Past

Recent foundational work by Ben-Bassat, Kelly, and Kremnitzer describes a model for derived analytic geometry as homotopical geometry relative to the infinity category of simplicial commutative complete bornological rings. In this talk, Rhiannon Savage will discuss a representability theorem for derived stacks in these contexts and will set out some new foundations for derived smooth geometry. Rhiannon will also briefly discuss the representability of the derived moduli stack of non-linear elliptic partial differential equations by an object we call a derived C∞-bornological affine scheme.

This tutorial will be a hands-on introduction to proving theorems in Lean, using its mathematical library Mathlib. It will not assume any previous knowledge about formal theorem provers. We will discover the Lean language, learn how to read a statement and a proof, and learn the essential "tactics" one can use to prove theorems in Lean.

Participants should come with a computer, and it would be best if they could install Lean before the tutorial by following the instructions at https://lean-lang.org/install/ . The installation should be easy and takes only a few minutes.

I will explain how the irreversibility of the renormalization group together with anomalies, including anomalies in the space of coupling constants, can be used to constrain the IR phases of defects in familiar quantum field theories. As an example, I will use these techniques to provide evidence for a conjectural "spin-flux duality" which describes how certain line operators are mapped across particle/vortex duality in 2+1d.

Einstein’s theory of general relativity reshaped our understanding of the universe. Instead of thinking of gravity as a force, Einstein showed it is the bending and warping of space and time caused by mass and energy. This radical idea not only explained how planets orbit stars, but also opened the door to astonishing predictions. In this seminar we will explore some of its most fascinating consequences from the expansion of the universe, to gravitational waves, and the existence of black holes.

The Ginzburg–Landau energy is often used to approximate the Dirichlet energy. As the perturbation parameter tends to zero, critical points of the Ginzburg–Landau energy converge, in an appropriate (bubbling) sense, to harmonic maps. In this talk I will first explain key analytical properties of this approximation procedure, then show that not every harmonic map can be approximated in this way. This is based on a rigidity theorem: under the energy threshold of 8pi, we classify all solutions of the associated nonlinear elliptic system from S2 to S2, thereby identifying exactly which harmonic maps can arise as Ginzburg–Landau limits in this regime.

A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification type has been restricted to lie in a given subset of the Galois group and provide an explicit formula for the leading constant. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG and how the existence of such number fields is controlled by a Brauer-Manin obstruction. No prior knowledge of stacks is needed for this talk!

TBC

I will present a collaborative project in which we formalized the construction of Brownian motion in Lean. Lean is an interactive theorem prover, with a large mathematical library called Mathlib. I will give an introduction to Lean and Mathlib, explain why one may want to formalize mathematics, and give a tour of the probability theory part of Mathlib. I will then describe the Brownian motion project, its organization, and some of the formalized results. For that project, we developed the theory of Gaussian measures and implemented a proof of Kolmogorov's extension theorem, as well as a modern version of the Kolmogorov-Chentsov continuity theorem based on Talagrand's chaining technique. Finally, I will discuss the next step of the project: formalizing stochastic integrals.

Symplectic implosion is an abelianisation construction in symplectic geometry. The implosion of the cotangent bundle of a group K plays a universal role in the implosion of manifolds with a K-action.  This universal implosion, which is usually a singular variety, can also be viewed as the non-reductive Geometric Invariant Theory quotient of the complexification G of K by its maximal unipotent subgroup. 

In this talk, we describe joint work with Johan Martens and Nick Proudfoot which uses point-counting techniques to calculate the intersection cohomology of the universal implosion.

Quantum tomography involves obtaining a full classical description of a prepared quantum state from experimental results. We propose a Langevin sampler for quantum tomography, that relies on a new formulation of Bayesian quantum tomography exploiting the Burer-Monteiro factorization of Hermitian positive-semidefinite matrices. If the rank of the target density matrix is known, this formulation allows us to define a posterior distribution that is only supported on matrices whose rank is upper-bounded by the rank of the target density matrix. Conversely, if the target rank is unknown, any upper bound on the rank can be used by our algorithm, and the rank of the resulting posterior mean estimator is further reduced by the use of a low-rank promoting prior density. This prior density is a complex extension of the one proposed in [Annales de l’Institut Henri Poincaré Probability and Statistics, 56(2):1465–1483, 2020]. We derive a PAC-Bayesian bound on our proposed estimator that matches the best bounds available in the literature, and we show numerically that it leads to strong scalability improvements compared to existing techniques when the rank of the density matrix is known to be small.

