Transfer learning is a machine learning technique that leverages knowledge acquired in one domain to improve learning in another, related task. It is a foundational method underlying the success of large language models (LLMs) such as GPT and BERT, which were initially trained for specific tasks. In this talk, I will demonstrate how reinforcement learning (RL), particularly continuous time RL, can benefit from incorporating transfer learning techniques, especially with respect to convergence analysis. I will also show how this analysis naturally yields a simple corollary concerning the stability of score-based generative diffusion models.

Based on joint work with Zijiu Lyu of UC Berkeley.

A 2-connected, rational homotopy 7-sphere is classified up to diffeomorphism by three invariants: its (finite) 4th cohomology group, its q-invariant and its Eells-Kuiper invariant.  The q-invariant is a quadratic refinement of the linking form and determines the homeomorphism type, while the Eells-Kuiper invariant then pins down the diffeomorphism type.  In this talk, I will discuss the diffeomorphism classification of a certain family of non-negatively curved, 2-connected, rational homotopy 7-spheres, discovered by Sebastian Goette, Krishnan Shankar and myself, which contains, in particular, all $S^3$-bundles over $S^4$ and all exotic 7-spheres.

Random Fields with posses the Markov Property have played an important role in the development of Constructive Field Theory. They are related to their relativistic counterparts through Nelson Reconstruction. In this talk I will describe an attempt to understand the Markov Property of the $\Phi^4$ measure in 3 dimensions. We will also discuss the Properties of its Generator (i.e) the $\Phi^4_3$ Hamiltonian. This is based on Joint work with T. Gunaratnam.

For families of Calabi-Yau threefolds, we derive an explicit formula to count the number of points over $\mathbb{F}_{q}$ in terms of the periods of the holomorphic three-form, illustrated by the one-parameter mirror quintic and the 5-parameter Hulek-Verrill family. The formula holds for conifold singularities and naturally incorporates p-adic zeta values, the Yukawa coupling and modularity in the local zeta function. I will give a brief introduction on the physics motivation and how this framework links arithmetic, geometric and physics.

The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity.  Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor.  I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow.

 This is joint work with Renato Velozo Ruiz (Imperial College London).

In this talk, David Luo will use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $p$-adic $\text{GL}(2N, F)$ which we call \textit{middle supercuspidal representations}. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, David will show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$. Furthermore, David poses a conjecture which refines the local converse theorem for general supercuspidal representations of $\text{GL}(n, F)$.

Major depressive disorder (MDD) is a heterogeneous mental disorder. International guidelines present overall symptom severity as the key dimension for clinical characterisation. However, additional layers of heterogeneity may reside within severity levels related to how symptoms interact with one-another in a patient, called symptom dynamics. We investigate these individual differences by estimating the proportion of patients that display differences in their symptom dynamics while sharing the same diagnosis and overall symptom severity. We show that examining symptom dynamics provides information about the person-specific psychopathological expression of patients beyond severity levels by revealing how symptoms aggravate each other over time. These results suggest that symptom dynamics may serve as a promising new dimension for clinical characterisation. Areas of opportunity are outlined for the field of precision psychiatry in uncovering disorder evolution patterns (e.g., spontaneous recovery; critical worsening) and the identification of granular treatment effects by moving toward investigations that leverage symptom dynamics as their foundation. Future work aimed at investigating the cascading dynamics underlying depression onset and maintenance using the large-scale (N > 5.5 million) CIPA Study are outlined. 

Given a set $A$ and a scalar $\lambda$, how large must the sum of dilate $A+\lambda\cdot A=\{a+\lambda a'\mid a,a'\in A\}$ be in terms of $|A|$? In this talk, we will discuss two different settings of this problem, and how they relate to each other.

• For transcendental $\lambda\in \mathbb{C}$ and $A\subset \mathbb{C}$, how does $|A+\lambda\cdot A|$ grow with $|A|$?
• For a fixed large $\lambda\in \mathbb{Z}$ and even larger prime $p$, with $A\subset \mathbb{Z}/p\mathbb{Z}$, how does the density of $A+\lambda\cdot A$ depend on the density of $A$?

Joint with David Conlon.

I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.

We study the fixed points of the Berezin transform on the Fock-type spaces F^{2}_{m} with the weight e^{-|z|^{m}}, m > 0. It is known that the Berezin transform is well-defined on the polynomials in z and \bar{z}. In this talk from Ghazaleh Asghari from Reading University, we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many m \in (0,\infty). We also show that, in some particular cases, the fixed point polynomials are harmonic for all m.

The 2d (massless) Sinh-Gordon model is amongst the simplest 2d quantum field theories that are expected to be integrable (= infinitely many symmetries), but without conformal symmetry. In this talk I will explain a rigorous construction of this model and its vertex correlations (= Laplace transforms) on the infinite cylinder using probability theory. A fundamental role is played by the Sinh-Gordon Hamiltonian and I will explain how the theory of Gaussian multiplicative chaos can be used to analyze this linear map. This talk will be based on joint work with Colin Guillarmou and Vincent Vargas.

