Past

In this talk, I will introduce a dynamic model of a banking network in which the value of interbank obligations is continuously adjusted to reflect counterparty default risk. An interesting feature of the model is that the credit value adjustments increase volatility during downturns, leading to endogenous distress contagion. The counterparty default risk can be computed backwards in time from the obligations' maturity date, leading to a specification of the model in terms of a forward-backward stochastic differential equation (FBSDE), coupled through the banks' default times. The singular nature of this coupling, makes a probabilistic analysis of the FBSDE challenging. So, instead, we derive a characterisation of the default probabilities through a cascade of partial differential equations (PDE). Each PDE represents a configuration with a different number of defaulted banks and has a free boundary that coincides with the banks' default thresholds. We establish classical well-posedness of this PDE cascade, from which we derive existence and uniqueness of the FBSDE.

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]