I will review notions of categorical complexity, and the more recent work of Biran, Cornea and Zhang on fragmentation in triangulated persistence categories (TPCs), then go on to discuss applications of this to filtered topology. In particular, we will consider a suitable category of filtered topological spaces and detail some constructions and properties, before showing that an associated 'filtered stable homotopy category' is a TPC. I will then give some interesting results relating to this.

Prof Ian Griffiths (a mental health first aider in the department) will lead a discussion about how to protect your mental health when studying an intense graduate degree and outline the support and resources available within the Mathematical Institute. 

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.

Motivated by pattern formations and cell movements, many evolution equations incorporating spatial convolution with suitable integral kernel have been proposed. Numerical simulations of these nonlocal evolution equations can reproduce various patterns depending on the shape and form of integral kernel.The solutions to nonlocal evolution equations are similar to the patterns obtained by reaction-diffusion system and Keller-Segel system models. In this talk, we classify nonlocal interactions into two types, and investigate their relationship with reaction-diffusion systems and Keller-Segel systems, respectively. In these partial differential equation systems, we introduce multiple auxiliary diffusive substances and consider the singular limit of the quasi-steady state to approximate nonlocal interactions. In particular, we introduce how the parameters of the partial differential equation system are determined by the given integral kernel. These analyses demonstrate that, under certain conditions, nonlocal interactions and partial differential equation systems can be treated within a unified framework.  
This talk is based on collaborations with Hiroshi Ishii of Hokkaido University and Hideki Murakawa of Ryukoku University. 

We develop a general framework for pathwise stochastic integration that extends Follmer's classical approach beyond gradient-type integrands and standard left-point Riemann sums and provides pathwise counterparts of Ito, Stratonovich, and backward Ito integration. More precisely, for a continuous path admitting both quadratic variation and Levy area along a fixed sequence of partitions, we define pathwise stochastic integrals as limits of general Riemann sums and prove that they coincide with integrals defined with respect to suitable rough paths. Furthermore, we identify necessary and sufficient conditions under which the quadratic variation and the Levy area of a continuous path are invariant with respect to the choice of partition sequences.

Given a von Neumann algebra M, we can consider the set of values of p such that Lp(M) has the approximation property: the identity on it is a limit of finite rank operators for a suitable topology. Apart from the case when p is infinite, which has been the subject of a lot of work initiated by Haagerup in the late 70s, this invariant has not been very much exploited so far. But ancient works in collaboration with Vincent Lafforgue and Tim de Laat suggest that, maybe, it can distinguish the factors of SL(n,Z) for different values of n. I will explain something that I realized only recently, and that explains why this is a difficult question: it implies some form of the classical Kakeya conjecture, which predicts the shape of sets in the Euclidean space in which a needle can be turned upside down. This talk from Mikael de la Salle will be an opportunity to discuss other connections between classical Fourier analysis and analysis in group von Neumann algebras, including in collaboration with Javier Parcet and Eduardo Tablate

We'll describe connections between probabilistic models for primes,
the Hardy-Littlewood k-tuples conjectures, the distribution of primes in
very short intervals, the interval sieve, and hypothetical Landau-Siegel
zeros of Dirichlet L-functions.  We will emphasize the role and limitations
of probabilistic ideas.

I will discuss a two-dimensional dilaton gravity theory with a sine potential. At the disk level, this theory admits a microscopic holographic realization as the double-scaled SYK model. Remarkably, in the open channel canonical quantization of the theory, the momentum conjugate to the length of two-sided Cauchy slices becomes periodic. As a result, the ERB length in sine dilaton gravity is discretized upon gauging this symmetry. For closed Cauchy slices, a similar discretization occurs in the physical Hilbert space, corresponding to a discrete spectrum for the length of the necks of trumpet geometries. By appropriately gluing two such trumpets together, one can then construct a wormhole geometry in sine dilaton gravity, whose amplitude matches the spectral correlation functions of a one-cut matrix integral. This correspondence suggests that the theory provides a path integral formulation of q-deformed JT gravity, where the matrix size is large but finite. Finally, I will describe how this theory of gravity can be regarded as a realization of q-deformed holography and propose a possible implementation of this framework to study the near-horizon dynamics of near-extremal de Sitter black holes.

