Past

TBA

We present Gromov's celebrated reconstruction theorem in Lorentzian geometry and show two applications. First, we introduce several notions of convergence of (isomorphism classes of) normalized bounded Lorentzian metric measure spaces, for which we describe several fundamental properties. Second, we state a version within the spacetime reconstruction problem from quantum gravity. Partly in collaboration with Clemens Sämann (University of Vienna).

Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators.  It is important to  to explore whether other problems in scientific computing, such as ODEs, PDEs, and  linear algebra that arise in both classical and quantum systems which are not unitary evolution,  can be handled by quantum computers.  

We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers non-autonomous ODEs/PDEs systems to autonomous ones, nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs—coined “Schrodingerization”—with uniform evolutions. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing which are easier to realize in the near term. We will also discuss  dimension lifting techniques for quantum simulation of stochastic DEs and PDEs with fractional derivatives. 

For homogeneous, defect-free TQFTs, (1) n+\epsilon-dimensional versions of the theories are relatively easy to construct; (2) an n+\epsilon-dimensional theory can be extended to n+1-dimensional (i.e. the top-dimensional path integral can be defined) if certain more restrictive conditions related to handle cancellation are satisfied; and (3) applying this path integral construction to a handle decomposition of an n+1-manifold yields a state sum description of the path integral.  In this talk, I'll show that the same pattern holds for defect TQFTs.  The adaptation of homogeneous results to the defect setting is mostly straightforward, with the only slight difficulty being the purely topological problem of generalizing handle theory to manifolds with defects.  If time allows, I'll describe two applications: a Verlinde-like dimension formula for the dimension of the ground state of fracton systems, and a generalization, to arbitrary dimension, of Ostrik's theorem relating algebra objects to modules (gapped boundaries).

Multiscale analysis has become a cornerstone of modern signal and image processing. Driven by the objective of representing data in a hierarchical fashion, capturing coarse-to-fine structures and revealing features across scales, multiscale transforms enable powerful techniques for a wide range of applications. In this talk, we will begin with a comprehensive overview of the construction of multiscale transforms via refinement operators, highlighting recent advances in the area. These operators serve as upsampling in the process of multiscaling. Once established, we will describe the adaptation of multiscale transforms to manifolds, and then focus on their extension to Wasserstein spaces. The talk will highlight both theoretical developments and practical implementations, illustrating the potential of multiscale methods in emerging data-driven applications. Lastly, we will explore how classical multiscaling tools such as wavelet transforms can be utilized for autoregressive image generation via large language models. We will show experimental results that indicate promising performance.

Wael Mattar

I will describe recent developments in information geometry (the study of optimal transport and entropy) for the setting of free probability.  One of the main goals of free probability is to model the large-n behavior of several $n \times n$ matrices $(X_1^{(n)},\dots,X_m^{(n)})$ chosen according to a sufficiently nice joint distribution that has a similar formula for each n (for instance, a density of the form constant times $e^{-n^2 \tr_n(p(x))}$ where $p$ is a non-commutative polynomial).  The limiting object is a tuple $(X_1,\dots,X_m)$ of operators from a von Neumann algebra.  We want the entropy and the optimal transportation distance of the probability distributions on $n \times n$ matrix tuples converge in some sense to their free probabilistic analogs, and so to obtain a theory of Wasserstein information geometry for the free setting.  I will present both negative results showing unavoidable difficulties in the free setting, and positive results showing that nonetheless several crucial aspects of information geometry do adapt.

We apply techniques of exponential asymptotics to the KdV equation to derive the small-time behaviour for dispersive waves that propagate in one direction.  The results demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of complex-plane singularities of the initial condition.  Using matched asymptotic expansions, we show how the small-time dynamics of complex singularities of the time-dependent solution are dictated by a Painlevé II problem with decreasing tritronquée solutions.  We relate these dynamics to the solution on the real line.

Mini Course: Topological Phenomena in the Cuntz semigroup 

Mathematical Institute, University of Oxford

10-12 Sept 2025

This short mini course aims to introduce participants to the interplay between algebraic and differential topology and the  Cuntz semigroup of C*-algebras. It will describe the use of the Cuntz semigroup to build C*-algebras outside the scope of the Elliott classification programme.  There will be opportunities for participants to offer contributed talks.

