This paper establishes a primal-dual formulation for continuous-time mean field games (MFGs) and provides a complete analytical characterization for the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player’s control problem with measurable coefficients is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation. Finally, a complete characterization of all NEs for MFGs is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by the solvability of the dual problem, and in particular through the regularity of the associated HJB equation.

A key new insight of our analysis for MFGs is that the dual variable within the primal-dual formulation is only required to coincide with the solution of the HJB equation on the support of the mean field flow, reminiscent of the adjoint process in the stochastic maximum principle which only needs to be defined along the optimal state trajectory.

Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization require neither the convexity of the associated Hamiltonian nor the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.

Venue: MCS2068

In this talk, I will present some results motivated by a model for transmission of an infectious disease with temporary immunity. The model is formulated in the literature either as a partial differential equation or, equivalently, as a delay differential equation (DDE) with distributed delay. Focusing on the DDE formulation, I will present an overview of the pseudospectral discretization approach to approximate a DDE with a system of ordinary differential equations. This method has recently been implemented in a DDE importer for the software package MatCont, allowing users to perform numerical bifurcation analysis of DDEs via the MatCont graphical user interface. I will show how waning and boosting of immunity give rise to very complex dynamics at the population level, including regions of parameters where multiple attractors exist (an equilibrium and a limit cycle or two stable limit cycles). Bistability phenomena can be particularly important to explain for instance the resurgence of pertussis cases after the COVID-19 pandemic, to levels different than pre-pandemic. I will then present some recent ongoing work to investigate the basins of attraction of DDEs.

Venue: MJC_2004 (Mountjoy Centre)

The interplay between diffusion, optimal transport and Ricci curvature has been an active theme at the interface of analysis, geometry and probability over the last two decades. On the analytic side, Bakry—Émery's \(Gamma\)-calculus characterises lower Ricci curvature bounds through gradient estimates for the heat semigroup, within the framework of Dirichlet forms.

On the geometric side, the work of Lott, Sturm and Villani shows that the same curvature information is encoded in the displacement convexity of the relative entropy along Wasserstein geodesics, leading to a synthetic notion of Ricci curvature that makes sense on general metric measure spaces.

The two viewpoints meet on the class of RCD spaces, developed in a series of papers with L. Ambrosio and N. Gigli. On such spaces the heat flow is linear --- equivalently, the Cheeger energy is quadratic --- and admits a double description as the \(L^2\)-gradient flow of the Dirichlet energy and as the Wasserstein EVI flow of the entropy. Together with stability under measured Gromov--Hausdorff convergence, this allows RCD spaces to inherit many of the structural properties and calculus tools available on smooth Riemannian manifolds.

The lecture will present the main ideas behind this circle of results and will close with some recent developments along related directions: gradient flows for convex functionals on RCD spaces, Hellinger--Kantorovich contractions, and metric Sobolev structures on spaces of random measures.

Venue: MCS0001

Studying subjects that contain maths can evoke a wide range of emotions. Whilst some students enjoy the challenge others may start to panic, leading to physical or psychological signs of stress. These negative feelings can create a cycle in which anxiety prevents them from studying maths effectively, which in turn increases the anxiety. This struggle can affect all aspects of study or may be limited to certain maths topics or time periods. They can arise at any point during study, even with those that had enjoyed mathematics previously. When the student is aware that they may become anxious, they often take steps to avoid engagement or display other signs that undermine progress.

During this presentation we will examine why and how these anxieties arise. Looking at the effect they have on both body and mind. Highlighting the similarities and differences between general and specific anxieties, we will examine key indicators such as avoidance behaviours, procrastination, emotional responses, and negative self-perception. We will discuss student interaction with various support mechanisms offered by our university including our Open Learn course which takes students on a step-by-step journey through the DEAL process (Describe, Explore, Act and Learn) which is an attainable way for students to work though maths anxiety and find strategies that will help them succeed. The session will highlight how small but purposeful changes in practice can have a meaningful impact on student outcomes and overall well-being.

Venue: MCS2053

We will discuss the initial motivations and our design philosophy for the Computational Mathematics II module, as well as how it was received by students. This will include expected and unexpected problems, especially in designing a coursework-only assessment structure in the age of AI. We will also highlight some incredible work by the students, and some lessons about creativity and student independence that we think are relevant to broader departmental discussions around group work, projects, and the future of our degree programmes.

Venue: MCS2050

Joint work with Zhengyao Huang.  Note that a divergence free horizontal vector current in Heisenberg space may be viewed as an element of the dual space of test vector horizontal fields. Using a horizontal Liouville theorem in Heisenberg space, the resulting flow lines of the divergence free vector field give rise to a family of horizontal curves and a measure on the collection of such. This proves a Smirnov-type decomposition as a current. As application we prove a result on horizontal free approximation in C^1 on compact subsets of Heisenberg space for which all rectifiable curves are constant.

Venue: MCS2068

Dvoretzky's theorem states that in real normed spaces there exist subspaces of sufficiently high dimension that are nearly Euclidean. Equivalently, every high-dimensional symmetric convex body has a section of dimension on the order of log(n) that is close to a Euclidean ball.

We study a functional version of this phenomenon in terms of radial functions on the sphere. We also give versions of Dvoretzky's Theorem in complex and quaternionic normed spaces. Finally, using the Hopf fibration, which lifts positive functions on complex or quaternionic projective spaces to radial functions on the sphere, we obtain functional versions of these extensions.

Venue: MCS2068