When doing arithmetic geometry, it is helpful to have invariants of the objects which we are studying that see both the arithmetic and the geometry. Motivic homotopy theory allows us to produce new invariants which generalise classical topological invariants, such as the Euler characteristic of a variety. These motivic invariants not only recover the classical topological ones, but also provide arithmetic information. In this talk, I'll review the construction of a motivic Euler characteristic, then study its arithmetic properties, and mention some applications. I'll then talk about work in progress with Ran Azouri, Stephen McKean and Anubhav Nanavaty which studies a "higher Euler characteristic", allowing us to produce an invariant of automorphisms valued in an arithmetically interesting group. I'll then talk about how to relate part of this invariant to a more classical invariant of quadratic forms.

TBA

Optimal damping consists of identifying a viscosity vector that maximizes the decay rate of a mechanical system's response. This can be rephrased as minimizing the trace of the solution of a Lyapunov equation whose coefficient matrix, representing the system dynamics, depends on the dampers' viscosities. The latter must be nonnegative for a physically meaningful solution, and the system must be asymptotically stable at the solution.

In this talk, we present conditions under which the system is never stable or may not be stable for certain values of the viscosity vector, and, in the latter case, discuss how to modify the constraints so as to guarantee stability. We show that the KKT conditions of our nonlinear optimization problem are equivalent to a viscosity-dependent nonlinear residual function that is equal to zero at an optimal viscosity vector. To minimize this residual function, we propose a Barzilai-Borwein residual minimization algorithm (BBRMA) and a spectral projection gradient algorithm (SPG). The efficiency of both algorithms relies on a fast computation of the gradient for BBRMA, and both the objective function and its gradient for SPG. By fully exploiting the low-rank structure of the problem, we show how to compute these in $O(n^2)$ operations, $n$ being the size of the mechanical system.

This is joint work with Qingna Li (Beijing Institute of Technology).

TBA

Cell and tissue movement during development, immune response, and cancer invasion depends on chemical or mechanical guidance cues. In many systems, this guidance arises not from long-range, pre-patterned cues but from self-generated gradients locally shaped by cells. However, how heterogeneous cell mixtures coordinate their migration by self-generated gradients remains largely unexplored. In this talk, I will first summarize our recent discovery that immune cells steer their long-range migration using self-generated chemotactic cues (Alanko et al., 2023). I will then introduce a multi-component Keller-Segel model that describes migration and patterning strategies of heterogeneous cell populations (Ucar et al., 2025). Our model predicts that the relative chemotactic sensitivities of different cell populations determine the shape and speed of traveling density waves, while boundary conditions such as external cell and attractant reservoirs substantially influence the migration dynamics. We quantitatively corroborate these predictions with in vitro experiments on co-migrating immune cell mixtures. Interestingly, immune cell co-migration occurs near the optimal parameter regime predicted by theory for coupled and colocalized migration. Finally, I will discuss the role of mechanical interactions, revealing a non-trivial interplay between chemotactic and mechanical non-reciprocity in driving collective migration.
 

This session will look at how you can get the most out of your lectures and tutorials. We’ll talk about how to prepare effectively, make lectures more productive, and understand what tutors expect from you during tutorials. You’ll leave with practical tips to help you study more confidently and make your learning time count.

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

Mathematicians often like to think of maths as objective. Science communicator Hana Ayoob joins us to discuss how the fact that humans do maths means that the ways maths is developed, used, and communicated are not neutral.

Modern deep learning methods provide the state-of-the-art in image reconstruction in most areas of computational imaging. However, such techniques are very data hungry and in a number of key imaging problems access to ground truth data is challenging if not impossible. This has led to the emergence of a range of self-supervised learning algorithms for imaging that attempt to learn to image without ground truth data. 

In this talk I will review some of the existing techniques and look at what is and might be possible in self-supervised imaging.

I will discuss an ascending filtration on the Chow group of zero-cycles on a smooth projective variety obtained roughly by considering the successive kernels of the iterates of some modified diagonal embedding of the variety. This filtration is particularly relevant in the case of abelian varieties and of hyper-Kähler varieties, where it is expected to be opposite to the conjectural Bloch-Beilinson filtration. In the case of abelian varieties, it can in fact be described explicitly in terms of the Beauville decomposition, while in the case of hyper-Kähler varieties, I conjecture (and prove in some cases) that it coincides with a filtration introduced earlier by Claire Voisin. As a by-product we obtain in joint work with Olivier Martin a criterion involving second Chern classes for two effective zero-cycles on a moduli space of stable objects on a K3 surface to be rationally equivalent, generalising a result of Marian-Zhao.