We talk about the global-in-time well-posedness of classical solutions to the vacuum free boundary problem of the 1D viscous Saint-Venant system for laminar shallow water with large data. Since the depth of the fluid vanishes on the moving boundary, the momentum equations become degenerate both in the time evolution and spatial dissipation, which may lead to singularities for the derivatives of the velocity of the fluid and then makes it challenging to study classical solutions. By exploiting the intrinsic degenerate-singular structures of the viscous Saint-Venant system, we are able to identify two classes of admissible initial depth profile and obtain the global well-posedness theory here: the first class of the initial depth profile satisfies the well-known BD entropy condition; the second class of the initial depth profile satisfies the well-known physical vacuum boundary condition, but violates the BD entropy condition. One of the key ingredients of the analysis here is to establish some new degenerate weighted estimates for the effective velocity via its transport properties, which do not require the initial BD entropy condition or the physical vacuum boundary condition. These new estimates enable one to obtain the upper bound for the first order spatial derivative of the flow map. Then the global-in-time regularity uniformly up to the vacuum boundary can be obtained by carrying out a series of singular or degenerate weighted energy estimates carefully designed for this system.

In this talk, we will explore three flow configurations that illustrate the behaviour of slow-moving viscous fluids in confined geometries: viscous gravity currents, fracturing of shear-thinning fluids in a Hele-Shaw cell, and rectangular channel flows of non-Newtonian fluids. We will first develop simple mathematical models to describe each setup, and then we will compare the theoretical predictions from these models with laboratory experiments. As is often the case, we will see that even models that are grounded in solid physical principles often fail to accurately predict the real-world flow behaviour. Our aim is to identify the primary physical mechanisms absent from the model using laboratory experiments. We will then refine the mathematical models and see whether better agreement between theory and experiment can be achieved.

The hot spots conjecture, proposed by Rauch in 1974, asserts that the second Neumann eigenfunction of the Laplacian achieves its global maximum (the hottest point) exclusively on the boundary of the domain. Notably, for triangular domains, the absence of interior critical points was recently established by Judge and Mondal in [Ann. Math., 2022]. Nevertheless, several important questions about the second Neumann eigenfunction in triangles remain open. In this talk, we address issues such as: (1) the uniqueness of non-vertex critical points; (2) the necessary and sufficient conditions for the existence of non-vertex critical points; (3) the precise location of the global extrema; (4) the position of the nodal line; among others. Our results not only confirm both the original theorem and Conjecture 13.6 proposed by Judge and Mondal in [Ann. Math., 2020], but also accomplish a key objective outlined in the Polymath 7 research thread 1 led by Terence Tao. Furthermore, we resolve an eigenvalue inequality conjectured by Siudeja [Proc. Amer. Math. Soc., 2016] concerning the ordering of mixed Dirichlet–Neumann Laplacian eigenvalues for triangles. Our approach employs the continuity method via domain deformation. 

TBA

I will present a pair of off-shell functionals in position space, localized on the self-dual and the anti-self-dual planes which naturally give the Parke-Taylor denominator. These can therefore be used: 
i) to compute scattering amplitudes of particles with different spins and helicities; and 
ii) develop a Lagrangian description. 
Using Witten's half-Fourier transform, I will express these functionals in twistor space and present the kernels in a closed compact form. For even multiplicities, I will show how to obtain this form geometrically which than then be “folded” to get the one-less odd-multiplicity result. 
 

I will review some recent developments in effective field theory of  composite higher-spin particles, namely, Zinoviev's massive gauge symmetry and 
the new chiral-field approach. The latter approach was inspired by a simple spinor-helicity structure first singled out by Arkani-Hamed, Huang and Huang, which encodes the higher-spin information of two massive particles. It turned out to be persistent in tree-level amplitudes with any number of additional identical-helicity gluons or gravitons, leading to the discovery of the chiral-field approach. I will mention the applications of massive higher-spin scattering amplitudes to classical gravitational dynamics of rotating black  holes. 
 

In this talk we review some recent applications of generalised symmetries to scattering amplitudes. We start in 4d by describing the connection between spontaneously broken higher-form symmetries and soft theorems for scattering amplitudes of the associated Nambu-Golstone bosons, and show a new soft theorem for theories with a so-called 2-group symmetry. Then, we switch gears and consider non-invertible symmetries in 2d theories. We show that the standard form of the S-matrix is incompatible with the non-invertible symmetry, and derive new S-matrices satisfying a modified crossing symmetry.

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.

Actions on CAT(0) cube complexes are powerful geometric tool for both algebraically decomposing groups and establishing subgroup separability results.  I will describe boundaries associated to hyperbolic and relatively hyperbolic groups. With a focus on (quotients of) free products, I will discuss variations on a boundary criteria of Bergeron—Wise for exhibiting cocompact actions on CAT(0) cube complexes.  I will explain some ideas on how to use these tools to show that most (small-cancellation or random density) quotients of free products preserve residual finiteness. This is based on multiple joint works with subsets of Einstein, Krishna MS, Montee, and Steenbock.

