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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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Professor Dr. Sebastian Pokutta will speak about; 'How Machines Explore, Conjecture, and Discover Mathematics'

Artificial Intelligence is increasingly becoming a genuine partner in mathematical research, not only as a computational tool, but as a driver of exploration, conjecture generation, and discovery. Under the umbrella of AI4Math, we develop methodologies that combine optimization, machine learning, and mathematical structure to navigate large, complex, and highly constrained search spaces that are inaccessible to traditional approaches.

In this talk, we illustrate this paradigm through a concrete case study: the Hadwiger–Nelson problem, a long-standing open problem in discrete geometry and extremal combinatorics concerning colorings of the plane without monochromatic unit-distance pairs. We show how neural networks can be used as expressive approximators to transform a mixed discrete–continuous geometric problem with hard constraints into a differentiable optimization problem with a probabilistic loss. This enables gradient-based exploration of admissible configurations and directly led to the discovery of two novel six-colorings, yielding the first improvement in thirty years for the off-diagonal variant of the problem.

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.

A recurring theme in differential geometry is understanding when a set of geometric conditions uniquely characterizes an object. In this talk, I will discuss this question for minimal surfaces in the unit ball, focusing on those meeting the boundary orthogonally. I will explain recent joint work with Jonathan Zhu in which we prove that every properly embedded, free-boundary, minimal annulus is congruent to the critical catenoid. I will also describe closely related results for axially symmetric Alt-Caffarelli cones and free-boundary minimal annuli in spherical caps, and discuss connections with other uniqueness results in differential geometry. 

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Cell proliferation has been widely studied across experimental systems and mathematical modelling frameworks, from early embryonic development to regenerating tissues. A characteristic feature of early embryonic development is that initially synchronous cell populations progressively lose synchrony after a species-specific number of divisions. Here, we use an age-dependent branching process to determine how variability in cell-cycle duration controls the persistence of synchrony. We derive analytical expressions for the mean and variance of cell number and show how they identify the underlying generation-time distribution. For gamma-distributed generation times, this establishes a direct relationship between cell-cycle variability and the number of synchronous divisions. We use these results to infer generation-time statistics from population-level cell counts and investigate how systematic changes in cell-cycle duration can bias inference under a stationary model. Our results provide a quantitative framework linking the precision of individual cell-cycle clocks to observable population-level dynamics.

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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.

We show that the group cohomology of simple connected Lie groups, and consequently their lattices and symmetric spaces, with L^p-coefficients vanishes below the rank. This confirms a conjecture of Gromov in the special case of the coefficient module L^p(G). We relate the vanishing of L^1-cohomology to a fixed-point theorem and confirm a conjecture of Farb. Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
In this talk, I will give a short introduction to group cohomology and discuss ideas from the proofs of both results. This talk is based on joint work with Saar Bader, Uri Bader, and Roman Sauer.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards.

In this talk we will discuss $G_2$-instantons and their relation to the heterotic $G_2$-system on 7-dimensional 2-step nilpotent Lie groups. We will first consider left-invariant coclosed $G_2$-structures and present necessary and sufficient conditions for their characteristic connection to be a $G_2$-instanton, expressed in terms of the torsion of the $G_2$-structure, the torsion of the connection, and the Lie group structure. We will then explain how these conditions imply that the corresponding left-invariant Riemannian metrics define naturally reductive structures.

As a continuation, we will consider solutions to the heterotic $G_2$-system on 2-step nilmanifolds. We will describe constructions of solutions with non-abelian gauge groups and discuss how the geometry of the underlying $G_2$-structure and the choice of gauge connection enter the equations. In particular, we will present families of solutions arising from the $G_2$-instantons described in the first part.

This is joint work with Andrew Clarke, Andrés J. Moreno, and Udhav Fowdar.

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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