Past

 Hyper-Kähler manifolds are rigid geometric structures. They have three different symplectic and complex structures, in direct analogy with the quaternions. Being Ricci-flat, they solve the vacuum Einstein equations, and so there has been considerable interest among physicists to explicitly construct such spaces. We will discuss in detail the examples arising from the Gibbons-Hawking ansatz. These give concrete descriptions of the metric, giving many examples to work with. They also lead to the generalised classification as hyper-Kähler quotients by P.B. Kronheimer, with one such space for each finite subgroup of SU(2). Finally, we will look at the McKay correspondence, relating the finite subgroups of SU(2) with the simple Lie algebras of type A,D,E.

Merge trees are a topological descriptor of a filtered space that enriches the degree zero barcode with its merge structure. The space of merge trees comes equipped with an interleaving distance d_I , which prompts a naive question: is the interleaving distance between two merge trees equal to the bottleneck distance between their corresponding barcodes? As the map from merge trees to barcodes is not injective, the answer as posed is no, but as proposed by Gasparovic et al., we explore intrinsic metrics d_I and d_B realized by infinitesimal path length in merge tree space, which do indeed coincide. This result suggests that in some special cases the bottleneck distance (which can be computed quickly) can be substituted for the interleaving distance (in general, NP-hard).

Join us on Friday Week 7 for some chill board games! Meet in N4.01 at 12pm for a Taylors sandwich lunch and positive end-of-term vibes.

The universal infinitesimal symmetry of a holomorphic field theory is the Lie algebra of holomorphic vector fields. We introduce the higher-dimensional Virasoro algebra and prove a local, universal, form of the Riemann–Roch theorem using Feynman diagrams. We use the concept of a (Jouanoulou) higher-dimensional chiral algebra as developed recently with Gui and Wang. We will remark on applications to superconformal field theory. This project is joint work with Zhengping Gui.

This session will explore practical ways to manage your time effectively as a student. We’ll discuss how to find the right balance between revising and working on problem sheets, tools and strategies to help you plan your workload, and how to set realistic priorities. We’ll also talk about what kind of study balance makes sense over the Christmas break. Come along to pick up useful tips for staying organised, focused, and on top of your studies.

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

Microbial communities contain many evolving and interacting bacteria, which makes them complex systems that are difficult to understand and predict. We use theory – including game theory, agent-based modelling, ecological network theory and metabolic modelling - and combine this with experimental work to understand what it takes for bacteria to succeed in diverse communities. One way is to actively kill and inhibit competitors and we study the strategies that bacteria use in toxin-mediated warfare. We are now also using our approaches to understand the human gut microbiome and its key properties including ecological stability and the ability to resist invasion by pathogens (colonization resistance). Our ultimate goal is to both stabilise microbiome communities and remove problem species without the use of antibiotics.

Pfaffian functions, and by extension Pfaffian and semi-Pfaffian sets, play a crucial role in various areas of mathematics, including o-minimal theory. Incidence combinatorics has recently experienced a surge of activity, fuelled by the introduction of the polynomial partitioning method of Guth and Katz. While traditionally restricted to simple geometric objects such as points and lines, focus has shifted towards incidence questions involving higher dimensional algebraic or semi-algebraic sets. We present a generalization of the polynomial partitioning method to semi-Pfaffian sets and illustrate how this leads to Pfaffian generalizations of classic results in incidence geometry, such as the Szemerédi-Trotter Theorem. Finally, we outline an application of semi-Pfaffian geometry and Khovanskii's bound to the robustness of neural networks.

We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. To account for liquidity dynamics, we introduce a stochastic volume effect governing the recovery of the deviation process, which represents the difference between the impacted and unaffected price. Additionally, we incorporate stochastic liquidity variations through a regime-switching Markov chain to capture abrupt shifts in market conditions. We study this singular control problem, where the trader optimally determines the timing and rate of purchases to minimize execution costs. The associated value function to this optimization problem is shown to satisfy a system of variational Hamilton–Jacobi–Bellman inequalities. Moreover, we establish that it is the unique viscosity solution to this HJB system and study the analytical properties of the free boundary separating the execution and continuation regions. To illustrate our results, we present numerical examples under different limit-order book configurations, highlighting the interplay between price impact, resilience dynamics, and stochastic liquidity regimes in shaping the optimal execution strategy.

