Past

This paper shows that a simple sale contract with a collection of options implements the full-information first-best allocation in a variety of continuous-time dynamic adverse selection settings with news. Our model includes as special cases most models in the literature. The implementation result holds regardless of whether news is public (i.e., contractible) or privately observed by the buyer, and it does not require deep pockets on either side of the market. It is an implication of our implementation result that, irrespective of the assumptions on the game played, no agent waits for news to trade in such models. The options here do not play a hedging role and are, thus, not priced using a no-arbitrage argument. Rather, they are priced using a game-theoretic approach.

Deflation is a technique to remove a solution to a problem so that other solutions to this problem can subsequently be found. The most prominent instance is deflation we see in eigenvalue solvers, but recent interest has been in deflation of rootfinding problems from nonlinear PDEs with many isolated solutions (spearheaded by Farrell and collaborators). 

In this talk I’ll show you recent results on deflation techniques for optimisation algorithms with many local minima, focusing on the Gauss—Newton algorithm for nonlinear least squares problems.  I will demonstrate advantages of these techniques instead of the more obvious approach of applying deflated Newton’s method to the first order optimality conditions and present some proofs that these algorithms will avoid the deflated solutions. Along the way we will see an interesting generalisation of Woodbury’s formula to least squares problems, something that should be more well known in Numerical Linear Algebra (joint work with Güttel, Nakatsukasa and Bloor Riley).

Main preprint: https://arxiv.org/abs/2409.14438. 

WoodburyLS preprint: https://arxiv.org/abs/2406.15120

We consider two problems in fluid dynamics: the collective locomotion of flying animals and the interaction of vortex rings with fluid interfaces. First, we present a model of formation flight, viewing the group as a material whose properties arise from the flow-mediated interactions among its members. This aerodynamic model explains how flapping flyers produce vortex wakes and how they are influenced by the wakes of others. Long in-line arrays show that the group behaves as a soft, excitable "crystal" with regularly ordered member "atoms" whose positioning is susceptible to deformations and dynamical instabilities. Second, we delve into the phenomenon of vortex ring reflections at water-air interfaces. Experimental observations reveal reflections analogous to total internal reflection of a light beam. We present a vortex-pair--vortex-sheet model to simulate this phenomenon, offering insights into the fundamental interactions of vortex rings with free surfaces.

A number of algorithms are now available---including Halko-Martinsson-Tropp, interpolative decomposition, CUR, generalized Nystrom, and QR with column pivoting---for computing a low-rank approximation of matrices. Some methods come with extremely strong guarantees, while others may fail with nonnegligible probability. We present methods for efficiently estimating the error of the approximation for a specific instantiation of the methods. Such certificate allows us to execute "responsibly reckless" algorithms, wherein one tries a fast, but potentially unstable, algorithm, to obtain a potential solution; the quality of the solution is then assessed in a reliable fashion, and remedied if necessary. This is joint work with Gunnar Martinsson. 

Time permitting, I will ramble about other topics in Randomised NLA. 

[This is the second in a series of two talks; the first talk will be in the Algebra Seminar of Tuesday Feb 4th https://www.maths.ox.ac.uk/node/70022]

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.

Originally posed in the 1950s, the Hadwiger-Nelson problem interrogates the ‘chromatic number of the plane’ via an infinite unit-distance graph. This question remains open today, known only to be 5,6, or 7. We may ask the same question of the hyperbolic plane; there the lack of homogeneous dilations leads to unique behaviour for each length scale d. This variance leads to other questions: is the d-chromatic number finite for all d>0? How does the d-chromatic number behave as d increases/decreases? In this talk, I will provide a summary of existing methods and results, before discussing improved bounds through the consideration of semi-regular tilings of the hyperbolic plane.

Given two von Neumann algebras A,B with an action by a locally compact (quantum) group G, one can consider its associated equivariant correspondences, which are usual A-B-correspondences (in the sense of Connes) with a compatible unitary G-representation. We show how the category of such equivariant A-B-correspondences carries an analogue of the Fell topology, which is preserved under natural operations (such as crossed products or equivariant Morita equivalence). If time permits, we will discuss one particular interesting example of such a category of equivariant correspondences, which quantizes the representation category of SL(2,R). This is based on joint works with Joeri De Ro and Joel Dzokou Talla. 

A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi-Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.

How to uniformly, or at least almost uniformly, choose an element from a finite group G? When G is too large to enumerate all its elements, direct (pseudo)random selection is impossible. However, if we have an explicit set of generators of G (e.g., as in the Rubik's cube group), several methods are available. This talk will focus on one such method based on the well-known product replacement algorithm. I will discuss how recent results on property (T) by Kaluba, Kielak, Nowak and Ozawa partially explain the surprisingly good performance of this algorithm.

Real-world networks have a complex topology with many interconnected elements often organized into communities. Identifying these communities helps reveal the system’s organizational and functional structure. However, network data can be noisy, with incomplete link observations, making it difficult to detect significant community structures as missing data weakens the evidence for specific solutions. Recent research shows that flow-based community detection methods can highlight spurious communities in sparse networks with incomplete link observations. To address this issue, these methods require regularization. In this talk, I will show how a Bayesian approach can be used to regularize flows in networks, reducing overfitting in the flow-based community detection method known as the map equation.

The normal covering number $\gamma(G)$ of a finite group $G$ is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of $\gamma(S_n)$ and $\gamma(A_n)$ as $n$ tends to infinity.

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.