Directed graph algebras have long been studied as tractable examples of C*-algebras, but they are limited by their inability to have torsion in their K_1 group. Graphs of groups, which are famed in geometric group theory because of their intimate connection with group actions on trees, are a more recent addition to the C*-algebra scene. In this talk, I will introduce the child of these two concepts – directed graphs of groups – and describe how their algebras inherit the best properties of its parents’, with a view to outlining how we can use these algebras to model a class of C*-algebras (stable UCT Kirchberg algebras) which is classified completely by K-theory.

In this talk we will construct a basis of quantum gravity states by cutting the Euclidean path integral. These states are made by inserting heavy dust shell operators on the asymptotic boundary. We will use this basis to resolve two puzzles : 

(1) The two boundary gravity Hilbert space seemingly does not factorise, which is in tension with holography. 

(2) Gibbons and Hawking proposed the gravity thermal partition function is computed by the euclidean path integral with a periodic time boundary condition. Why is does this perform a trace over gravity states?

To resolve these puzzles we will introduce some tricks that simply the evaluation of the gravity path integral in the saddle point approximation. 

In this talk, we present a topological framework for interpreting the latent representations of Multilayer Perceptrons (MLPs) [1] using tools from Topological Data Analysis. Our approach constructs a simplicial tower, a sequence of simplicial complexes linked by simplicial maps, to capture how the topology of data evolves across network layers. This construction is based on the pullback of a cover tower on the output layer and is inspired by the Multiscale Mapper algorithm. The resulting commutative diagram enables a dual analysis: layer persistence, which tracks topological features within individual layers, and MLP persistence, which monitors how these features transform across layers. Through experiments on both synthetic and real-world medical datasets, we demonstrate how this method reveals critical topological transitions, identifies redundant layers, and provides interpretable insights into the internal organization of neural networks.

Let p be a prime. In this talk we look at the bounded derived category of modules over the Rubik’s cube group and show that the faithful action on the corners and edges is a progenerator for the coadmissible subcategory.

The motion of a particle suspended in a fluid flow is governed by hydrodynamic interactions. In this talk, I will present the rich nonlinear dynamics that arise from particle-fluid interactions for two different setups: (i) passive particles in 3D channel flows where fluid inertia is important, and (ii) active particles in 3D channel flows in the Stokes regime (i.e. without fluid inertia).

For setup (i), the particle-fluid interactions result in focusing of particles in the channel cross section, which has been exploited in biomedical microfluidic technologies to separate particles by size. I will offer insights on how dynamical system features of bifurcations and tipping phenomena might be exploited to efficiently separate particles of different sizes. For setup (ii), microswimmers routinely experience unidirectional flows in confined environment such as sperm cells swimming in fallopian tubes, pathogens moving through blood vessels, and microrobots programed for targeted drug delivery applications. I will show that our minimal model of the system exhibits rich nonlinear and chaotic dynamics resulting in a diverse set of active particle trajectories.

Odd systems, characterised by broken time-reversal or parity symmetry, 
exhibit striking transport phenomena due to transverse responses. In this 
talk, I will introduce the concept of odd diffusion, a generalisation of 
diffusion in two-dimensional systems that incorporates antisymmetric tensor 
components. Focusing on systems of interacting particles, I adapt a 
geometric approach to derive effective transport equations and show how 
interactions give rise to unusual transport in odd systems. I present 
effects like enhanced self-diffusion, reversed Hall drift and even absolute 
negative mobility that solely originate in odd diffusion. These results 
reveal how microscopic symmetry-breaking gives rise to emergent, equilibrium 
and non-equilibrium transport, with implications for soft matter, chiral 
active systems, and topological materials.

We introduce a non-probabilistic, path-by-path framework for continuous-time, path-dependent portfolio allocation. Extending the self-financing concept recently introduced in Chiu & Cont (2023), we characterize self-financing portfolio allocation strategies through a path-dependent PDE and provide explicit solutions for the portfolio value in generic markets, including price paths that are not necessarily continuous or exhibit variation of any order.

As an application, we extend an aggregating algorithm of Vovk and the universal algorithm of Cover to continuous-time meta-algorithms that combine multiple strategies into a single strategy, respectively tracking the best individual and the best convex combination of strategies. This work extends Cover’s theorem to continuous-time without probability.

