Past

When tensor products of N minimal models accumulate at central charge N, they also admit relevant operators arbitrarily close to marginality. This raises the tantalizing possibility that they can be use to reach purely Virasoro symmetric CFTs where the breaking of extended chiral symmetry can be seen in a controlled way. This talk will give an overview of the theories where this appears to be the case, according to a brute force check at low lying spins. We will also encounter an interesting non-example where the same type of analysis can be used to give a simpler proof of integrability.

In 2005, Faltings initiated a p-adic analogue of the complex Simpson correspondence, a theory that has since been explored by various authors through different approaches. In this two-lecture series (part I in the Algebra Seminar and part II in the Arithmetic Geometry Seminar), I will present a joint work in progress with Michel Gros and Takeshi Tsuji, motivated by the goal of comparing the parallel approaches we have developed and establishing a robust framework to achieve broader functoriality results for the p-adic Simpson correspondence.

The approach I developed with M. Gros relies on the choice of a first-order deformation and involves a torsor of deformations along with its associated Higgs-Tate algebra, ultimately leading to Higgs bundles. In contrast, T. Tsuji's approach is intrinsic, relying on Higgs envelopes and producing Higgs crystals. The evaluations of a Higgs crystal on different deformations differ by a twist involving a line bundle on the spectral variety.  A similar and essentially equivalent twisting phenomenon occurs in the first approach when considering the functoriality of the p-adic Simpson correspondence by pullback by a morphism that may not lift to the chosen deformations.
We introduce a novel approach to twisting Higgs modules using Higgs-Tate algebras, similar to the first approach of the p-adic Simpson correspondence. In fact, the latter can itself be reformulated as a twist. Our theory provides new twisted higher direct images of Higgs modules, that we apply to study the functoriality of the p-adic Simpson correspondence by higher direct images with respect to a proper morphism that may not lift to the chosen deformations. Along the way, we clarify the relation between our twisting and another twisting construction using line bundles on the spectral variety that appeared recently in other works.

The Riemann problem is an IVP having simple piecewise constant initial data that is invariant under scaling. In 1D, the problem was originally considered by Riemann during the 19th century in the context of gas dynamics, and the general theory was more or less completed by Lax and Glimm in the mid-20th century. In 2D and MD, the situation is much more complicated, and very few analytic results are available. We discuss a shock reflection problem for the Euler equations for potential flow, with initial data that generates four interacting shockwaves. After reformulating the problem as a free boundary problem for a nonlinear PDE of mixed hyperbolic-elliptic type, the problem is solved via a sophisticated iteration procedure. The talk is based on joint work with G-Q Chen (Oxford) et. al. arXiv:2305.15224, to appear in JEMS (2025).

The Keating-Snaith conjecture for quadratic twists of elliptic curves predicts the central values should have a log-normal distribution. I present recent progress towards establishing this in the range of large deviations of order of the variance. This extends Selberg’s Central Limit Theorem from ranges of order of the standard deviation to ranges of order of the variance in a variety of contexts, inspired by random walk theory. It is inspired by recent work on large deviations of the zeta function and central values of L-functions.
 

Kasparov's bivariant K-theory (or KK-theory) is an extremely powerful invariant for both C*-algebras and C*-dynamical systems, which was originally motivated for a tool to solve classical problems coming from topology and geometry. Its paramount importance for classification theory was discovered soon after, impressively demonstrated within the Kirchberg-Phillips theorem to classify simple nuclear and purely infinite C*-algebras. Since then, it can be said that every methodological novelty about extracting information from KK-theory brought along some new breakthrough in classification theory. Perhaps the most important example of this is the Lin-Dadarlat-Eilers stable uniqueness theorem, which forms the technical basis behind many of the most important articles written over the past decade. In the recent landmark paper of Carrion et al, it was demonstrated how the stable uniqueness theorem can be upgraded to a uniqueness theorem of sorts under extra assumptions. It was then posed as an open problem whether the statement of a desired "KK-uniqueness theorem" always holds.

