Past

We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics.  This is a joint work with Shreyasi Datta.

In this talk I will review some recent results concerning the qualitative behaviour of symplectic integrators applied to Hamiltonian PDEs, such as the nonlinear wave equation or Schrödinger equations. 

Additionally, I will discuss the problem of numerical resonances, the existence of modified energy and the existence and stability of numerical solitons over long times. 

These are works with B. Grébert, D. Bambusi, G. Maierhofer and K. Schratz. 

In string theory, elementary particles correspond to the various oscillation modes of fundamental strings, whose dynamics in spacetime is described by a two-dimensional conformal field theory on the worldsheet of the propagating strings. While the theory enjoys several desirable features - UV-finiteness, the presence of the graviton in the closed-string spectrum, a pathway to unification - several aspects remain elusive or unsatisfactory, including the on-shell and perturbative nature of string scattering amplitudes and the presence of infrared divergences. String field theory - the formulation of string theory as a quantum field theory - provides a unique and complete framework for describing string dynamics, allowing for example to compute off-shell amplitudes and non-perturbative contributions, to regulate infrared divergences and to approach background independence. This talk will be concerned with the construction of string field theories. Following a brief review of string theory, I will introduce the string fields, and discuss the construction of a string field action and the associated Feynman diagrams. Finally, I will mention some applications before concluding.

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.

This talk is about a classical problem in differential geometry and global analysis: the isometric immersions of Riemannian manifolds into Euclidean spaces. We focus on the PDE approach to isometric immersions, i.e., the analysis of Gauss--Codazzi--Ricci equations, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.

The Ciarlet definition of a finite element has been core to our understanding of the finite element method since its inception. It has proved particularly useful in structuring the implementation of finite element software. However, the definition does not encapsulate all the details required to uniquely implement an element, meaning each user of the definition (whether a researcher or software package) must make further mathematical assumptions to produce a working system. 

The talk presents a new definition built on Ciarlet’s that addresses these concerns. The novel definition forms the core of a new piece of software in development, FUSE, which allows the users to consider the choice of finite element as part of the data they are working with. This is a new implementation strategy among finite element software packages, and we will discuss some potential benefits of the development.

Determining stationary compact configurations of vorticity described by the 2D Euler equations is a classic problem dating back to the late 19th century. The aim is to find equilibrium distributions of vorticity, in the form  of point vortices, vortex sheets, vortex patches, and hollow vortices. This endeavour has driven the development of mathematical and numerical techniques such as Hamiltonian vortex dynamics and contour dynamics.

In the case of vortex sheets, methods and results are presented for finding rotating equilibria, some in the presence of point vortices. To begin, a numerical approach based on that recently developed by Trefethen, Costa, Baddoo, and others for solving Laplace's equation in the complex plane by series and rational approximation is described. The method successfully reproduces the exact vortex sheet solutions found by O'Neil (2018) and Protas & Sakajo (2020). Some new solutions are found.

The numerical approach suggests an analytical method based on conformal mapping for finding exact closed-form vortex sheet equilibria. Examples are presented.

Finally, new numerical solutions are computed for steady, doubly-connected vortex layers of uniform vorticity surrounding a solid object such that the fluid velocity vanishes on the outer free boundary. While dynamically unrelated, these solutions have mathematical analogy and application to the industrial free boundary problem arising in the dip-coating of objects by a viscous fluid.

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

In this talk, we consider Dirichlet forms related to stochastic quantisation for the exp(φ)_{2}-model on the torus. We show strong uniqueness of the corresponding Dirichlet operators by applying an idea of (singular) SPDEs. This talk is based on ongoing joint work with Hirotatsu Nagoji (Kyoto University).

We consider two approximation schemes for the construction of a class of non-Gaussian multiplicative chaos, and show that they give rise to the same limit in the entire subcritical regime. Our approach uses a modified second moment method with the help of a new coupling argument, and does not rely on any Gaussian approximation or thick point analysis. As an application, we extend the martingale central limit theorem for partial sums of random multiplicative functions to L^1 twists. This is a joint work with Ofir Gorodetsky.

Rota’s Alternierende Verfahren theorem in classical probability theory, which examines the convergence of iterates of measure preserving Markov operators, relies on a dilation technique. In the noncommutative setting of von Neumann algebras, this idea leads to the notion of absolute dilation.  