Time-series datasets are central in numerous fields of science and engineering, such as biomedicine, Earth observation, and network analysis. Extensive research exists on state-space models (SSMs), which are powerful mathematical tools that allow for probabilistic and interpretable learning on time series. Estimating the model parameters in SSMs is arguably one of the most complicated tasks, and the inclusion of prior knowledge is known to both ease the interpretation but also to complicate the inferential tasks. In this talk, I will introduce a novel joint graphical modeling framework called DGLASSO (Dynamic Graphical Lasso) [1], that bridges the static graphical Lasso model [2] and the causal-based graphical approach for the linear-Gaussian SSM in [3]. I will also present a new inference method within the DGLASSO framework that implements an efficient block alternating majorization-minimization algorithm. The algorithm's convergence is established by departing from modern tools from nonlinear analysis. Experimental validation on synthetic and real weather variability data showcases the effectiveness of the proposed model and inference algorithm.

[1] E. Chouzenoux and V. Elvira. Sparse Graphical Linear Dynamical Systems. Journal of Machine Learning Research, vol. 25, no. 223, pp. 1-53, 2024

[2] J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse covariance estimation with the graphical LASSO. Biostatistics, vol. 9, no. 3, pp. 432–441, Jul. 2008.

[3] V. Elvira and E. Chouzenoux. Graphical Inference in Linear-Gaussian State-Space Models. IEEE Transactions on Signal Processing, vol. 70, pp. 4757-4771, Sep. 2022.

In the numerical simulation of incompressible flows and elastic materials, it is often desirable to design discretisation schemes that preserve key structural properties of the underlying physical model. In particular, the conservation of angular momentum plays a critical role in accurately capturing rotational effects, and is closely tied to the symmetry of the stress tensor. Classical formulations such as the Stokes equations or linear elasticity can exhibit significant discrepancies when this symmetry is weakly enforced or violated at the discrete level.

This work focuses on mixed finite element methods that impose the symmetry of the stress tensor strongly, thereby ensuring exact conservation of angular momentum in the absence of body torques and couple stresses. We systematically study the effect of this constraint in both incompressible Stokes flow and linear elasticity, including anisotropic settings inspired by liquid crystal polymer networks. Through a series of benchmark problems—ranging from rigid body motions to transversely isotropic materials—we demonstrate the advantages of angular-momentum-preserving discretisations, and contrast their performance with classical elements.

Our findings reveal that strong symmetry enforcement not only leads to more robust a priori error estimates and pressure-independent velocity approximations, but also more reliable physical predictions in scenarios where angular momentum conservation is critical.

These insights advocate for the broader adoption of structure-preserving methods in computational continuum mechanics, especially in applications sensitive to rotational invariants.

When a specimen of non-trivial shape undergoes deformation under a dead load or during an active process, finite element simulations are the only technique for evaluating the deformation. Classical books describe complicated techniques for evaluating stresses and strains in semi-infinite, circular or cylindrical objects.  However, the results obtained are limited, and it is well known that elasticity (linear or nonlinear) is strongly intertwined with geometry. For the simplest geometries, it is possible to determine the exact deformation, essentially for low loading values, and prove that there is a threshold above which the specimen loses stability. The next step is to apply perturbation techniques (linear and nonlinear bifurcation theory).
 

In this talk, I will demonstrate how many aspects can be simplified or revealed through the use of complex analysis and conformal mapping techniques for shapes, strains, and active stresses in thin samples. Examples include leaves and embryonic jellyfish.

I will talk about some recent work with Tingxiang Zou on higher-dimensional Elekes-Szabó problems in the case of an Ind-constructible action of a group G on a variety X. We expect nilpotent algebraic subgroups N of G to be responsible for any such; this roughly means that if H and A are finite subsets with non-expansion |H*A| <= |A|^{1+\eta}, then H concentrates on a coset of some such N.

A natural example is the action of the Cremona group of birational transformations of the plane. I will talk about a recent result which confirms the above expectation when we restrict to the group of polynomial automorphisms of the plane, using Jung's description of this group as an amalgamated free product, as well as some work in progress which combines weak polynomial Freiman-Ruzsa with effective Mordell-Lang, after Akshat Mudgal, to handle some further special cases.