In this talk I want to present the affirmative answer to this question: If A and B are separable C*-algebras and (f,g) is a Cuntz pair of absorbing representations whose induced class in KK(A,B) vanishes, then f and g are strongly asymptotically unitarily equivalent. The talk shall focus on the main conceptual ideas towards this theorem, and I plan to discuss variants of the theorem if time permits. It turns out that the analogous KK-uniqueness theorem is true in a much more general context, which covers equivariant and/or ideal-related and/or nuclear KK-theory.

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

ALF gravitational instantons, of which the Taub-NUT and Atiyah-Hitchin metrics are prototypes, are the complete non-compact hyperkähler 4-manifolds with cubic volume growth. Examples have been known since the 1970's, but a complete classification was only given around 10 years ago. In this talk, I will present joint work with Haskins and Nordström where we extend some of these results to complete non-compact 7-manifolds with holonomy G2 and an asymptotic geometry, called ALC (asymptotically locally conical), that generalises to higher dimension the asymptotic geometry of ALF spaces.

Inverse problems involve reconstructing unknown physical quantities from indirect measurements. They appear in various fields, including medical imaging (e.g., MRI, Ultrasound, CT), material sciences and molecular biology (e.g., electron microscopy), as well as remote sensing just to name a few examples. While deep neural networks are currently able to achieve state-of-the-art performance in many imaging tasks, in this talk we argue that  many inverse imaging problems cannot be solved convincingly using a black-box solution. Instead, they require a well-crafted combination of computational tools taking the underlying signal, the physical constraints and acquisition characteristics into account.

In the first part of the talk, we introduce INDigo+, a novel INN-guided probabilistic diffusion algorithm for arbitrary image restoration tasks. INDigo+ combines the perfect reconstruction property of invertible neural networks (INNs) with the strong generative capabilities of pre-trained diffusion models. Specifically, we leverage the invertibility of the network to condition the diffusion process and in this way we generate high quality restored images consistent with the measurements.

In the second part of the talk, we discuss the unfolding techniques which is an approach that allows embedding priors and models in the neural network architecture. In this context we discuss the problem of monitoring the dynamics of large populations of neurons over a large area of the brain. Light-field microscopy (LFM), a type of scanless microscopy, is a particularly attractive candidate for high-speed three-dimensional (3D) imaging which is needed for monitoring neural activity. We review fundamental aspects of LFM and then present computational methods based on deep learning for neuron localization and activity estimation from light-field data.
Finally, we look at the multi-modal case and present an application in art investigation. Often X-ray images of Old Master paintings contain information of the visible painting and of concealed sub-surface design, we therefore introduce a model-based neural network capable of separating from the “mixed X-ray”  the X-ray image of the visible painting and the X-ray of the concealed design.

This is joint work with  A. Foust, P. Song, C. Howe, H. Verinaz, J. Huang, Di You and Y. Su from Imperial College London, M. Rodrigues and W. Pu from University College London, I. Daubechies from Duke University, Barak Sober from the Hebrew University of Jerusalem and C. Higgitt and N. Daly from The National Gallery in London.

Black holes play a central role in our understanding of quantum gravity, but identifying their precise counterparts in a dual QFT remains a tricky business. These states are heavy, chaotic, and encode various universal aspects — but are also notoriously hard to characterise. In this talk, we’ll explore how supersymmetric field theories provide a controlled setting to study black hole states. In particular, we’ll introduce the idea of fortuitous states as a useful criterion for identifying BPS black hole states. We’ll then illustrate this concept with concrete examples, including the (supersymmetric) SYK model and the D1-D5 CFT.

The discussion will be based on the following recent papers:
arXiv:2402.10129, arXiv:2412.06902, and arXiv:2501.05448.

Exploring fascinating mathematics more independently by doing a Part B project or dissertation can be one of the most exciting and rewarding parts of undergraduate study. Supervision meetings are one of the main tools for making the most of this experience.

In this Fridays@2 session, a panel of staff and students with experience in Part B projects and dissertations will share practical tips on how to prepare, communicate effectively, and tackle common challenges. Whether you’re currently working on a project, planning one, or just curious, join us for insights and an interactive Q&A.

In recent years, a novel approach to studying integrable models has emerged which leverages a higher-dimensional gauge theory, specifically a holomorphic-topological theory. This new framework provides alternative methods for investigating quantum aspects of integrability and for constructing integrable models in more than two dimensions. This talk will review the foundations of this approach, its applications, and the exciting possibilities it opens up for future research in the field of integrable systems. 