In this talk, we explore when a Fourier multiplier on a group von Neumann algebra is absolutely dilatable. We discuss conditions that guarantee absolute dilatability and present an explicit counterexample—a Fourier multiplier that does not satisfy this property. This talk is based on a joint work with Christian Le Merdy.

Profinite rigidity explores the extent to which non-isomorphic groups can be distinguished by their finite quotients. Many interesting examples of this phenomenon arise in the context of group extensions—short exact sequences of groups with a fixed kernel and quotient. This talk will outline two main mechanisms that govern profinite rigidity in this setting and provide concrete examples of families of extensions that cannot be distinguished by their finite quotients.

The talk is based on my DPhil thesis.

The class of $t$-perfect graphs consists of graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. These were first studied by Chvátal in 1975, motivated by the related and well-studied class of perfect graphs. While perfect graphs are easy to colour, the same is not true for $t$-perfect graphs; numerous questions and conjectures have been posed, and even the most basic, on whether there exists some $k$ such that every $t$-perfect graph is $k$-colourable, has remained open since 1994. I will talk about joint work with Maria Chudnovsky, Linda Cook, James Davies, and Sang-il Oum in which we establish the first finite bound and show that a little less than 200 000 colours suffice.

The discrete Hodge Laplacian offers a way to extract network topology and geometry from higher-ordered networks. The operator is inspired by concepts from algebraic topology and differential geometry and generalises the graph Laplacian. In particular, it allows to relate global structure of networks to the local properties of nodes. In my talk, I will talk about some general behaviour of the Hodge Laplacian and then continue to show how to use the extracted information to a) to use trajectory data infer the topology of the underlying network while simultaneously classifying the trajectories and b) to extract cell differentiation trees from single-cell data, an exciting new application in computational genomics.    

The Gelfand--Kirillov dimension is a classical invariant that measures the size of smooth representations of p-adic groups. It acquired particular relevance in the mod p Langlands program because of the work of  Breuil--Herzig--Hu--Morra--Schraen, who computed it for the mod p cohomology of GL_2 over totally real fields, and used it to prove several structural properties of the cohomology. In this talk, we will present a simplified proof of this result, which has the added benefit of working unchanged for nonsplit inner forms of GL_2. This is joint work with Bao V. Le Hung.

Come chill out after a busy term and play some board games with us. We'll provide some games but feel free to bring your own!

I will describe a non-commutative version of the Zariski topology and explain how to use it to produce a functorial spectrum for all derived rings. If time permits I will give some examples and show how a weak form of Gelfand duality for non-commutative rings can be deduced from this. This work is in collaboration with Simone Murro and Matteo Capoferri.

In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from the microscopic Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains.  This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the global existence and stability of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibrium solutions. This is a joint work with Pierre Germain (Imperial College London) and Joonhyun La (KIAS).

A base-g Niven number is an integer divisible by the sum of its digits in base-g. We show that any sufficiently large integer can be written as the sum of three base-3 Niven numbers, and comment on the extension to other bases. This is an application of the circle method, which we use to count the number of ways an integer can be written as the sum of three integers with fixed, near-average, digit sum. 

When and how well can a high-dimensional system of stochastic differential equations (SDEs) be approximated by one with independent coordinates? This fundamental question is at the heart of the theory of mean field limits and the propagation of chaos phenomenon, which arise in the study of large (many-body) systems of interacting particles. This talk will present recent sharp quantitative answers to this question, both for classical mean field models and for more recently studied non-exchangeable models. Two high-level ideas underlie these answers. The first is a simple non-asymptotic construction, called the independent projection, which is a natural way to approximate a general SDE system by one with independent coordinates. The second is a "local" perspective, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit a surprising connection with first-passage percolation.

Namikawa has shown that the functor of flat graded Poisson deformations of a conic symplectic singularity is unobstructed and pro-representable. In a subsequent work, Losev showed that the universal Poisson deformation admits, a quantization which enjoys a rather remarkable universal property. In a recent work, we have repackaged the latter theorem as an expression of the representability of a new functor: the functor of quantizations. I will describe how this theorem leads to an easy proof of the existence of a universal equivariant quantizations, and outline a work in progress in which we describe a presentation of a rather complicated quantum Hamiltonian reduction: the finite W-algebra associated to a nilpotent element in a classical Lie algebra. The latter result hinges on new presentations of twisted Yangians.