A 2024 collection of articles in the Bulletin of the AMS asked "Will machines change mathematics?", suggesting that  "Pure mathematicians are used to enjoying a great degree of research autonomy and intellectual freedom, a fragile and precious heritage that might be swept aside by a mindless use of machines." and challenging readers to  "decide upon our subject’s future direction.”

This was a response to the mathematical capabilities of emerging technologies, alone or in combination. These techniques include  software such as LEAN for  providing formal proofs; use of LLMs to produce credible, if derivative, research papers with expert human guidance; specialist algorithms such as AlphaGeometry; and sophisticated use of machine learning to search for examples.  Their development (at huge cost in compute power and energy) has been accompanied by an unfamiliar and exuberant level of hype from well-funded start-ups claiming to “solve mathematics” and the like. And it raises questions beyond the technical concerning governance, funding and the nature of the mathematical profession.

To try and understand what’s going on we look historical examples of changes in mathematical practice - as an example we consider key developments in the early days of computational group theory.

The speaker is keen to hear of colleagues using LLMs, LEAN or similar things in research, even if they can’t come to the talk.

Strong chaos, the butterfly effect, is a ubiquitous phenomenon in physical systems. In quantum mechanical systems, one of the diagnostics of quantum chaos is an out-of-time-order correlation function, related to the commutator of operators separated in time. In this talk we will review the work of Maldacena, Shenker and Stanford (arxiv:1503.01409), who conjectured that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λL ≤ 2πkBT/\hbar. We will then discuss a system that displays a maximal Lyapunov exponent: the SYK model. 

In the first half of this talk, I will provide a brief introduction to Isomonodromic deformations with the one-point torus as my main example, and show the relation to the elliptic form of Painlevé VI equation as well as the Lamé equation. In the second half of this talk, I will present an overview of my results in the past few years concerning the associated tau-functions, conformal blocks, and accessory parameters. Finally, I will motivate how probabilistic methods in conformal field theory help us understand the data within Lamé type equations.

Nearly $G_2$-structures in dimension seven are, up to scaling, critical points of a geometric flow called (modified) Laplacian co-flow. Moreover, since nearly $G_2$-structures define Einstein metrics, they can also be associated to critical points of the volume-normalised Ricci flow. In this talk, we will discuss a recent joint work with Jason Lotay, showing that many of these nearly $G_2$ critical points are unstable for the modified co-flow, and giving a lower bound on the index.

A 1977 conjecture of Erdős and Hajnal asserts that for every hereditary class of graphs not containing all graphs, every graph in the class has a polynomial-sized clique or stable set. Fox, Pach, and Suk and independently Chernikov, Starchenko, and Thomas asked whether this conjecture holds for every class of graphs of bounded VC-dimension. In joint work with Alex Scott and Paul Seymour, we resolved this question in the affirmative. The talk will introduce the Erdős–Hajnal conjecture and discuss some ideas behind the proof of the bounded VC-dimension case.

A finite tensor category is a suitably nice abelian category with a compatible monoidal structure. It makes perfect sense to define the representation type of such a category, as a measure of how complicated the category is in terms of its indecomposable objects. For example, finite representation type means that the category contains only finitely many indecomposable objects, up to isomorphism.  

In this talk from Petter Bergh, we shall see that if a finite tensor category has finitely generated cohomology, and the Krull dimension of its cohomology ring is at least three, then the category is of wild representation type. This is a report on recent joint work with K. Erdmann, J. Plavnik, and S. Witherspoon. 

It is well known to mathematicians that there is a deep relationship between the arithmetic of algebraic varieties and their geometry.  

The dynamics of spatially-structured networks of N interacting stochastic neurons can be described by deterministic population equations in the mean-field limit. While this is known, a general question has remained unanswered: does synaptic weight scaling suffice, by itself, to guarantee the convergence of network dynamics to a deterministic population equation, even when networks are not assumed to be homogeneous or spatially structured? In this work, we consider networks of stochastic integrate-and-fire neurons with arbitrary synaptic weights satisfying a O(1/N) scaling condition. Borrowing results from the theory of dense graph limits, or graphons, we prove that, as N tends to infinity, and up to the extraction of a subsequence, the empirical measure of the neurons' membrane potentials converges to the solution of a spatially-extended mean-field partial differential equation (PDE). Our proof requires analytical techniques that go beyond standard propagation of chaos methods. In particular, we introduce a weak metric that depends on the dense graph limit kernel and we show how the weak convergence of the initial data can be obtained by propagating the regularity of the limit kernel along the dual-backward equation associated with the spatially-extended mean-field PDE. Overall, this result invites us to reinterpret spatially-extended population equations as universal mean-field limits of networks of neurons with O(1/N) synaptic weight scaling. This work was done in collaboration with Pierre-Emmanuel Jabin (Penn State) and Datong Zhou (Sorbonne Université).