We begin with the Fontaine--Wintenberger isomorphism, which gives an example of an extension of Q_p and of F_p((t)) with isomorphic absolute Galois groups. We explain how by trying to lift maps on mod p reductions one encounters Witt vectors. Next, by trying to apply the theory of Witt vectors to the two extensions, we encounter the idea of tilting. Perfectoid fields are then defined more-or-less so that tilting may be reversed. We indicate the proof of the tilting correspondence for perfectoid fields following the Witt vectors approach, classifying the untilts of a given characteristic p perfectoid field along the way. To end, we touch upon the Fargues--Fontaine curve and the geometrization of l-adic local Langlands as motivation for globalizing the tilting correspondence to perfectoid spaces.

In this talk I will give a number of short vignettes of work that has been undertaken in my group over the last 15 years. Mathematically, the theme that underlies our work is the importance of randomness to biological systems. I will explore a number of systems for which randomness plays a critical role. Models of these systems which ignore this important feature do a poor job of replicating the known biology, which in turn limits their predictive power. The underlying biological theme of the majority our work is development, but the tools and techniques we have built can be applied to multiple biological systems and indeed further afield. Topics will be drawn from, locust migration, zebrafish pigment pattern formation, mammalian cell migratory defects, appropriate cell cycle modelling and more. I won't delve to deeply into anyone area, but am happy to take question or to expand upon of the areas I touch on.

We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads -- these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals.

We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur.  These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group.  Joint work in progress with Tom Fisher.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint for learning time-dependent PDEs. 

In this talk, we will first explore connections with low-rank matrix recovery problems in numerical linear algebra. Then, we will show that one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis without querying the adjoint.

I will introduce basic concepts from category theory that are relevant to the study of generalized symmetries. Then, I will focus on constructions known as equivariantization and de-equivariantization, which allow one to move between categories with a group G-action and those with a Rep(G)-action. I will also discuss their relation to the concept of gauging, if time permits.

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.

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the p-enstrophies, p > 2. The talk is based on a recent publication with Christian Seis and Emil Wiedemann.

An adaptive regularization algorithm for unconstrained nonconvex optimization is presented in
which the objective function is never evaluated, but only derivatives are used and without prior knowledge of Lipschitz constant.  This algorithm belongs to the class of adaptive regularization methods, for which optimal worst-case complexity results are known for the standard framework where the objective function is evaluated. It is shown in this paper that these excellent complexity bounds are also valid for the new algorithm. Theoretical analysis of both exact and stochastic cases are discussed and  new probabilistic conditions on tensor derivatives are proposed.  Initial experiments on large binary classification highlight the merits of our method.

Biological matter has the fascinating ability to autonomously generate material deformations via intrinsic active forces, where the latter are often present within effectively two-dimensional structures. The dynamics of such “active surfaces” inevitably entails a complex, self-organized interplay between geometry of a surface and its mechanical interactions with the surrounding. The impact of these factors on the self-organization capacity of surfaces made of an active material, and how related effects are exploited in biological systems, is largely unknown.

In this talk, I will first discuss general numerical challenges in analysing self-organising active surfaces and the bifurcation structure of emergent shape spaces. I will then focus on active surfaces with broken up-down symmetry, of which the eukaryotic cell cortex and epithelial tissues are highly abundant biological examples. In such surfaces, a natural interplay arises between active stresses and surface curvature. We demonstrate that this interplay leads to a comprehensive library of spontaneous shape transformations that resemble stereotypical morphogenetic processes. These include cell-division-like invaginations and the autonomous formation of tubular surfaces of arbitrary length, both of which robustly overcome well-known shape instabilities that would arise in analogue passive systems.

Congruence Subgroup Property is a characterisation of finite-index subgroups of automorphism groups. It first arose from the study of subgroups of linear groups. In this talk, I will show a few examples where it holds and where it fails, and give an overview of what is known about the family $SL_n\mathbb{Z}$, $Out(F_n)$, $MCG(\Sigma)$. Then I will describe some related results in the case of Mapping Class Groups, and explain their relation to profinite rigidity of 3-manifolds.