Past

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 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. 

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 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.

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.

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.

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.

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.

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.

For over 150 years, the study of phase transitions—such as water freezing into ice or magnets losing their magnetism—has been a cornerstone of statistical physics. In this talk, we explore the critical behavior of two-dimensional percolation models, which use random graphs to model the behavior of porous media. At the critical point, remarkable symmetries and emergent properties arise, providing precise insights into the nature of these systems and enriching our understanding of phase transitions. The presentation is designed to be accessible and does not assume any prior background in percolation theory.

About the Speaker: 

Hugo Duminil-Copin is is a French mathematician recognised for his groundbreaking work in probability theory and mathematical physics. He was appointed full professor at the University of Geneva in 2014 and since 2016 has also been a permanent professor at the Institut des Hautes Études Scientifiques (IHES) in France. In 2022 he was awarded the Fields Medal, the highest distinction in mathematics.

In this talk, we discuss the condition for the marginal distribution of the solution to singular SPDEs on the d-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearized equation. As applications of our result, we see the singularity of the Phi^4_3-measure with respect to the Gaussian free field measure and the border of parameters for the fractional Phi^4-measure to be singular with respect to the base Gaussian measure. This talk is based on a joint work with Martin Hairer and Seiichiro Kusuoka.

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. 
In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. 
 

Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net. 

In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). If time permits, we will briefly outline ongoing work on bicommutant category modules (which are the structures assigned by the Segal Chiral CFT at the level of 1-manifolds), hinting towards a categorified analogue of Connes fusion of von Neumann algebra modules.

(Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces)

A trace on a group is a positive-definite conjugation-invariant function on it. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.

Valulations (of polytopes or matroids) are very useful and very mysterious. After taking some time to explain this concept, I will categorify it, with the aim of making it both more useful and less mysterious.

"The question of whether an impurity can be screened by bulk degrees of freedom is central to the study of defects and to (variations of) the Kondo problem. In this talk I discuss how symmetry, generalized or not, can give serious constraints on the possible scenarios at long distances. These can be quantified in the UV where the defect is weakly coupled. I will give some examples of interesting symmetric defect RG flows in (1+1) and (2+1)d.

Based on https://arxiv.org/pdf/2412.18652 and work in progress."

Minimal discrete energy problems arise in a variety of scientific contexts – such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few.     Our goal is to analyze the structure of configurations generated by optimal (and near optimal)-point configurations that minimize the Riesz s-energy over a sphere in Euclidean space R^d and, more generally, over a bounded manifold. The Riesz s-energy potential, which is a generalization of the Coulomb potential, is simply given by 1/r^s, where r denotes the distance between pairs of points. We show how such potentials for s>d and their minimizing point configurations are ideal for use in sampling surfaces.

Connections to the results by Field's medalist M. Viazovska and her collaborators on best-packing and universal optimality in 8 and 24 dimensions will be discussed. Finally we analyze the minimization of a "k-nearest neighbor" truncated version of Riesz energy that reduces the order N^2 computation for energy minimization to order N log N , while preserving global and local properties.

The security of many widely used communication systems hinges on the presumed difficulty of factoring integers or computing discrete logarithms. However, Shor's celebrated algorithm from 1994 demonstrated that quantum computers can perform these tasks in polynomial time. In 2023, Regev proposed an even faster quantum algorithm for factoring integers. Unfortunately, the correctness of his new method is conditional on an ad hoc number-theoretic conjecture. Using tools from analytic number theory, we establish a result in the direction of Regev's conjecture. This enables us to design a provably correct quantum algorithm for factoring and solving the discrete logarithm problem, whose efficiency is comparable to Regev's approach. In this talk, we will give an accessible account of these developments.

 While extremal black hole microstates are reproduced by index calculations, the study of near-BPS black holes requires special care to account for quantum fluctuations. A semiclassical analysis indicates that the spectrum of such black holes has a large extremal degeneracy followed by a mass gap up to a continuum of non-BPS states. The inclusion of a theta angle term alters the properties of the spectrum (Witten effect shifting the mass gap and mixed 't Hooft anomaly). This journal club will study two papers by Toldo and Heydeman, [2412.03695] and [2412.03697] where they study 4d near-BPS black holes. As we shall see, a key point of their derivation is the reduction to 2d JT gravity. The dual CFTs are ABJM and some class R (non lagrangian) theories. Since these theories are strongly coupled, the gravity analysis offers a powerful tool to describe their specturm at finite temperature.