Week 17.02.2025 – 23.02.2025

The dynamics of many-body systems, such as gases of particles or lattices of spins, often display, at large scales of space and time, a high degree of universality. Indeed, this dynamics is usually described by a few equations, those of hydrodynamics, representing
the flows of conserved currents such as those of particles and energy. This is because other "degrees of freedom" thermalise much more quickly, and the full dynamics projects onto that of conserved currents. In fact, surprisingly, even correlations between
local observables at large separations in time, and large-scale fluctuations, can be described by hydrodynamics. This is the object of various theories of hydrodynamic fluctuations, such as macroscopic fluctuation theory (for systems where diffusion dominates),
and its ballistic counterpart (for systems where persistent currents exist). I will introduce the main ideas behind such theories, restricting to systems in one dimension of space for simplicity. I will concentrate on perhaps the simplest and newest, ballistic
macroscopic fluctuation theory, taking simple examples such as the gas of classical hard rods (hard spheres, but in one dimension) - but many concepts are general.

Causal optimal transport and the related adapted Wasserstein distance have recently been popularized as a more appropriate alternative to the classical Wasserstein distance in the context of stochastic analysis and mathematical finance. In this talk, we establish some interesting consequences of causality for transports between laws of continuous time stochastic processes, such as SDEs and Gaussian processes. In particular, these (bi-)causal transports admit stochastic integral representations, from which we can establish topological properties and compute explicitly the adapted Wasserstein distance between Gaussian Volterra processes. Time permitting, we will discuss the stability and approximation of the adapted Wasserstein distance to address the cases where an explicit computation is not known. This talk is based on joint works with Prof. Rama Cont and Y. Jiang.

Title: Computing torsion points on Jacobians of Curves
Abstract: Points on the Jacobian of a curve can easily be constructed when points on the curve are known. We may ask whether one could compute points on the Jacobian without prior knowledge of any points on the curve. In this talk we discuss a method for computing the group of 3-torsion points on the Jacobian of a genus 3 curve. We use elementary geometry to derive a system of equations whose equations parametrise 3-torsion points and use complex analysis and lattice reduction to find precise expressions for the solutions. Time permitting, I will discuss an application of this to computing the local conductor exponent at 2.

I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the  congruence subgroup property, relates the finite-index subgroups of mapping class groups of surfaces to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. If there’s time, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.

How does a biological system produce long time scales that vastly outlast intrinsic biochemical rates, yet are not infinite? This challenge features in various biological tasks involving memory and sensing. We uncover how this also manifests in the cellular assembly of a C. elegans embryo. High-resolution imaging reveals that the formation of the cell’s actin cortex is preceded by a stage where thousands of highly branched actin structures transiently grow and disassemble [1]. Many structures grow orders of magnitude past intrinsic degradation time scales before disassembling, yet without proliferating. We uncover how an overlooked bifurcation in the underlying biochemical dynamics can account for this huge lifetime disparity. We find that a simple mechanism based on resource competition can guide the system towards this dynamical bifurcation without the need for parameter fine-tuning or a biological regulatory mechanism. If time allows I will mention

[1] Victoria Tianjing Yan, Arjun Narayanan, Tina Wiegand, Frank Jülicher, Stephan W. Grill, Nature (2022).