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In mathematical models of opinion dynamics, individuals interact with each other and adjust their opinions based on their interactions. In opinion models, network structures determine which individuals can interact with each other and thereby affect how opinions change with time. In this talk, I will introduce opinion models and why scientists study them. I will also discuss several variants of bounded-confidence models (BCMs), in which opinions take continuous values in a region, and I will examine how network structure affects the formation of consensus, polarization, and fragmentation of populations.

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We study model-free policy learning for discrete-time mean-field control (MFC) problems with finite state space and compact action space. In contrast to the extensive literature on value-based methods for MFC, policy-based approaches remain largely unexplored due to the intrinsic dependence of transition kernels and rewards on the evolving population state distribution, which prevents the direct use of likelihood-ratio estimators of policy gradients from classical single-agent reinforcement learning. We introduce a perturbation scheme on the state-distribution flow and prove that the gradient of the resulting perturbed value function converges to the true policy gradient as the perturbation magnitude vanishes. This construction yields a model-free estimator based solely on simulated trajectories and an auxiliary estimate of the sensitivity of the state distribution. Building on this framework, we develop MF-REINFORCE, a model-free policy gradient algorithm for MFC, and establish quantitative bounds on its bias and mean-squared error.

We'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

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Professor Raj Shukla will talk about: 'Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces'

Adaptive Topological DeepONets: Functional Measurements in Locally Convex Spaces
Deep Operator Networks (DeepONets) typically encode an input function through its point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov, we replace point samples with continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space. The topology of this space is generated by a point-separating family of seminorms rather than a single norm. Within this setting, we develop both fixed and adaptive functional measurement systems. These measurements are combined with the coefficient-space Two-Step training procedure of Lee and Shin, and a training-only decoder with regularization stabilizes the adaptive coordinates.
On the theoretical side, we derive a discrete error decomposition that separates measurement error, output-basis error, and neural-approximation error, together with a refinement based on Barron-type approximation rates.
We evaluate the framework on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and both fixed-time and time-evolving Navier–Stokes vorticity operators. For heterogeneous Darcy flow, the functional models keep nearly resolution-independent errors of 5.5–5.6% on unseen grids. For the controlled problem, adaptive measurements reduce the mean error below 1.2%. For fixed-time Navier–Stokes, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, reaching a mean relative L2 error of 1.685% ± 0.017% with only 128 functional coordinates. A Fourier neural operator (FNO) of comparable size achieves a lower error of 0.832% ± 0.172%, but it needs the full 64 × 64 input field, twice the training time, and 10.7 times the peak GPU memory.
Overall, the formulation yields compact, interpretable, and discretization-portable coordinates in the continuous dual space, including for non-normable input spaces.

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Professor Sara Shashaani is going to talk about: 'Adaptive Sampling and Regularization for Stochastic Trust-region Methods'

Trust-region methods have proven highly effective for unconstrained nonconvex stochastic optimization problems where objective and gradient information are available only through noisy stochastic oracles. ASTRO is a class of adaptive sampling trust-region methods that dynamically determine sampling effort while constructing local quadratic models from noisy function and gradient observations. By exploiting dependence among samples and the stochastic structure of the problem, ASTRO achieves strong convergence and complexity guarantees. Its derivative-free variant, ASTRO-DF, relies solely on noisy function evaluations and also enjoys almost-sure convergence guarantees.

Control policies optimized in simulation can perform poorly in the real system when the simulator's parameters are estimated from limited data but the resulting parameter uncertainty is not represented inside the simulation. A common way to incorporate such ambiguity is to simulate each trajectory under a randomly drawn and unobservable parameter value. This forces the policy to act robustly at the start of the control episode. Over time, however, the policy can often infer the parameter value from its observations and specialize its actions accordingly, which is undesirable in systems where latent factors are expected to shift. In financial markets, for example, a policy hedging a derivative payoff should remain robust to changes in the volatility regime. To induce such continual robustness, we propose training policies in simulators where ambiguity can vary with the system's state but does not systematically vanish over time. We formalize this as stationary ambiguity: the simulator induces a stationary filter process over the latent state. We show how to construct such simulators and demonstrate, on hedging problems, that policies trained under stationary ambiguity maintain robustness to latent factors over time, leading to strong performance on real market data. As a modeling principle, stationary ambiguity informs which models make realistic simulators, how their parameters should be randomized, and how simulator and policy should be initialized. While our experiments focus on hedging, stationary ambiguity may also be useful for other control problems driven by exogenous stochastic processes with shifting latent structure.