Recently, work has been done to understand higher-form symmetries in linear gravity. Just like Maxwell theory, which has both electric and magnetic U(1) higher form symmetries, linearised gravity exhibits analogous structure. The authors of
[https://arxiv.org/pdf/2409.00178] investigate electric and magnetic higher form symmetries in linearised gravity, which correspond to shift symmetries of the graviton and the dual graviton respectively. By attempting to gauge the two symmetries, the authors investigate the mixed ’t Hooft anomalies anomaly structure of linearised gravity. Furthermore, if a specific shift symmetry is considered, the corresponding charges are related to Roger Penrose's quasi-local charge construction.

Based on: [https://arxiv.org/pdf/2410.08720][https://arxiv.org/pdf/2409.00178][https://arxiv.org/pdf/2401.17361]

How do mobile organisms situate themselves in space?  This is a fundamental question in both ecology and cell biology but, since space use is an emergent feature of movement processes operating on small spatio-temporal scales, it requires a mathematical approach to answer.  In recent years, increasing empirical research has shown that non-locality is a key aspect of movement processes, whilst mathematical models have demonstrated its importance for understanding emergent space use patterns.  In this talk, I will describe a broad class of models for modelling the space use of interacting populations, whereby directed movement is in the form of non-local advection.  I will detail various methods for ascertaining pattern formation properties of these models, fundamental for answering the question of how organisms situate themselves in space, and describe some of the rich variety of patterns that emerge. I will also explain how to connect these models to data on animal and cellular movement.

Riemannian manifolds with holonomy G_2 form an exceptional class of Ricci-flat manifolds occurring only in dimension 7. In the compact setting, their moduli spaces are known to be smooth (unobstructed), finite-dimensional, and to carry a natural Riemannian structure induced by the L^2-metric; but besides this very little is known about the global properties of G_2-moduli spaces. In this talk, I will review the basics of G_2-geometry and present new results concerning the distance theory and the geometry of the moduli spaces.
 

We resolve Manin's conjecture for all Châtelet surfaces over Q
(surfaces given by equations of the form x^2 + ay^2 = f(z)) -- in other
words, we establish asymptotics for the number of rational points of
increasing height. The key analytic ingredient is estimating sums of
Fourier coefficients of modular forms along polynomial values.

Can one recover a matrix from only matrix-vector products? If so, how many are needed? We will consider the matrix recovery problem for the class of hierarchical rank-structured matrices. This problem arises in scientific machine learning, where one wishes to recover the solution operator of a PDE from only input-output pairs of forcing terms and solutions. Peeling algorithms are the canonical method for recovering a hierarchical matrix from matrix-vector products, however their recursive nature poses a potential stability issue which may deteriorate the overall quality of the approximation. Our work resolves the open question of the stability of peeling. We introduce a robust version of peeling and prove that it achieves low error with respect to the best possible hierarchical approximation to any matrix, allowing us to analyze the performance of the algorithm on general matrices, as opposed to exactly hierarchical ones. This analysis relies on theory for low-rank approximation, as well as the surprising result that the Generalized Nystrom method is more accurate than the randomized SVD algorithm in this setting. 

Placing M-theory on a non-compact Calabi-Yau threefold allows us to construct low energy field theories in 5d with minimal supersymmetry, in a limit in which gravity is decoupled.  We venture into this topic by introducing all the building blocks we hope to capture in a 5d SCFT. Next, from the geometric perspective we realise the 5d gauge theory data from the objects within the Calabi-Yau geometry, given by curves, divisors, rulings, and singularities. After seeing how the geometry captures all the possible field theory data, we illustrate how to build some simple 5d SCFTs by placing M-theory on toric Calabi-Yau threefolds.

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 a new ensemble framework for boosting the performance of overparameterized unfolding networks solving the compressed sensing problem. We combine a state-of-the-art overparameterized unfolding network with a continuation technique, to warm-start a crucial quantity of the said network's architecture; we coin the resulting continued network C-DEC. Moreover, for training and evaluating C-DEC, we incorporate the log-cosh loss function, which enjoys both linear and quadratic behavior. Finally, we numerically assess C-DEC's performance on real-world images. Results showcase that the combination of continuation with the overparameterized unfolded architecture, trained and evaluated with the chosen loss function, yields smoother loss landscapes and improved reconstruction and generalization performance of C-DEC, consistently for all datasets.

