Past

Entanglement entropy has long served as a key diagnostic of topological order in (2+1) dimensions. In particular, the topological entanglement entropy captures a universal quantity (the total quantum dimension) of the underlying topological order. However, this information alone does not uniquely determine which topological order is realized, indicating the need for more refined probes. In this talk, I will present a family of quantities formulated as multi-entropy measures, including examples such as reflected entropy and the modular commutator. Unlike the conventional bipartite setting of topological entanglement entropy, these multi-entropy measures are defined for tripartite partitions of the Hilbert space and capture genuinely multipartite entanglement. I will discuss how these measures encode additional universal data characterizing topologically ordered ground states.

Please register for the event via https://sites.google.com/view/oxwmathbio2025

The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.

We introduce the class of exponentially-algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of their types in a reduct of the theory of differentially closed fields.

We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions and give some new decidability results.

This is joint work with Rémi Jaoui, Lyon

Understanding the relationship between expectation and price is central to applications of mathematical finance, including algorithmic trading, derivative pricing and hedging, and the modelling of margin and capital. In this presentation, the link is established via dynamic entropic risk optimisation, which is promoted for its convenient integration into standard pricing methodologies and for its ability to quantify and analyse model risk. As an example of the versatility of entropic pricing, discrete models with classical and quantum information are compared, with studies that demonstrate the effectiveness of quantum decorrelation for model fitting.

Two different, almost orthogonal approaches to QFT are: (1) the study of von Neumann algebras of local observables in flat space, and (2) the study of extended and topological defects in general spacetime manifolds. While naively the two focus on different aspects, it has been recently pointed out that some of the axioms of approach (1) clash with certain expectations from approach (2). In this JC talk, I’ll give a brief introduction to both approaches and review the recent discussion in [2008.11748], [2503.20863], and [2509.03589], explaining (i) what the tensions are, (ii) a recent proposal to solve them, and (iii) why it can be useful.

The Whitney forms on a simplicial triangulation are piecewise affine differential forms that are dual to integration over chains.  The so-called blow-up Whitney forms are piecewise rational generalizations of the Whitney forms.  These differential forms, which are also called shadow forms, were first introduced by Brasselet, Goresky, and MacPherson in the 1990s.  The blow-up Whitney forms exhibit singular behavior on the boundary of the simplex, and they appear to be well-suited for constructing certain novel finite element spaces, like tangentially- and normally-continuous vector fields on triangulated surfaces.  This talk will discuss the blow-up Whitney forms, their properties, and their applicability to PDEs like the Bochner Laplace problem.  

The nematic Helmholtz-Korteweg equation is a fourth-order scalar PDE modelling time-harmonic acoustic waves in nematic Korteweg fluids, such as nematic liquid crystals. Conforming discretizations typically require C1-conforming elements, for example the Argyris element, whose implementation is notoriously challenging - especially in three dimensions - and often demands a high polynomial degree. 
In this talk, we consider an alternative non-conforming C0-hybrid interior penalty method that is both stable and convergent for any polynomial degree greater than two. Classical C0-interior penalty methods employ an H1-conforming subspace and treat the non-conformity with respect to H2 with discontinuous Galerkin techniques. Building on this idea, we use hybridization techniques to improve the computational efficiency of the discretization. We provide a brief overview of the numerical analysis and show numerical examples, demonstrating the method's ability to capture anisotropic propagation of sound in two and three dimensions. 

For $\ell$ an odd prime number and $d$ a squarefree integer, a notable problem in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record due to Ellenberg—Venkatesh is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Peter Koymans.

Let \( G = \exp(\mathfrak{g}) \) be a connected, simply connected nilpotent Lie group with Lie algebra \( \mathfrak{g} \), and let \( H = \exp(\mathfrak{h}) \) be a closed subgroup with Lie algebra \( \mathfrak{h} \). Consider a unitary character \( \chi \) of \( H \), given by \(\chi(\exp X) = \chi_{f}(\exp X) = e^{i f(X)}, \  X \in \mathfrak{h}, \) for some \( f \in \mathfrak{g}^{\ast} \). Let \( \tau = \operatorname{Ind}_{H}^{G} \chi \) denote the monomial representation of \( G \) induced from \( \chi \).

The object of interest is the algebra \( D_{\tau}(G/H) \) of \( G \)-invariant differential operators acting on the homogeneous line bundle associated with the data \( (G, H, \chi) \). Under the assumption that \( \tau \) has finite multiplicities, it is known that \( D_{\tau}(G/H) \) is commutative.

In this talk, I will discuss the Polynomial Conjecture for the representation \( \tau \), which asserts that the algebra \( D_{\tau}(G/H) \) is isomorphic to  
\(\mathbb{C}[\Gamma_{\tau}]^{H}\),  the algebra of \( H \)-invariant polynomial functions on \( \Gamma_{\tau} \). Here, \( \Gamma_{\tau} = f + \mathfrak{h}^{\perp} \) denotes the affine subspace of \( \mathfrak{g}^{\ast} \).