I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the congruence subgroup property, relates the finite-index subgroups of mapping class groups to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. Time permitting, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.

We consider targeted intervention problems in dynamic network and graphon games. First, we study a general dynamic network game in which players interact over a graph and seek to maximize their heterogeneous, concave goal functionals. We establish the existence and uniqueness of a Nash equilibrium in both the finite-player network game and the corresponding infinite-player graphon game, and prove its convergence as the number of players tends to infinity. We then introduce a central planner who implements a dynamic targeted intervention. Given a fixed budget, the central planner maximizes the average welfare at equilibrium by perturbing the players' heterogeneous goal functionals. Using a novel fixed-point argument, we prove the existence and uniqueness of an optimal intervention in the graphon setting, and show that it achieves near-optimal performance in large finite networks. Finally, we study the special case of linear-quadratic goal functionals and derive semi-explicit solutions for the optimal intervention.

 

This is a joint work with Sturmius Tuschmann.  

Equations of fluctuating hydrodynamics, also called Dean-Kawasaki type equations, are stochastic PDEs describing the evolution of finitely many interacting particles which obey a Langevin equation. First, we give a mathematical derivation for such equations. The focus is on systems of interacting particles described by second order Langevin equations. For such systems,  the equations of fluctuating hydrodynamics are a stochastic variant of Vlasov-Fokker-Planck equations, where the noise is white in space and time, conservative and multiplicative. We show a dichotomy previously known for purely diffusive systems holds here as well: Solutions exist only for suitable atomic initial data, but provably not for any other initial data. The class of systems covered includes several models of active matter. We will also discuss regularisations, where existence results hold under weaker assumptions. 

to follow

Consider a tail trivial, positively associated site percolation process such that the set of open vertices is stochastically dominated by the set of closed ones. We show that, for any planar graph $G$, such a process must contain zero or infinitely many infinite connected components. The assumptions cover Bernoulli site percolation at parameter $p$ less than or equal to one half, resolving a conjecture of Benjamini and Schramm. As a corollary, we prove that $p_c$ is greater than or equal to $1/2$ for any unimodular, invariantly amenable planar graphs.

We will then apply this percolation statement to the loop $O(n)$ model on the hexagonal lattice, and show that, whenever $n$ is between $1$ and $2$ and $x$ is between $1/\sqrt{2}$ and $1$, the model exhibits infinitely many loops surrounding every face of the lattice, giving strong evidence for conformally invariant behavior in the scaling limit (as conjectured by Nienhuis).

This is joint work with Alexander Glazman (University of Innsbruck) and Nathan Zelesko (Northeastern University).

What do the roots of random polynomials look like? Classical works of Erdős-Turán and others show that most roots are near the unit circle and they are approximately rotationally equidistributed. We will begin with an understanding of why this happens and see how ideas from extremal combinatorics can mix with analytic and probabilistic arguments to show this. Another main feature of random polynomials is that their roots tend to "repel" each other. We will see various quantitative statements that make this rigorous. In particular, we will study the smallest separation $m_n$ between pairs of roots and show that typically $m_n$ is on the order of $n^{-5/4}$. We will see why this reflects repulsion between roots and discuss where this repulsion comes from. This is based on joint work with Oren Yakir.

to follow

A discussion on how race and ethnicity interact with the concept of merit in academia, based on sections from the book 'Misconceiving Merit' by Blair-Loy and Cech. 

In this talk, we will explore three flow configurations that illustrate the behaviour of slow-moving viscous fluids in confined geometries: viscous gravity currents, fracturing of shear-thinning fluids in a Hele-Shaw cell, and rectangular channel flows of non-Newtonian fluids. We will first develop simple mathematical models to describe each setup, and then we will compare the theoretical predictions from these models with laboratory experiments. As is often the case, we will see that even models that are grounded in solid physical principles often fail to accurately predict the real-world flow behaviour. Our aim is to identify the primary physical mechanisms absent from the model using laboratory experiments. We will then refine the mathematical models and see whether better agreement between theory and experiment can be achieved.

TBA

TBC

Several interesting and emerging problems in statistics, machine learning and optimal transport can be cast as minimisation of (entropy-regularised) objective functions defined on an appropriate space of probability distributions.  Numerical methods have historically focused on linear objective functions, a setting in which one has access to an unnormalised density for the distributional target.  For nonlinear objectives, numerical methods are relatively under-developed; for example, mean-field Langevin dynamics is considered state-of-the-art.  In the nonlinear setting even basic questions, such as how to tell whether or not a sequence of numerical approximations has practically converged, remain unanswered.  Our main contribution is to present the first computable measure of sub-optimality for optimisation in this context.  

Joint work with Clémentine Chazal, Heishiro Kanagawa, Zheyang Shen and Anna Korba.