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form \si(𝑥)=𝑔𝑥\si(x)=gx, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

The BNSR Invariant is a classical geometric invariant that encodes the finite generation of all coabelian subgroups of a given finitely generated group. The aim of this talk is to present a conjecture about the structure of the BNSR invariant of an Artin group and to present a new family in which the conjecture is true in terms of graph colorings.

In this talk, I will present the generalization of scaling limit results for stochastic transport equations on torus by Flandoli, Galeati and Luo, to compact manifolds. We consider the stochastic transport equations driven by colored space-time noise(smooth in space, white in time) on a compact Riemannian manifold without boundary. Then we study the scaling limits of stochastic transport equations, tuning the noise in such a way that the space covariance of the noise on the diagonal goes to identity matrix but the covariance operator itself goes to zero, which includes the large scale analysis regime with diffusive scaling.

We obtain different scaling limits depending on the initial data. With space white noise as initial data, the solutions converge in distribution to the solution of a stochastic heat equation with additive noise. With square integrable initial data, the solutions of transport equation converge to the solution of the deterministic heat equation, and we give quantitative estimates on the convergence rate.

We propose a natural version of Connes' Rigidity Conjecture (1982) that involves property (T) groups with infinite centre. Using methods at the rich intersection between von Neumann algebras and geometric group theory, we identify several instances where this conjecture holds. This is joint work with Ionut Chifan, Denis Osin, and Hui Tan.

Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. 

This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek. 

A group is free-by-cyclic if it is an extension of a free group by a cyclic group. Knowing that a group is virtually free-by-cyclic is often quite useful; it implies that the group is coherent and that it is cohomologically good in the sense of Serre. In this talk we will give a homological characterisation of when a finitely generated RFRS group is virtually free-by-cylic and discuss some generalisations.

In recent years there has been a great deal of interest in detecting properties of the fundamental group $\pi_1M$ of a $3$-manifold via its finite quotients, or more conceptually by its profinite completion.

This motivates the study of the profinite completion $\widehat {\pi_1M}$ of the fundamental group of a $3$-manifold. I shall discuss a description of the  finitely generated prosoluble subgroups of the profinite completions of all 3-manifold groups and of related groups of geometric nature.

In this talk I will review recent results on the development of a form factor program for integrable quantum field theories (IQFTs) perturbed by irrelevant operators. It has been known for a long time that under such perturbations integrability is preserved and that the two-body scattering phase gets deformed in a simple manner. The consequences of such a deformation are stark, leading to theories that exhibit a so-called Hagedorn transition and no UV completion. These phenomena manifest physically in several distinct ways. In our work we have mainly asked the question of how the deformation of the S-matrix translates into the correlation functions of the deformed theory. Does the scaling of correlators at long and short distances capture any of the "pathologies" mentioned above? Can our understanding of irrelevant perturbations tell us something about the space of IQFTs and about their form factors? In this talk I will answer these questions in the afirmative, summarising work in collaboration with Stefano Negro, Fabio Sailis and István M. Szécsényi.

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between $H^{-1}$ and $H^{1}$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions. This is a joint work with J. A. Carrillo and J. Warnett. 

The large sieve inequality for Dirichlet characters is a central result in analytic number theory, which encodes a strong orthogonality property between primitive characters of varying conductors. This can be viewed as a statement about $\mathrm{GL}_1$ automorphic representations, and it is a key open problem to prove similar results in the higher $\mathrm{GL}_m$ setting; for $m \ge 2$, our best bounds are far from optimal. We'll outline two approaches to such results (sketching them first in the elementary case of Dirichlet characters), and discuss work-in-progress of Thorner and the author on an improved $\mathrm{GL}_m$ large sieve. No prior knowledge of automorphic representations will be assumed.

Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.

I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure. The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. This is joint work with Partha Ghosh and, in the Spin(7) setting, Ragini Singhal.

This brief pedagogical talk introduces key concepts of Carroll geometries, which arise as the limit of relativistic spacetimes in the vanishing speed of light regime. In this limit, light cones collapse along a timelike direction, resulting in a manifold equipped with a degenerate metric. Consequently, physics in such spacetimes exhibits peculiar properties. Despite this, the Carroll contraction is relevant to a wide range of applications, from flat-space holography to condensed matter physics. To complement this introduction, and depending on the audience’s interests, I can discuss Carroll affine connections, symmetry groups, conservation laws, and Carroll-invariant field theories.