It is well-known that one can attach Galois representations to modular forms. In the case of cusp forms, the corresponding l-adic Galois representations are irreducible for every prime l, while in the case of Eisenstein series, the corresponding Galois representations are reducible. The Langlands correspondence is expected to generalise this picture, with cuspidal automorphic representations always giving rise to irreducible Galois representations. In the cuspidal, polarized, regular algebraic setting over a CM field, a construction of Galois representations is known, but their irreducibility is still an open problem in general. I will discuss recent joint work with Zachary Feng establishing new instances of irreducibility, and outline how our methods extend some previous approaches to this problem.

Linear discrete inverse problems arise in many areas of science and engineering, from medical imaging and geophysics to atmospheric modelling. Their numerical solution often relies on iterative algorithms, particularly Krylov subspace methods, that can efficiently handle large-scale, ill-posed systems. In many practical settings, however, exact computations of matrix–vector products, preconditioners, or right-hand sides are either infeasible or unnecessary, leading to inexact iterations. This talk explores the interplay between inexactness and the regularizing behaviour of Krylov subspace methods for inverse problems. We discuss how approximate computations influence the regularization effect inherent in early iterations, as well as  semiconvergence, and how controlled inexactness may be exploited to improve computational efficiency. The aim is to provide a broad perspective on recent insights and open questions at the interface of inverse problems, iterative solvers, and computational inexactness.

Maximum likelihood estimation is a ubiquitous task in statistics and its applications. The task is: given some observations of a random variable, find the distribution(s) in your statistical model which best explains these observations. A modern perspective on this classical problem is to study the "likelihood geometry" of a statistical model. By focusing on models which have a polynomial parametrization, i.e., lie on an algebraic variety, this perspective brings in tools, algorithms and invariants from algebraic geometry and combinatorics.

In this talk, I will explain some of the key ideas in likelihood geometry and discuss its recent application to the study of likelihood asymptotics, i.e., understanding likelihood estimation for very large or very small observation counts. Agostini et al. showed that these asymptotics can be modeled and understood using tools from tropical geometry, and they used this to completely describe the asymptotics for linear models. In our work, we use the same approach to treat the class of log-linear models (also known as Gibbs distributions or maximum entropy models) and give a complete and combinatorial description of the likelihood asymptotics under some conditions.

This talk is based on joint work with Emma Boniface (UC Berkeley) and Serkan Hoşten (San Francisco SU), available at: https://epubs.siam.org/doi/full/10.1137/24M1656839

An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the pth-order tensor, based on lower-order derivatives. Between each recalculation of the pth-order derivative approximation, a high-order secant equation can be used to update the pth-order tensor as proposed in (Welzel 2022) or the approximation can be kept constant in a lazy manner. When refreshing the pth-order tensor approximation after m steps, an exact evaluation of the tensor or a finite difference approximation can be used with an explicit discretization stepsize. For all the newly adaptive regularization variants, we retrieve standard complexity bound to reach a second-order stationary point.  Discussions on the number of oracle calls for each introduced variant are also provided. When p = 2, we obtain a second-order method that uses quasi-Newton approximations with optimal number of iterations bound. 

Elekes and Szabó established non-trivial incidence bounds for binary algebraic relations in characteristic 0, generalizing the Szemerédi-Trotter theorem for point-line-incidence. This was later generalized to binary relations defined in reducts of so-called distal structures in a result of Chernikov, Peterzil and Starchenko. For fields of positive characteristic, such bounds fail to hold in general. Bays and Martin apply the bounds for distal structures in the context of valued fields to derive incidence bounds in the sense of Szemerédi-Trotter in fields admitting valuations with finite residue field, such as F_p(t). We show that this result can be made uniform in the size of the finite residue field, by making precise in some sense the intuition that ACVF is distal relative to the residue field. In this talk, I will introduce the relevant notions from incidence combinatorics and distality, before outlining a proof of the uniform-in-p result.

