Past

In seminal work, Harper showed that the partial sums of random multiplicative functions enjoy "better than square-root cancellation". This surprising phenomenon gives rise to many questions; for example, recent research has focused on whether certain subsums possess this property. In this talk, I will introduce smooth numbers, discuss their "anatomy", and relate this to recent progress on random multiplicative functions over smooth numbers. This is based on joint work with Max Wenqiang Xu.

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.

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.

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.

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.

From contours of 2D shapes to knotted proteins, many areas of statistics and physical sciences are concerned with data in the form of geometric loops - embeddings of the circle into a Euclidean space. It is desirable to find a representation of the geometry of these loops that is invariant under Euclidean symmetries, and indifferent to how the loops are parametrised. In this joint work with Ka Man Yim (Oxford) and Otto Sumray (MPI Dresden), we propose a novel representation of a loop’s geometry by rep resenting the distance matrix of the loop as a Morse function on a Möbius band - the second finite subset space of the circle. By considering the persistent homology of this function, we obtain a representation of the loop’s geometry that has the desired symmetry invariance. We show that this feature map is stable with respect to perturbations, give a geometric interpretation of the features obtained and explore analogues involving higher order finite subset spaces.

In this talk, I will discuss the challenges of solving string theory through the two-dimensional quantum field theory on the string worldsheet. Focusing on the spectrum of superstrings in AdS3, I will explain how conformal field theory techniques yield exact results for special backgrounds, and how methods from integrable models allow us to go beyond these cases.

Mapping class groups arise in an elementary way in the study of surfaces, and their properties are used in many areas of group theory and low-dimensional topology. This talk will be an introduction to the study of mapping class groups, in particular via a geometric space known as the curve complex. This approach is classical and yields strong properties and a deep understanding of mapping class groups. In the second half of the talk, we will discuss a broadening of this construction to a family of groups known as Artin groups, whose properties are still very mysterious and are the subject of an active area of research.

Renewal equations are one of the most popular ways to model outbreaks of infectious diseases and estimate key control indicators for decision-makers. This talk will focus on new mathematical methods that address where traditional renewal equations often fail to capture the human, biological, and environmental complexities of epidemics. These methods cover both directly transmitted and vector-borne diseases, describe epidemics on complex human mobility networks, and enable more targeted interventions across space and time.

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.

Let K be a degree d number field. A famous theorem of Faltings states that a "nice" curve C of genus at least 2 defined over K has only finitely many points defined over K. However, C can of course admit infinitely algebraic points of degree d. For example, a hyperelliptic curve defined over the rationals admits infinitely many quadratic points. In this talk, we will see many examples of curves which have infinitely many points of degree 2 or 3 and, moreover, we shall see the number fields arising from these points are relatively rare.

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.

Tim van Beeck (University of Göttingen) is going to talk about: "An Embedded Trefftz DG Approach to Least-Squares Finite Element Methods"

Least-squares finite element methods (LSFEM) minimise the residual of the PDE directly. This gives discrete problems that are stable by construction, have symmetric positive definite system matrices, and come with a built-in reliable and efficient error estimator. In this talk, we present an embedded Trefftz DG approach to LSFEM. Trefftz methods use basis functions that locally lie in the kernel of the differential operator, which greatly reduces the number of degrees of freedom. The embedded Trefftz DG method obtains these spaces through an element-local embedding into a standard DG space, so no explicit Trefftz basis is needed. We apply the method to a range of linear second-order problems, including Helmholtz-type equations and PDEs in non-divergence form, and show that it keeps the properties above under suitable mesh resolution conditions, while requiring far fewer degrees of freedom than standard DG or first-order least-squares formulations. We conclude with an outlook on nonlinear problems such as the Hamilton-Jacobi-Bellman equation.

Self-propelled particles such as swimming micro-organisms often accumulate at boundaries. Active Brownian particles (ABPs), that is, particles moving at constant speed with an orientation undergoing rotational diffusion, provide a simple model. I will discuss the distribution of ABPs inside a channel. 

Stallings core graphs are a powerful graphical representation of the finitely generated subgroups of a free group. They bring geometric insight to the study of free groups, make many properties and algorithmic tasks transparent, and are a ubiquitous tool in combinatorial and geometric group theory.
Free group algebras are strikingly analogous to free groups. Most notably, just as the Nielsen-Schreier theorem says that every subgroup of a free group is free, the Cohn-Lewin theorem says that every right (or left) ideal of a free group algebra is free. Yet free group algebras have had no counterpart to core graphs.
In this talk I will present one. We define the core of a module over a free group algebra, and show that it is unique whenever the module is algebraic (also known as bound). Many of the concepts of Stallings' theory, such as folding and pullbacks, generalize to these new objects, and this yields simple proofs and algorithms for many natural problems about free group algebras. For example, I will present an algorithm for testing primitivity in free modules which generalizes (Wilton's simplification of) Whitehead's classical algorithm for testing primitivity in free groups.
This is joint work with Doron Puder.
 

Scale and conformal symmetries play a central role in various areas of theoretical physics, ranging from statistical mechanics and quantum field theory to string theory. A natural question is under what conditions scale invariance implies the larger symmetry of conformal invariance.  In this talk, I will discuss non-perturbative arguments for this enhancement in two and four dimensions, highlighting the assumptions under which they hold. I will first explain how Zamolodchikov's $c$-theorem, under suitable assumptions, can be used to establish the enhancement of scale invariance to local conformal invariance in two dimensions. I will then turn to four dimensions and discuss the argument of Dymarsky, Komargodski, Schwimmer, and Theisen based on dilaton scattering amplitudes and unitarity. I will conclude by commenting on possible generalisations to other dimensions and, time permitting, recent progress on the question.