This is a joint work with Amira Akkari, Ben Wood, and Lukas Gonon.

Phenotypic variation is a ubiquitous feature of biological cell populations, even in genetically identical cells growing in uniform environments. Such variability can have profound consequences for population-level behaviour, particularly under stress, yet it is often neglected in classical modelling frameworks.

In the first part of this talk, I consider mathematical models of bacterial population growth that explicitly incorporate non-heritable variation in individual cell growth rates. I examine how phenotypic heterogeneity and environmental selection shape population growth and the dynamics of phenotypic subpopulations. We derive theoretical results for population growth rates and compare them with predictions from homogeneous models, identifying regimes in which variability qualitatively alters population outcomes.

 In the second part of the talk, I turn to self-generated chemotaxis, a collective process in which cells modify their chemical environment to guide movement. Using a hybrid discrete–continuum model that couples stochastic cell motion with a continuum description of the chemoattractant, I investigate how phenotypic variation in motility, sensing, and chemical degradation affects the robustness of collective migration. The results and tools developed have broader implications for collective behaviour in cell biology, ecology, and evolution.

Diffusion models and flow-based methods have achieved strong results in image generation, but often rely on costly iterative sampling. We introduce W-Flow, a framework that compresses a Wasserstein gradient flow into a one-step neural generator. By minimizing the Sinkhorn divergence, W-Flow transports a reference distribution toward the data distribution through efficient optimal-transport updates that capture global distributional discrepancies. We prove that, under suitable assumptions, the finite-sample training dynamics converge to the continuous-time distributional dynamics. Empirically, W-Flow sets a new state of the art in one-step ImageNet generation, with improved mode coverage and domain transfer. We will also discuss extensions using alternative energy functionals and applications to post-training.

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Underneath large glaciers and ice sheets, water flows through a permeable network of cavities and channels, held open by melting the ice above balancing the downwards flow of ice. Dissipation within the water flow is a significant enough source of heat that instabilities can develop if the flow rate is high enough, eroding large channels that rapidly drain water from the glacier bed. I will present a model for the system, and discuss the linear problem, observational evidence for the stability criterion, and the non-linear interactions that rapidly become the dominant control on channel spacing. Recently, there has been some discussion of slowing glaciers down by pumping water out from under them - I will consider the viability of this strategy in view of the results in this talk.

Dr Peter Braam is going to talk about; 'State time geometry: causal performance profiles and optimal data transport in parallel execution'

The increasing complexity of parallel architectures and heterogeneous microarchitectures makes predicting and optimising program performance notoriously difficult. For Optimal Data Transport, we present a discrete variant of the Wasserstein–Fisher–Rao metric that quantifies the true cost of data layout transformations and movement across memory hierarchies. For Causal Performance Profiles, we introduce the Lyons–Gregg Signature, which combines the ideas of Terry Lyons' rough path signatures with hardware performance counters (eBPF) to capture cross-correlated, causal bottlenecks in execution streams. Both arose from State Time Geometry (STG), a model for stateful program execution on computing infrastructure, first modelled as a dynamical system governing state-values over the space of memory addresses. The address space generalises to geometric objects defining infrastructure and leads to the metric. The state transitions of parallel executions become a Grothendieck quantum field theory over the infrastructure and carry the statistical model for the Lyons-Gregg Signature. The central theme is that an intuitive faithful model is not doomed by complexity but forms a geometric domain in which both theoretical and engineering perspectives are simplified.

(In a companion lecture in the Computing Laboratory at 11:00 on Nov 13, we will discuss STG's underlying categorical and geometric structure and its relationship to programming languages and formal methods)