Let $p$ be an odd prime. Let $K/\mathbf{Q}_p$ be a finite unramified extension. Let $\rho: G_K \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ be a continuous representation. We prove that $\rho$ has a crystalline lift of small irregular weight if and only if it has multiple crystalline lifts of certain specified regular weights. The inspiration for this result comes from recent work of Diamond and Sasaki on geometric Serre weight conjectures. We also discuss applications to partial weight one modularity.

Ever wondered how ideas from physics can used in real-world scenarios? Come to this talk to understand what is an option and how they are traded in markets. I will recall some basic notions of stochastic calculus and derive the Black-Scholes (BS) equation for plain vanilla options. The BS equation can be solved using standard path integral techniques, that also allow to price more exotic derivatives. Finally, I will discuss whether the assumptions behind Black-Scholes dynamics are reasonable in real-world markets (spoiler: they're not), volatility smiles and term structures of the implied volatility.

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 will discuss local regularity properties for solutions to non-local equations naturally arising in kinetic theory. We will focus on the Strong Harnack inequality for global solutions to a non-local kinetic equation in divergence form. We will explain the connection to the Boltzmann equation and we will mention a few consequences on the asymptotic behaviour of the solutions.

Building on Leo’s talk last week, I will present the full Galois characterisation of henselianity and introduce some of the ‘explicit’ ingredients he referred to during his presentation. In particular, I will describe a Galois cohomology-inspired criterion for distinguishing between different characteristics. I will then outline the full proof of the Galois characterisation of p-adically closed fields, indicating how each of the ingredients enters the argument.

Is it possible to tell the isomorphism type of an infinite group from its collection of finite quotients? This question, known as profinite rigidity, has deep roots in various areas of mathematics, ranging from arithmetic geometry to group theory. In this talk, I will introduce the question, its history and context. I will explain how profinite rigidity is studied using the machinery of profinite completions, including elementary proofs and counterexamples. Then I will outline some of the key results in the field, ranging from 1970 to the present day. Time permitting, I will elaborate on recent results of myself on the profinite rigidity of certain classes of solvable groups. 

Structural mechanics plays a crucial role in soft matter physics, mechanobiology, metamaterials, pattern formation, active matter, and soft robotics. What unites these seemingly disparate topics is the natural balance that emerges between elasticity, geometry, and stability. This seminar will serve as a high-level overview of our work on several problems concerning the stability of structures. I will cover three topics: (1) shapeshifting shells; (2) mechanical metamaterials; and (3) elastogranular mechanics.

I will begin by discussing our development of a generalized, stimuli-responsive shell theory. (1) Non-mechanical stimuli including heat, swelling, and growth further complicate the nonlinear mechanics of shells, as simultaneously solving multiple field equations to capture multiphysics phenomena requires significant computational expense. We present a general shell theory to account for non-mechanical stimuli, in which the effects of the stimuli are
generalized into three forms: those that add mass to the shell, those that increase the area of the shell through the natural stretch, and those that change the curvature of the shell through the natural curvature. I will show how this model can capture the morphogenesis of the optic cup, the snapping of the Venus flytrap, leaf growth, and the buckling of electrically active polymer plates. (2) I will then discuss how cutting thin sheets and shells, a process
inspired by the art of kirigami, enables the design of functional mechanical metamaterials. We create linear actuators, artificial muscles, soft robotic grippers, and mechanical logic units by systematically cutting and stretching thin sheets. (3) Finally, if time permits, I will introduce our work on the interactions between elastic and granular matter, which we refer to as elastogranular mechanics. Such interactions occur across all lengths, from morphogenesis, to root growth, to stabilizing soil against erosion. We show how combining rocks and string in the absence of any adhesive we can create large, load bearing structures like columns, beams, and arches. I will finish with a general phase diagram for elastogranular behavior.

I will describe a measure of quantum state complexity defined by minimizing the spread of the wavefunction over all choices of basis. We can efficiently compute this measure, which displays universal behavior for diverse chaotic systems including spin chains, the SYK model, and quantum billiards.  In the minimizing basis, the Hamiltonian is tridiagonal, thus representing the dynamics as if they unfold on a one-dimensional chain. The recurrent and hopping matrix elements of this chain comprise the Lanczos coefficients, which I will relate through an integral formula to the density of states. For Random Matrix Theories (RMTs), which are believed to describe the energy level statistics of chaotic systems, I will also derive an integral formula for the covariances of the Lanczos coefficients. These results lead to a conjecture: quantum chaotic systems have Lanczos coefficients whose local means and covariances are described by RMTs. 
 

The hydrodynamic description for emergent behavior of interacting agents is governed by Euler alignment equations, driven by different protocols of pairwise communication kernels. A main question of interest is how short- vs. long-range interactions dictate the large-crowd, long-time dynamics. 

