The theory of characters for infinite groups, initiated by Thoma, is a natural generalization of the representation theory of finite groups. More precisely, a character on a discrete group is a normalised positive definite function which is conjugation invariant and extremal. Connes conjectured a rigidity result for characters of an important family of discrete groups, namely, irreducible lattices in higher-rank semisimple Lie groups. The conjecture states that every character is either the trace of a finite-dimensional representation, or vanishes off the center. This rigidity property implies the Stuck-Zimmer conjecture for such lattices, namely, ergodic actions are either essentially transitive or essentially free. I will present a recent joint result with Michael Glasner, Yuval Gorfine, Liam Hanany and Arie Levit in which we prove that non-uniform irreducible lattices in higher-rank semisimple groups are character rigid. As a result, we also obtain a resolution of the Stuck-Zimmer conjecture for all non-uniform lattices.

The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.

We introduce the class of exponentially-algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of their types in a reduct of the theory of differentially closed fields.

We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions and give some new decidability results.

This is joint work with Rémi Jaoui, Lyon

Random Fields with posses the Markov Property have played an important role in the development of Constructive Field Theory. They are related to their relativistic counterparts through Nelson Reconstruction. In this talk I will describe an attempt to understand the Markov Property of the $\Phi^4$ measure in 3 dimensions. We will also discuss the Properties of its Generator (i.e) the $\Phi^4_3$ Hamiltonian. This is based on Joint work with T. Gunaratnam.

Understanding the relationship between expectation and price is central to applications of mathematical finance, including algorithmic trading, derivative pricing and hedging, and the modelling of margin and capital. In this presentation, the link is established via dynamic entropic risk optimisation, which is promoted for its convenient integration into standard pricing methodologies and for its ability to quantify and analyse model risk. As an example of the versatility of entropic pricing, discrete models with classical and quantum information are compared, with studies that demonstrate the effectiveness of quantum decorrelation for model fitting.

We study  dynamic 𝑁-player noncooperative games called 𝛼-potential games, where the change of a player’s objective function upon her unilateral deviation from her strategy is equal to the change of an 𝛼-potential function up to an error 𝛼. Analogous to the static potential game (which corresponds to 𝛼=0), the 𝛼-potential game framework is shown to reduce the challenging task of finding 𝛼-Nash equilibria for a dynamic game to minimizing the 𝛼-potential function. Moreover, an analytical characterization of 𝛼-potential functions is established, with 𝛼 represented in terms of the magnitude of the asymmetry of objective functions’ second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, 𝛼 is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely, distributed games and games with mean field interactions, are analyzed to highlight the dependence of 𝛼 on general game characteristics that are beyond the mean field paradigm, which focuses on the limit of 𝑁 with homogeneous players. To analyze the 𝛼-NE (Nash equilibrium), the associated optimization problem is embedded into a conditional McKean–Vlasov control problem. A verification theorem is established to construct 𝛼-NE based on solutions to an infinite-dimensional Hamilton–Jacobi–Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.

Surfactants are chemicals that preferentially reside at interfaces. Once surfactant molecules have adsorbed to an interface, they reduce the surface tension between the two neighbouring fluids and may induce fluid flow. Surfactants have many household applications, such as in cleaning products and cosmetics, as well as industrial applications, like mineral processing and agriculture. Thus, understanding the dynamics of surfactant solutions is particularly important with regards to improving the efficacy of their applications as well as highlighting how they work. In this seminar, we will explore the spreading of a droplet over a substrate, in which there is constant injection of liquid and soluble surfactant through a slot in the substrate. Firstly, we will see how the inclusion of surfactant alters the spreading of the droplet. We will then investigate the early- and late-time behaviour of our model and compare this with numerical simulations. We shall conclude by briefly examining the effect of changing the geometry of the inflow slot.

Cellular adhesion is a fundamental mechanism underlying diverse collective cell behaviours, from tissue self-organisation in developmental biology to the formation of directional queues that guide cell migration. Modelling such interactions has also proven mathematically rich, motivating the use of continuum partial differential equation models that capture adhesion through nonlocal interaction kernels. These models can, for instance, reproduce classical cell-sorting patterns arising from differential adhesion in mixtures of cell populations. In this talk, we briefly review such models and explain how a local approximation of nonlocal aggregation–diffusion equations can be derived in the limit of short-range interactions. We then discuss recent advances in the field and highlight new results on pattern formation driven by adhesive interactions in migrating and proliferating cell populations, as well as in systems of nonreciprocally interacting cells.