Week 11.02.2024 – 17.02.2024

We will explore the interplay between thermodynamic cost, in terms of energy dissipated, and precision of a physical system whose only accessible information is a time series of discrete events. The analytical derivations - based on variational methods of large deviations - reveal universal bounds, extending beyond the thermodynamic uncertainty relation to diverse nonequilibrium driven systems and general time-asymmetric observables. Additionally, we will see how optimal precision saturating the bounds can be physically achieved and showcase practical applications. This includes distinguishing voluntary actions controlled by the sensorimotor cortex of rats and detecting coherence effects in atomic clocks.

I will talk about the history of the Erdos-Renyi random graph model G(n,p), and its critical point, p=1/n. The critical window is a range of p, in which the largest components have the same scaling but not exactly the same distributional limit as at criticality itself. I will discuss several aspects of the behaviour of these graphs within the critical window, including a surprising connection to the mixing of random permutations.

We will overview a construction of p-adic L-functions for GSp(4) by Loeffler-Pilloni-Skinner-Zerbes and discuss some recent work on interpolating them in a family of Hecke character twists.

I will discuss the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3-manifold M, and describe the resulting PSL(2,R) invariant measures on the Grassmann bundle of M. (Joint work with J. Kahn and I. Smilga.)

The concept of non-ergodicity in quantum many body systems can be discussed in the context of the wave functions of the many body system or as a property of the dynamical observables, such as time-dependent spin correlators. In the former approach the non-ergodic delocalized state is defined as the one in which the wave functions occupy a volume that scales as a non-trivial power of the full phase space. In this work we study the simplest quantum spin glass model and find that in the delocalized non-ergodic regime the spin–spin correlators decay with the characteristic time that scales as non-trivial power of the full Hilbert space volume. The long time limit of this correlator also scales as a power of the full Hilbert space volume. We identify this phase with the glass phase whilst the many body localized phase corresponds to a ’hyperglass’ n which dynamics is practically absent. We discuss the implications of these findings to quantum information problems.

Hadwiger's conjecture in convex geometry, formulated in 1957, states that every convex body in $\mathbb{R}^n$ can be covered by $2^n$ translations of its interior. Despite significant efforts, the best known bound related to this problem was $\mathcal{O}(4^n \sqrt{n} \log n)$ for more than sixty years. In 2021, Huang, Slomka, Tkocz, and Vritsiou made a major breakthrough by improving the estimate by a factor of $\exp\left(\Omega(\sqrt{n})\right)$. Further, for $\psi_2$ bodies they proved that at most $\exp(-\Omega(n))\cdot4^n$ translations of its interior are needed to cover it.

Through a probabilistic approach we show that the bound $\exp(-\Omega(n))\cdot4^n$ can be obtained for convex bodies with sufficiently many well-behaved sub-gaussian marginals. Using a small diameter approximation, we present how the currently best known bound for the general case, due to Campos, Van Hintum, Morris, and Tiba can also be deduced from our results.

Within the mathematical sciences there exist particular challenges associated with the provision of timely and detailed feedback, both of which are important given the widespread use of formative, and typically weekly, problem sheet assessments to aid and structure the mathematical development of learners. In this talk I will report on the outcomes from a cycle of action research that was designed to enhance the feedback received by students and their subsequent engagement with it in a large research-intensive mathematical sciences department along with more recent work to explore how students engage with additional opportunities for support and feedback to aid their mathematical learning. Student views on the current feedback they receive will be discussed, but more broadly the findings offer insight into alternative feedback practices that mathematical sciences departments might wish to explore.

I will discuss models for longitudinal data, where the data consists of noisy measurements taken at several different time points for each individual, and the aim is to model how each individual's underlying response varies over time. If we assume linear variation of the responses over time, we could use a linear mixed model for this task. In this talk, I will discuss more flexible modelling approaches, which allow the variation of the response over time to be any smooth curve. There is a strong link with models for functional data, and I will describe previous work on adapting methods designed for functional data (where measurements are typically taken very frequently) to longitudinal data (with typically only a few measurements on each individual). However, these existing methods sometimes give fitted mean functions which are more complex than needed to provide a good fit to the data. I will describe a new penalised likelihood approach to flexibly model longitudinal data, with a penalty term to control the balance between fit to the data and smoothness of the subject-specific mean curves. I will show that the new method substantially improves the quality of inference relative to existing methods across a range of simulated examples, and apply the method to data on changes in body composition in adolescent girls.