A d-dimensional TQFT is a topological invariant which assigns (d-1)-dimensional manifolds to vector spaces and d-dimensional cobordisms to linear maps. In the early 90s, Reshetikhin and Turaev constructed examples of these in the case d=3, using the data of certain types of linear categories. In this talk, I will provide an overview of this construction, and then explore how this might be meaningfully extended downwards to assign 1-manifolds to "2-vector spaces". Minimal knowledge of category theory assumed!

Witten’s seminal 1988 work revealed the connection between 3-dimensional Chern-Simons theory and knot invariants. In this talk, I will provide a physically motivated overview and explain how skein relations manifest from a path-integral/partition-function perspective on 3-manifolds with Wilson lines inserted. There will also be some fun topological brain-twisters for the audience. If time permits, I will comment on recent developments involving factorization homology and its relation to correlators for logarithmic CFTs.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

We present an adjoint method for optimization of the spatially inhomogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a "discretize then optimize" approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian.  The boundary conditions that are included in this analysis include spectral reflection, thermal reflection, and inflow boundary conditions. For thermal reflection, a "score function" is included as a statistical regularization. This is joint work with Yunan Yang (Cornell). This special seminar is jointly held with the Keble Complexity Research Cluster.

I'll discuss some of the history of the use of random matrices for studying the structure of operator algebras, starting with Voiculescu's notion free independence. We'll see that the original notions of convergence of random matrix models to certain infinite-dimensional operators is actually fairly weak, and discuss the more recent "strong convergence" phenomenon and its applications to C*-algebras. Finally, I'll touch upon some ongoing work, joint with A. Shiner and S. White, for continuing to use random matrix tools to prove structural properties of C*-algebras.

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.

Given a finite poset $P$ we ask how small a family of subsets of $[n]$ can be such that it does not contain an induced copy of the poset, but adding any other subset creates such a copy. This number is called the saturation number of $P$, denoted by $\operatorname{sat}^*(n,P)$. Despite the apparent similarity to the saturation for graphs, this notion is vastly different. For example, it has been shown that the saturation numbers exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2 n^{1/2}$. In fact, it is believed that the saturation number is always bounded or exactly linear. In this talk we will be discussing the most recent advances in this field, with the focus on the diamond poset, whose saturation number was unknown until recently.

Joint with Sean Jaffe.

Adam Jones will explore the relationship between the category of smooth representations of a semisimple p-adic Lie group G and the module category over its associated pro-p Iwahori-Hecke algebra via the canonical invariance adjunction. This relationship is well understood in characteristic 0, in fact it yields a category equivalence equivalence, but in characteristic p it is very mysterious and largely defies understanding. We will explore methods of constructing an appropriate subcategory of Hecke modules which is well behaved under the adjunction, and which can be shown to contain all parabolic inductions. He will give examples of this yielding results when G has rank 1, and more recently when G = SL_3 in certain cases.

One of the great powers of global symmetry is its ability to constrain the possible phases of many-body quantum systems. In this talk, we will present a symmetry that enforces every symmetric model to be in a phase with a Fermi surface. This constraint is entirely non-perturbative and a strong form of symmetry-enforced gaplessness. We construct this symmetry in fermionic quantum lattice models on a $d$-dimensional Bravais lattice, and it is generated by a U(1) fermion-number symmetry and Majorana translation symmetry. The resulting symmetry group is an infinite-dimensional non-abelian Lie group closely related to the Onsager algebra. We will comment on the topology of these symmetry-enforced Fermi surfaces and the UV symmetry's relation to the IR LU(1) symmetry of ersatz Fermi liquids. (This talk is based on ongoing work with Shu-Heng Shao and Luke Kim.)

Mathematical models for elastic materials undergoing growth will be considered. The characteristic feature is a multiplicative decomposition of the deformation gradient into an elastic part a growth-related part. Approaches towards the existence of solutions will be discussed in
various settings, including models with and without codimension. This is joint work with Kira Bangert and Julian Blawid.

