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.

Einstein’s theory of general relativity reshaped our understanding of the universe. Instead of thinking of gravity as a force, Einstein showed it is the bending and warping of space and time caused by mass and energy. This radical idea not only explained how planets orbit stars, but also opened the door to astonishing predictions. In this seminar we will explore some of its most fascinating consequences from the expansion of the universe, to gravitational waves, and the existence of black holes.

The electrochemical processes in electrolytic cells are the basis for modern energy technology such as batteries. Electrolytic cells consist of an electrolyte (an salt dissolved in solution), two electrodes, and a battery. The Poisson–Nernst–Planck equations are the simplest mathematical model of steady state ionic transport in an electrolytic cell. We find the matched asymptotic solutions for the ionic concentrations and electric potential inside the electrolytic cell with mass conservation and known flux boundary conditions. The mass conservation condition necessitates solving for a higher order solution in the outer region. Our results provide insight into the behaviour of an electrochemical system with a known voltage and current, which are both experimentally measurable quantities.

Join bestselling author Simon Singh and Oxford mathematician turned educator Junaid Mubeen for a session on maths communication! Learn how to present mathematics in a way that is both accessible and engaging, and how to apply these principles in a teaching context. Simon and Junaid will draw on their experiences in the Parallel Academy https://parallel.org.uk, an online initiative they set up in 2023, which has since grown to support thousands of keen and talented students to pursue maths beyond the curriculum. 

With the classification theory of simple and nuclear C*-algebras of real rank zero advanced to a level which may very well be final, it is natural to wonder what happens when one allows ideals, but not too many of them. Contrasting the simple case, the K-theoretical classification theory for real rank zero C*-algebras with finitely many ideals is only satisfactorily developed in subcases, and in many settings it is even unclear and/or disputed which flavor of K-theory to use.

Restricting throughout to the setting of real rank zero, Søren Eilers will compare what is known of the classification of graph C*-algebras and of approximately subhomogeneous C*-algebras, with an emphasis on what kind of conclusion can be extracted from restrictions on the complexity of the ideal lattice. The results presented are either more than a decade old or joint with An, Liu and Gong.