L6

Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.

A fundamental result in 3-manifold topology is the loop theorem: Given a null-homotopic simple closed curve in the boundary of a compact 3-manifold $M$, it bounds an embedded disk in $M$. The standard topological proof of this uses the tower construction due to Papakyriakopoulos. In this talk, I will introduce basic existence and regularity results on minimal surfaces, and show how to use the tower construction to prove a geometric version of the loop theorem due to Meeks--Yau: Given a null-homotopic simple closed curve in the boundary of a compact Riemannian 3-manifold $M$ with convex boundary, it bounds an embedded disk of least area. This also gives an independent proof of the (topological) loop theorem.

Many classes of finitely generated groups have been studied using formal language theory techniques. One historical example is the study of geodesics, which gives rise to the strict growth series of a group. Properties of languages associated to groups can provide insight into the nature of the growth series.

In this talk we will introduce languages associated to conjugacy classes, rather than elements of the group. This will lead us to define an analogous series, namely the conjugacy growth series of a group, which has become a popular topic in recent years. After discussing the necessary group theoretic and language tools needed, we will focus on how these conjugacy languages behave in graph products. We will finish with some new results which look at when these properties can extend to virtual graph products.

Group cohomology is a powerful tool which has found many applications in modern group theory. It can be calculated and interpreted through geometric, algebraic and topological means, and as such it encodes the relationships between these different aspects of infinite groups. The aim of this talk is to introduce a circle of ideas which link group cohomology with the theory of BNS invariants, and the property of being subgroup separable. No prior knowledge of any of these topics will be assumed.

Condensed Mathematics is a tool recently developed by Clausen and Scholze and it is proving fruitful in many areas of algebra and geometry. In this talk, we will cover the definition of condensed sets, the analogues of topological spaces in the condensed setting. We will also talk about condensed modules over a ring and some of their nice properties like forming an abelian category. Finally, we'll discuss some recent results that have been obtained through the application of Condensed Mathematics.

In order for one to infer reasonable predictions from a model, it must be calibrated to reproduce observations in the market. We use the semimartingale optimal transport methodology to formulate this calibration problem into a constrained optimisation problem, with our model calibrated using a finite number of European options observed in the market as constraints. Given such a PDE formulation, we are able to then derive a dual formulation involving an HJB equation which we can numerically solve. We focus on two cases: (1) The stochastic interest rate is known and perfectly matches the observed term structure in the market, however the asset local volatility and correlation are not known and must be calibrated; (2) The dynamics of both the stochastic interest rate and the underlying asset are unknown, and we must jointly calibrate both to European options on the interest rate and on the asset.

We study the impact of transition scenario uncertainty, and in particular, the uncertainty about future carbon price and electricity demand, on the pace of decarbonization of the electricity industry. To this end, we build a discrete time mean-field game model for the long-term dynamics of the electricity market subject to common random shocks affecting the carbon price and the electricity demand. These shocks depend on a macroeconomic scenario, which is not observed by the agents, but can be partially deduced from the frequency of the shocks. Due to this partial observation feature, the common noise is non-Markovian. We consider two classes of agents: conventional producers and renewable producers. The former choose an optimal moment to exit the market and the latter choose an optimal moment to enter the market by investing into renewable generation. The agents interact through the market price determined by a merit order mechanism with an exogenous stochastic demand. We prove the existence of Nash equilibria in the resulting mean-field game of optimal stopping with common noise, developing a novel linear programming approach for these problems. We illustrate our model by an example inspired by the UK electricity market, and show that scenario uncertainty leads to significant changes in the speed of replacement of conventional generators by renewable production.

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. We develop a deep transport map that is suitable for sampling concentrated distributions defined by an unnormalised density function. We approximate the target distribution as the push-forward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a tensor train decomposition. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. We propose two bridging strategies suitable for wide use: tempering the target density with a sequence of increasing powers, and smoothing of an indicator function with a sequence of sigmoids of increasing scales. The latter strategy opens the door to efficient computation of rare event probabilities in Bayesian inference problems.

Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.
 

Elkem is developing a new method for categorising the reactivity between silicon carbide (SiC) powder and silicon monoxide gas (SiO(g)). Experiments have been designed which pass SiO gas through a powdered bed of SiC inside of a heated crucible, resulting in a reaction between the two. The SiO gas is produced via a secondary reaction outside of the SiC bed. Both reactions require specific temperature and pressure constraints to occur. Therefore, we would like to mathematically model the temperature distribution and gas flow within the experimental set-up to provide insight into how we can control the process.

Complexities arise from:

• Endothermic reactions causing heat sinks
• Competing reactions beyond the two we desire
• Dynamically changing properties of the bed, such as permeability