In this talk, I will give an account of the measure of large values where |ζ(1/2 + it)| > exp(V), with t ∈ [T,2T] and V ∼ αloglogT. This is the range that influences the moments of the Riemann zeta function. I will present previous results on upper bounds by Arguin and Bailey, and new lower bounds in a soon to be completed paper, joint with Louis-Pierre Arguin, and explain why, with current machinery, the lower bound is essentially optimal. Time permitting, I will also discuss adaptations to other families of L-functions, such as the central values of primitive characters with a large common modulus.

We consider particles following a diffusion process in a multi-well potential and attracted by their barycenter (corresponding to the particle approximation of the Wasserstein flow of a suitable free energy). It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Exactly at the critical temperature below which multiple equilibria appear, the situation is somewhat degenerate and we can get uniform in N convergence estimates, but polynomial instead of exponential.

The spectral gap (or bass note) of a closed hyperbolic surface is the smallest non-zero eigenvalue of its Laplacian. This invariant plays an important role in many parts of hyperbolic geometry. In this talk, I will speak about joint work with Will Hide on the question of which numbers can appear as spectral gaps of closed arithmetic hyperbolic surfaces.

A conjecture of Holzegel, Schmelzer and Warnick states that there is a Ricci flow with surgery connecting the two Ricci flat metrics Taub-Bolt and Taub-NUT. We will present some recent progress towards proving this conjecture. This includes showing for the first time the existence of a Ricci flow with surgery with local topology change $\mathbb{CP}^2\setminus\{ \mathrm{pt}\} \rightarrow \mathbb{R}^4$.

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.

The development of a complex functional multicellular organism from a single cell involves tightly regulated and coordinated cell behaviours coupled through short- and long-range biochemical and mechanical signals. To truly comprehend this complexity, alongside experimental approaches we need mathematical and computational models, which can link observations to mechanisms in a quantitative, predictive, and experimentally verifiable way. In this talk I will describe our efforts to model aspects of embryonic development, focusing in particular on the planar polarised behaviours of cells in epithelial tissues, and discuss the mathematical and computational challenges associated with this work. I will also highlight some of our work to improve the reproducibility and re-use of such models through the ongoing development of Chaste (https://github.com/chaste), an open-source C++ library for multiscale modelling of biological tissues and cell populations.

What is the common theory of all finite fields equipped with an additive and/or multiplicative character? Hrushovski answered this question in the additive case working in (a mild version of) continuous logic. Motivated by natural number-theoretic examples we generalise his results to the case allowing for both (non-trivial) additive character and (sufficiently generic) multiplicative character. Apart from answering the above question we obtain a quantifier elimination result and a generalisation of the definability of the Chatzidakis-Macintyre-van den Dries counting measure to this context. The proof relies on classical results on bounds of character sums following from the work of Weil.

An epsilon-representation of a discrete group G is a map from G to the unitary group U(n) that is epsilon-multiplicative in norm uniformly across the group. In the 1980s, Kazhdan showed that surface groups of genus at least 2 are not uniform-to-local stable in the sense that they admit epsilon-representations that cannot be perturbed, even locally (on the generators), to genuine representations.
 

In this talk, Marius Dadarlat of Purdue University will discuss the role of bounded 2-cohomology in Kazhdan's construction and explain why many rank-one lattices in semisimple Lie groups are not uniform-to-local stable, using certain K-theory properties reminiscent of bounded cohomology.

I will briefly introduce the Bloch-Kato conjecture, a very general conjecture relating special values of L-functions to arithmetic, and explain how it generalises many more familiar theorems and conjectures such as the BSD conjecture for elliptic curves. I will then introduce the concept of an "Euler system", which is a powerful tool in proving cases of these conjectures, and survey some recent constructions of Euler systems using the geometry of Shimura varieties.

