and annihilation operators of the gravitons. The case of soft gravitons can be treated following the Bloch-Nordsieck approach and, in this case, I will be computing two other observables: the zero-frequency limit (ZFL) of the spectrum dE/d\omega of the emitted radiation and the angular momentum loss at 2PM and 3PM. I will consider also the case in which there are static modes localised at $\omega=0$. In the third part of the seminar I will be discussing soft theorems with one graviton emission, first briefly at tree level, and then at loop level following the paper by Weinberg from 1965. Assuming the eikonal resummation and that all infrared divergences in the case of gravity come only from one loop diagrams, I will compute the universal soft terms, corresponding to $\frac{1}{\omega}$, $\log \omega$ and $\omega \log^2 \omega$, first at the tree and one-loop level and then for the last two observables also at two-loop level. I will then use them to compute their contribution to the spectrum of emitted energy. Finally, if I have time left, I I will study the high energy limit. In particular, since the graviton is the massless particle with the highest spin, we expect universality at high energy. I will show that universality at high energy is satisfied both in the elastic and inelastic case, but this happens in the inelastic case in a very non trivial way. I will end with some conclusions and with a list of open problems.

TBA

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.

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.

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 study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence to define invariance notion of stochastic integrals. We introduce the concept of quadratic roughness of a path along a partition sequence and show that for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Using these results we derive a formulation of the pathwise Föllmer-Itô calculus which is invariant with respect to the partition sequence.

We further present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic variation with respect to the sequence of partitions for these constructions. We identify a class of paths whose quadratic variation along a partition sequence is invariant under coarsening