I will present problems a stochastic variant of the classic optimal transport problem as well as a large deviation question for a mean field system of interacting particles. We shall see that those problems can be analyzed by means of a Hamilton-Jacobi equation on the space of probability measures. I will then present the main challenge on such equations as well as the current known techniques to address them. In particular, I will show how the notion of relaxed controls in this setting naturally solve an important difficulty, while being clearly interpretable in terms of geometry on the space of probability measures.

In the 1990s, Goodwillie developed a theory of calculus for homotopical functors. His idea was to approximate a functor by a tower of ‘polynomial functors’, similar to how one approximates a function by its Taylor series. The role of linear polynomials is played by functors that behave like homology theories, in the sense that there is a Mayer-Vietoris sequence computing their homotopy groups. As such, the Goodwillie tower interpolates between stable and unstable homotopy theory. The theory has application to the computation of the homotopy groups of spheres, higher algebra, and algebraic K-theory. In my talk, I will give an introduction to this topic. In particular, I will explain that Goodwillie's calculus reveals a deep connection between the homotopy theory of spaces and Lie algebras and how this is related to a chain rule for the derivatives of functors.
 

I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

I will discuss the Brill-Noether theory of a general elliptic $K3$ surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general $k$-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)

In this talk, I will discuss my joint paper with Olivier Biquard and Paul Gauduchon on ALF Ricci-flat Riemannian  4-manifolds. My collaborators had previously classified all such spaces that are toric and Hermitian, but not Kaehler. Our main result uses an open curvature condition to prove a rigidity result of the following type: any Ricci-flat metric that is sufficiently close to a non-Kaehler, toric, Hermitian ALF solution (with respect to a norm that imposes reasonable fall-off at infinity) is actually  one of the  known Hermitian toric  solutions. 
 

Extra-chromosomal DNA (ecDNA) is a genetic error found in more than 30% of tumour samples across various cancer types. It is a key driver of oncogene amplification promoting tumour progression and therapeutic resistance, and is correlated to the worse clinical outcomes. Different from chromosomal DNA where genetic materials are on average equally divided to daughter cells controlled by centromeres during mitosis, the segregation of ecDNA copies is random partition and leads to a fast accumulation of cell-to-cell heterogeneity in copy numbers.  I will present our analytical and computational modeling of ecDNA dynamics under random segregation, examining the impact of copy-number-dependent versus -independent fitness, as well as the maintenance and de-mixing of multiple ecDNA species or variants within single cells. By integrating experimental and clinical data, our results demonstrate that ecDNA is not merely a by-product but a driving force in tumor progression. Intra-tumor heterogeneity exists not only in copy number but also in genetic and phenotypic diversity. Furthermore, ecDNA fitness can be copy-number dependent, which has significant implications for treatment.

We study  dynamic 𝑁-player noncooperative games called 𝛼-potential games, where the change of a player’s objective function upon her unilateral deviation from her strategy is equal to the change of an 𝛼-potential function up to an error 𝛼. Analogous to the static potential game (which corresponds to 𝛼=0), the 𝛼-potential game framework is shown to reduce the challenging task of finding 𝛼-Nash equilibria for a dynamic game to minimizing the 𝛼-potential function. Moreover, an analytical characterization of 𝛼-potential functions is established, with 𝛼 represented in terms of the magnitude of the asymmetry of objective functions’ second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, 𝛼 is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely, distributed games and games with mean field interactions, are analyzed to highlight the dependence of 𝛼 on general game characteristics that are beyond the mean field paradigm, which focuses on the limit of 𝑁 with homogeneous players. To analyze the 𝛼-NE (Nash equilibrium), the associated optimization problem is embedded into a conditional McKean–Vlasov control problem. A verification theorem is established to construct 𝛼-NE based on solutions to an infinite-dimensional Hamilton–Jacobi–Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.