The path signature is a powerful tool for solving regression problems on path space, i.e., for computing conditional expectations $\mathbb{E}[Y | X]$ when the random variable $X$ is a stochastic process -- or a time-series. We provide new theoretical convergence guarantees for two different, complementary approaches to regression using signature methods. In the context of global regression, we show that linear functionals of the robust signature are universal in the $L^p$ sense in a wide class of examples. In addition, we present a local regression method based on signature semi-metrics, and show universality as well as rates of convergence. 

Based on joint works with Davit Gogolashvili, Luca Pelizzari, and John Schoenmakers.

Please note: The MCF seminar usually takes place on Thursdays from 16:00 to 17:00 in L5. However, for this week, the timing will be changed to 15:00 to 16:00.

The machinery of Euler systems (originating in the work of Kolyvagin and Thaine in the late 1980s) is an extremely powerful tool for studying the cohomology of Galois representations, and hence for attacking big conjectures such as Birch–Swinnerton-Dyer. However, current approaches to this theory require the Galois representation to satisfy some sort of "ordinarity" condition, which is a serious restriction in applications. I will discuss recent joint work with Sarah Zerbes in which we extend the Euler system machine to cover situations where this ordinary condition doesn't hold, using a surprising new ingredient (adapted from earlier work of Naomi Sweeting): non-principal ultrafilters, which serve to keep track of the sequences of auxiliary primes arising in Kolyvagin's argument. Applications of this theory, including new cases of the Iwasawa main conjecture, will be discussed in Sarah's talk later the same afternoon.

Renormalization group (RG) flow is a central aspect of our modern understanding of QFT. We may wonder about the relationship of renormalization to some of the other properties of a QFT, and if we can reconstruct RG flow from these properties. It has recently been proposed by Chavda, McLoughlin, Mizera and Staunton in [2510.25822] and [2511.10613] that unitarity can give us at least a part of RG flow, which is known as the Unitarity Flow Conjecture. In this talk, I will summarize the central ideas of this conjecture, and provide some evidence for it.

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.

I will present a parametric finite element formulation for structure-preserving numerical methods. The approach introduces two scalar Lagrange multipliers and evolution equations for surface energy and volume, ensuring that the resulting schemes maintain the underlying geometric and physical structures. To illustrate the method, I will discuss two applications: surface diffusion and two-phase Stokes flow. By combining piecewise linear finite elements in space with structure-preserving second-order time discretizations, we obtain fully discrete schemes of high temporal accuracy. Numerical experiments confirm that the proposed methods achieve the expected accuracy while preserving surface energy and volume.

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.

The theory of characters for infinite groups, initiated by Thoma, is a natural generalization of the representation theory of finite groups. More precisely, a character on a discrete group is a normalised positive definite function which is conjugation invariant and extremal. Connes conjectured a rigidity result for characters of an important family of discrete groups, namely, irreducible lattices in higher-rank semisimple Lie groups. The conjecture states that every character is either the trace of a finite-dimensional representation, or vanishes off the center. This rigidity property implies the Stuck-Zimmer conjecture for such lattices, namely, ergodic actions are either essentially transitive or essentially free. I will present a recent joint result with Michael Glasner, Yuval Gorfine, Liam Hanany and Arie Levit in which we prove that non-uniform irreducible lattices in higher-rank semisimple groups are character rigid. As a result, we also obtain a resolution of the Stuck-Zimmer conjecture for all non-uniform lattices.

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. 

The superconformal index is one of the most powerful tools at the disposal of a supersymmetric field theorist. It counts protected states, is an RG flow invariant, and can be used to test for UV duality. Furthermore, it can be used to detect symmetry enhancements in the IR that are usually inaccessible by use of standard compactification or quiver techniques. The goal of this talk is to provide a practical introduction to computing indices. We will start with the supersymmetric harmonic oscillator to get some intuition, before building up a toolkit to compute indices for your favourite 4d N=1 SCFTs. Time permitting, we will discuss indices with N=2 supersymmetry.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Matrix-product state (MPS) skeletons are connected networks of local one-dimensional quantum lattice models with ground states admitting an MPS representation with finite bond dimension. In this talk, I will discuss how such skeletons underlie certain families of models obeying the Onsager algebra, and how these simple ground states provide a route to explicitly computing correlation functions.