In our summer research project, we used recent developments in the classification of *-homomorphisms to construct isomorphisms between certain infinite amalgamated reduced free products. We used an intertwining argument to build the isomorphisms from the maps realised through classification. In this talk, we will give an overall view of the argument and tools used and give some examples.

When we look at a graph, one of the first questions that might naturally come to mind is "Does this graph have any symmetry?" In fancier language, is the automorphism group trivial? The quantum automorphism group is a compact matrix quantum group which is meant to capture the quantum symmetries of the graph. Although we've ramped up the jargon, we still have the same question: Given a graph, does it have any quantum symmetries? In this talk, I'll explain why this question is undecidable, thanks to an Adian-Rabin theorem for quantum automorphism groups. Consequently, there are graphs where the quantum automorphism group is non-trivial but has no non-trivial finite-dimensional representations (or in other words, the discrete quantum automorphism group is trivial).

Joint work with Josse van Dobben de Bruyn. 
 

Since the early 20th century, it was speculated that signed networks tend to display "structural balance". This is where certain configurations of positive and negative edges capture the property that a "friend of a friend is a friend" and the "friend of an enemy is an enemy". Originating in mathematical psychology for the purpose of studying triadic interactions, the notion of balance has since been generalised, and is believed to manifest outside of social systems.

Efforts have been made to quantify structural balance, which is rarely fully displayed in real world networks. In this talk, we will discuss current measures of balance and their fallacies, which tend to arise from computing what balance is not – as opposed to what it is. In turn, we propose a probabilistic model for structural balance, and suggest an inferential framework to robustly assess it's manifestation in real data.

We study the minimum size of a maximal independent set in a regular graph, that is, the smallest sets which are both independent and dominating. In a balanced complete bipartite graph, every maximal independent set contains exactly half of the vertices, so large degree alone does not guarantee a small maximal independent set. 

In this talk I will show that regular graphs with bounded codegrees admit much smaller maximal independent sets. Specifically, we show that every $d$-regular graph on $n$ vertices with a moderate codegree bound contains a maximal independent set of size at most $(1+o(1))n (\log d)/d$, as $d$ goes to infinity, and this bound is sharp. The proof is based on a nibble method where, in each step, we sample from the hard-core model. I will also discuss applications.

This is joint work with Matthew Jenssen and Andrey Shapiro.

The de Rham cohomology of a smooth manifold roughly measures the failure of the fundamental theorem of calculus on such a manifold. When such a manifold is a compact homogenous space $G/R$ this cohomology can be calculated via a differential graded algebra in the  exterior powers of its cotangent space $Lie(G)/Lie(R)^*$. When this exterior algebra is replaced by a Clifford algebra, one creates a filtered deformation of de Rham cohomology, which we call Clifford cohomology. I will explain this theory and give results on the description of this cohomology and provide links to Dirac cohomology and nilpotent Lie algebra cohomology. Joint work with Karmen Grizelj, Andrey Krutov, Kyo Nishiyama and Pavle Pandzic.

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.

Allen-Cahn equations are popular phenomenological models for phase transitions that describe materials on a mesoscopic scale. The macroscopic scale observed by us humans corresponds to a singular scaling limit, the so-called sharp-interface limit, in which transition layers collapse onto hypersurfaces. Since Ilmanen’s pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand this sharp-interface limit of systems of coupled Allen-Cahn equations. In this talk, I will present the first global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke. 

This is joint work with Keisuke Takasao (Kyoto University).
 

Calabi-Yau varieties constitute a wide class which (strictly) includes abelian varieties and K3 surfaces. In this talk, I will discuss  Brantner-Taelman's work on the mixed characteristic deformation theory of Calabi-Yau varieties, and discuss my recent work characterising their "canonical lifts" via $p$-adic Hodge theory. Time permitting, I will also discuss the (in general conjectural) modularity of Calabi-Yau varieties over number fields and its relation to the aforementioned lifts.

The dual Thurston norm ball of a 3-manifold is a polytope in its first real homology group. In this talk, I will introduce a natural generalization of this polytope to sutured 3-manifolds, constructed using L^2-torsion theory. The resulting invariant generalizes both the sutured Floer homology polytope and the taut foliation polytope, and yields beautiful decomposition and covering formulas. This is joint work with Yaoping Xie.

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. 

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.

We study the geometry of simplicial cochains on Vietoris–Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper-bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

This is joint work with Darrick Lee: https://arxiv.org/pdf/2608.11137

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.

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.

Manin and his collaborators initiated a program in 1989 with the goal of understanding the distribution of rational points of bounded height on Fano varieties, using methods from algebraic geometry and analytic number theory. Since then, there has been substantial progress on this program, including the formulation of a precise conjecture. I will discuss the achievements of this program since then, with a focus on surfaces, including some of my own work on quintic del Pezzo surfaces.

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.

Joseph Webb (Mathematical Institute, University of Oxford) is going to talk about "An optimisation framework for the well-conditioned training of physics-informed neural networks"

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.