I will present recent advances toward proving this conjecture, with a particular emphasis on Duflo's Polynomial Conjecture concerning the Poisson center of \( \Gamma_{\tau} \). Furthermore, I will discuss the case where \( \tau \) has discrete-type multiplicities in the exponential setting, shedding light on a counterexample to Duflo's conjecture.
 

The Cuntz semigroup of a C*-algebra A consists of equivalence classes of positive elements, where equivalence means roughly that two positive elements have the same rank relative to A.  It can be thought of as a generalization of the Murray von Neumann semigroup to positive elements and is an incredibly sensitive invariant. We present a calculation of the homotopy groups of these Cuntz classes as topological subspaces of A when A is classifiable in the sense of Elliott.  Remarkably, outside the case of compact classes, these spaces turn out to be contractible.  

In this talk, I will present a representation-theoretical approach to constructing a non-commutative analogue of the classical Laplace transform on Lie groups. I will begin by discussing the motivations for such a generalization, emphasizing its connections with harmonic analysis, probability theory, and the study of evolution equations on non-commutative spaces. I will also outline some of the key challenges that arise when extending the Laplace transform to the setting of Lie groups, including the non-commutativity of the group operation and the complexity of its dual space.

The main part of the talk will focus on an explicit construction of the Laplace transform in the framework of connected, simply connected nilpotent Lie groups. This construction relies on Kirillov’s orbit method, which provides a powerful bridge between the geometry of coadjoint orbits and the representation theory of nilpotent groups.

As an application, I will describe an operator-theoretic analogue of the classical Müntz–Szász theorem, establishing a density result for a family of generalized polynomials in associated with the group setting. This result highlights the strength of the representation-theoretical approach and its potential for solving classical approximation problems in a non-commutative context.

Symmetry Topological Field Theories (SymTFTs) are topological field theories that encode the symmetry structure of global symmetries in terms of a theory in one higher dimension. While SymTFTs for internal (global) symmetries have been highly successful in characterizing symmetry aspects in the last few years, a corresponding framework for spacetime symmetries remains unexplored. We propose an extension of the SymTFT framework to include spacetime symmetries. In particular, we propose a SymTFT for the conformal symmetry in various spacetime dimensions. We demonstrate that certain BF-type theories, closely related to topological gravity theories, possess the correct topological operator content and boundary conditions to realize the conformal algebra of conformal field theories living on boundaries. As an application, we show how effective theories with spontaneously broken conformal symmetry can be derived from the SymTFT, and we elucidate how conformal anomalies can be reproduced in the presence of even-dimensional boundaries.
 

I will explain how to obtain local L^\infty estimates for optimal transport problems. Considering entropic optimal transport and optimal transport with p-cost, I will show how such estimates, in combination with a geometric linearisation argument, can be used in order to obtain ε-regularity statements. This is based on recent work in collaboration with M. Goldman (École Polytechnique) and R. Gvalani (ETH Zurich).

I will talk about the problem of recognising when a manifold admits a finite cover that fibres over the circle, with emphasis on the case of hyperbolic manifolds in odd dimensions. I will survey the state-of-art, and discuss the role that group theory plays in the problem. Finally, I will discuss a recent result that sheds light on the analogous group-theoretic problem, that is, virtual algebraic fibring of Poincaré-duality groups. The final theorem is joint with Sam Fisher and Giovanni Italiano.

The well-known Milnor-Wood inequality gives a bound on the Toledo invariant of a representation of the fundamental group of a compact surface in a non-compact Lie group of Hermitian type. While a lot is known regarding the counting of maximal Toledo components, and their role in higher Teichmueller theory, the non-maximal case remains elusive. In this talk, I will present a strategy to count the number of such non-maximal Toledo connected components. This is joint work in progress with Brian Collier and Jochen Heinloth, building on previous work with Olivier Biquard, Brian Collier and Domingo Toledo.

We provide an iterative solution approach for the indefinite Helmholtz equation discretised using finite elements, based upon a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to the shifted operator, and prove that the iteration is k- and mesh-robust when O(k) HSS iterations are performed. The HSS iterations involve solving a shifted operator that is suitable for approximation by multigrid using standard smoothers and transfer operators, and hence we can use O(N) parallel processors in a high performance computing implementation, where N is the total number of degrees of freedom. We argue that the algorithm converges in O(k) wallclock time when within the range of scalability of the multigrid. We provide numerical results verifying our proofs and demonstrating this claim, establishing a method that can make use of large scale high performance computing systems.

This talk is hosted by Rutherford Appleton Laboratory and will take place @ Harwell Campus, Didcot, OX11 0QX

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.