The equations lack closure for the pressure away thermal equilibrium. We identify a distinctive feature of Euler alignment -- a reversed direction of entropy. We discuss the role of a reversed entropy inequality in selecting mono-kinetic closure for emergence of strong solutions, prove the existence of such solutions, and characterize their related invariants which extend the 1-D notion of an “e” quantity.

The hydrodynamic description for emergent behavior of interacting agents is governed by Euler alignment equations, driven by different protocols of pairwise communication kernels. A main question of interest is how short- vs. long-range interactions dictate the large-crowd, long-time dynamics. 

The equations lack closure for the pressure away thermal equilibrium. We identify a distinctive feature of Euler alignment -- a reversed direction of entropy. We discuss the role of a reversed entropy inequality in selecting mono-kinetic closure for emergence of strong solutions, prove the existence of such solutions, and characterize their related invariants which extend the 1-D notion of an “e” quantity.

Roe algebras were introduced in the late 1990's in the study of indices of elliptic operators on (locally compact) Riemannian manifolds. Roe was particularly interested in coarse equivalences of metric spaces, which is a weaker notion than that of quasi-isometry. In fact, soon thereafter it was realized that the isomorphism class of these class of C*-algebras did not depend on the coarse equivalence class of the manifold. The converse, that is, whether this class is a complete invariant, became known as the 'Rigidity Problem for Roe algebras'. In this talk we will discuss an affirmative answer to this question, and how to approach its proof. This is based on joint work with Federico Vigolo.

Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Pierrick Bousseau.

Lagrangian mean curvature flow (LMCF) is a way to deform Lagrangian submanifolds inside a Calabi-Yau manifold according to the negative gradient of the area functional. There are influential conjectures about LMCF due to Thomas-Yau and Joyce, describing the long-time behaviour of the flow, singularity formation, and how one may flow past singularities. In this talk, we will show how to flow past a conically singular Lagrangian by gluing in expanders asymptotic to the cone, generalizing an earlier result by Begley-Moore. We solve the problem by a direct P.D.E.-based approach, along the lines of recent work by Lira-Mazzeo-Pluda-Saez on the network flow. The main technical ingredient we use is the notion of manifolds with corners and a-corners, as introduced by Joyce following earlier work of Melrose.

Do there exist universal sequences for all mazes on the two-dimensional integer lattice? We will give background on this question, as well as some recent results. Joint work with Mariaclara Ragosta.

Let X be a smooth rigid-analytic variety. Ardakov and Wadsley introduced the sheaf D-cap of infinite order differential operators on X, along with the category of coadmissible D-cap-modules. In this talk, we present a Riemann–Hilbert correspondence for these coadmissible D-cap-modules. Specifically, we interpret a coadmissible D-cap-module as a p-adic differential equation, explain what it means to solve such an equation, and describe how to reconstruct the module from its solutions.

Einstein’s equations are difficult to solve and if you want to compute something in holography knowing an explicit metric seems to be essential. Or is it? For some theories, observables, such as on-shell actions and free energies, are determined solely in terms of topological data, and an explicit metric is not needed. One of the key tools that has recently been used for this programme is equivariant localization, which gives a method of computing integrals on spaces with a symmetry. In this talk I will give a pedestrian introduction to equivariant localization before showing how it can be used to compute the on-shell action of 6d Romans Gauged supergravity. 
 

Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of (infinitedimensional) objects. For instance, being able to flow labeled datasets is a core task for applications ranging from domain adaptation to transfer learning or dataset distillation. In this setting, we propose to represent each class by the associated conditional distribution of features, and to model the dataset as a mixture distribution supported on these classes (which are themselves probability distributions), meaning that labeled datasets can be seen as probability distributions over probability distributions. We endow this space with a metric structure from optimal transport, namely the Wasserstein over Wasserstein (WoW) distance, derive a differential structure on this space, and define WoW gradient flows. The latter enables to design dynamics over this space that decrease a given objective functional. We apply our framework to transfer learning and dataset distillation tasks, leveraging our gradient flow construction as well as novel tractable functionals that take the form of Maximum Mean Discrepancies with Sliced-Wasserstein based kernels between probability distributions.

Six-functor formalisms are ubiquitous in mathematics, and I will start this talk by giving a quick introduction to them. A three-functor formalism is, as the name suggests, (the better) half of a six-functor formalism. I will discuss what it means for such a three-functor formalism to be unitary, and why commutative Von Neumann algebras (and hence, by the Gelfand-Naimark theorem, measure spaces) admit a unitary three-functor formalism that can be viewed as mixing sheaf theory with functional analysis. Based on joint work with André Henriques.