Past

Please join us for a fireside chat, hosted by OSE, between PQShield founder and visiting professor, Dr Ali El Kaafarani, and Sami Walter, associate at Oxford Sciences Enterprises (OSE). 

Dr Ali El Kaafarani is the founder and CEO of PQShield, a post-quantum cryptography (PQC) company empowering organisations, industries and nations with quantum-resistant cryptography that is modernising the vital security systems and components of the world's technology supply chain.  

In this chat, we’ll discuss Dr Ali El Kaafarani’s experience founding PQShield and lessons learned from spinning a company out from the Oxford ecosystem.

This week’s Fridays@2 session is intended to provide advice on exam preparation and how to approach the Part A, B, and C exams.  A panel consisting of past examiners and current students will answer any questions you might have as you approach exam season.

In the theory of perfectoid fields, the tilting operation takes a perfectoid field K (a densely normed complete field of positive residue characteristic p for which the map which sends x to its p-th power is surjective as a self-map on O/pO where O is the ring of integers) to its tilt, which is computed as the limit in the category of multiplicative monoids of K under repeated application of the map sending x to its p-th power, and then a natural normed field structure is constructed. It may happen that two non-isomorphic perfectoid fields have isomorphic tilts. The family of characteristic zero untilts of a complete nontrivially normed complete perfect field of positive characteristic are parameterized by the Fargues-Fontaine curve.

Taking into account these parameters, we show that this correspondence between perfectoid fields of mixed characteristic and their tilts may be regarded as a quantifier-free bi-interpretation in continuous logic. The existence of this bi-interpretation allows for some soft proofs of some features of tilting such as the Fontaine-Wintenberger theorem that a perfectoid field and its tilt have isomorphic absolute Galois groups, an approximation lemma for the tilts of definable sets, and identifications of adic spaces.

This is a report on (rather old, mostly from 2016/7) joint work with Silvain Rideau-Kikuchi and Pierre Simon available at https://arxiv.org/html/2505.01321v1 .

I will talk about intersection theory over any globally valued field and how it is connected to some model-theoretic problems.

In recent joint work with J. Merikoski, we developed a new way to employ $\mathrm{SL}_2(\mathbb{R})$  spectral methods to number-theoretical counting problems, entirely avoiding Kloosterman sums and the Kuznetsov formula. The main result is an asymptotic formula for an automorphic kernel, with error terms controlled by two new kernels. This framework proves particularly effective when averaging over the level and leads to improvements in equidistribution results involving quadratic polynomials. In particular, we show that the largest prime divisor of $n^2 + h$ is infinitely often larger than $n^{1.312}$, recovering earlier results that had relied on the Selberg eigenvalue conjecture. Furthermore, we obtain, for the first time in this setting, strong uniformity in the parameter $h$.
 

Parameters in mathematical models are often impossible to determine fully or accurately, and are hence subject to uncertainty. By modelling the input parameters as stochastic processes, it is possible to quantify the uncertainty in the model outputs. 

In this talk, we employ the multilevel Monte Carlo (MLMC) method to compute expected values of quantities of interest related to partial differential equations with random coefficients. We make use of the circulant embedding method for sampling from the coefficient, and to further improve the computational complexity of the MLMC estimator, we devise and implement the smoothing technique integrated into the circulant embedding method. This allows to choose the coarsest mesh on the  first level of MLMC independently of the correlation length of the covariance function of the random  field, leading to considerable savings in computational cost.

 

Please note; this talk is hosted by Rutherford Appleton Laboratory, Harwell Campus, Didcot, OX11 0QX

Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. However, when the domain is infinite dimensional and the range is finite dimensional, the result is not true – even under the assumption that the map is “polynomial” – and a general theory is still lacking. In this seminar, I will provide sharp quantitative criteria for the validity of Sard’s theorem in this setting, obtained combining a functional analysis approach with new tools in semialgebraic geometry. As an application, I will present new results on the Sard conjecture in sub-Riemannian geometry. Based on a joint work with A. Lerario and L. Rizzi.

A beautiful example of a nonlinear eigenvalue problem is the determination of complex eigenvalues for wave scattering. This talk will show how nicely this can be done by applying AAA rational approximation to a scalarized resolvent sampled at a few real frequencies.  Even for a domain as elementary as a circle with a gap in it, such computations do not seem to have been done before. This is joint work with Oscar Bruno and Manuel Santana at Caltech.

A fundamental question in neuroscience is to understand how information is represented in the activity of  tens of thousands of neurons in the brain. Towards this end, low-rank matrix and tensor decompositions are commonly used to identify correlates of behavior in high-dimensional neural data. In this talk I will first present a novel tensor decomposition based on the slice rank which is able to disentangle mixed modes of covarying patterns in data tensors. Second, to compliment this statistical approach, I will present our recent dynamical systems modelling of neural activity over learning. Rather than factorizing data tensors themselves, we instead fit a dynamical system to the data, while constraining the tensor of parameters to be low rank. Together these projects highlight how applications in neural data can inspire new classes of low-rank models.

We shall explain how to represent a couple of basic notions in model theory by standard simplicial diagrams from homotopy theory. Namely, we shall see that the notions of a {definable/invariant type}, {convergence}, and {contractibility} are defined by the same simplicial formula, and so are that of a {complete E-M type} and an {idempotent of an oo-category}.  The first reformulation makes precise Hrushovski's point of view that a definable/invariant type is an operation on types rather than a property of a type depending on the choice of a model, and suggests a notion of a type over a {space} of parameters. The second involves the nerve of the category with a single idempotent non-identity morphism, and leads to a reformulation of {non-dividing} somewhat similar to that of lifting idempotents in an oo-category. If time permits, I shall also present simplicial reformulations of distality, NIP, and simplicity.

We do so by associating with a theory the simplicial set of its n-types, n>0. This simplicial set, or rather its symmetrisation, appeared earlier in model theory under the names of {type structure}  (M.Morley. Applications of topology to Lw1w. 1974), {type category} (R.Knight, Topological Spaces and Scattered Theories. 2007), {type space functors} (Haykazyan. Spaces of Types in Positive Model Theory. 2019; M.Kamsma. Type space functors and interpretations in positive logic. 2022).

There is a well-known class of knots, called torus knots, which are those that can be drawn on a "standardly embedded" torus (one that separates the 3-sphere into two solid tori). A fairly natural property of other knots to consider is the genus necessary for that knot to be drawn on a standardly embedded genus g surface. This knot invariant has been studied under the name "embeddability". The goal of this talk is to introduce the invariant, look at some upper and lower bounds in terms of other invariants, and examine its behavior under connected sum.

In this talk, we discuss the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions, or on the one-dimensional torus. To approximate the Schrödinger map equation, we employ the stochastic  Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamic of the deterministic equation. We explore the relationship between the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove existence of statistically stationary solutions to the binormal curvature flow.[https://arxiv.org/abs/2501.16499]

This is a joint work with Professor M. Hofmanová.

The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.

 The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.

One curious little fact about the Riemann zeta function is that if you evaluate its derivatives at the zeros of zeta, then on average this is real and positive (even though the function is complex). This has been proven for some time now, but the aim of this talk is to generalise the question further (higher derivatives, complex moments) and gain insight using random matrix theory. The takeaway message will be that there are a multitude of different proof techniques in RMT, each with their own advantages

Given a Hamiltonian circle action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the equivariant Fukaya category to the Fukaya category of a symplectic reduction.  Yanki Lekili and I have some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories.

I will present some contributions to the quasiisometry classification of solvable Lie groups of exponential growth that we obtain using sublinear bilipschitz equivalences, which are generalized quasiisometries. This is joint work with Ido Grayevsky.

At most how many edge-disjoint Hamilton cycles does a given directed graph contain? It is easy to see that one cannot pack more than the minimum in-degree or the minimum out-degree of the digraph. We show that in the random directed graph $D(n,p)$ one can pack precisely this many edge-disjoint Hamilton cycles, with high probability, given that $p$ is at least the Hamiltonicity threshold, up to a polylog factor.

Based on a joint work with Asaf Ferber.

The lambda-calculus was invented to formalise arithmetic by encoding numbers and operations as abstract functions. We will introduce the lambda-calculus and present two encodings of modular arithmetic: the first is a recipe to quotient your favourite numeral system, and the second is purpose-built for modular arithmetic. A highlight of the second approach is that it does not require recursion i.e., it is defined without fixed-point operators. If time allows, we will also give an implementation of the Chinese remainder theorem which improves computational efficiency. 

We consider the weak-error rate of the SPDE approximation by regularized Dean-Kawasaki equation with Itô noise, for particle systems exhibiting mean-field interactions both in the drift and the noise terms. Global existence and uniqueness of solutions to the corresponding SPDEs are established via the variational approach to SPDEs. To estimate the weak error, we employ the Kolmogorov equation technique on the space of probability measures. This work generalizes previous results for independent Brownian particles — where Laplace duality was used. In particular, we recover the same weak error rate as in that setting. This paper builds on joint work with X. Ji., H. Kremp and  N.  Perkowski.

This talk will focus on systems-theoretic and control theory tools that help characterize the responses of nonlinear systems to external inputs, with an emphasis on how network structure “motifs” introduce constraints on finite-time, transient behaviors.  Of interest are qualitative features that are unique to nonlinear systems, such as non-harmonic responses to periodic inputs or the invariance to input symmetries. These properties play a key role as tools for model discrimination and reverse engineering in systems biology, as well as in characterizing robustness to disturbances. Our research has been largely motivated by biological problems at all scales, from the molecular (e.g., extracellular ligands affecting signaling and gene networks), to cell populations (e.g., resistance to chemotherapy due to systemic interactions between the immune system and tumors; drug-induced mutations; sensed external molecules triggering activations of specific neurons in worms), to interactions of individuals (e.g., periodic or single-shot non-pharmaceutical “social distancing'” interventions for epidemic control). Subject to time constraints, we'll briefly discuss some of these applications.

This session, led by the Counselling Service,  will guide you through a CBT informed understanding of anxiety, which may arise about exams. The session includes:

• Psychoeducation - what is happening in the brain and body when we worry about exams
• Paradox of worry – how more pressure, makes studying less likely
• Experiential exercises – management strategies to maintain focus and engagement
• Takeaway tools – a collection of managing stress skills for self-guided practice
• Practical tips – enhance your exam preparation

The Morse-Conley complex is a central object in information compression in topological data analysis, as well as the application of homological algebra to analysing dynamical systems. Given a poset-graded chain complex, its Morse-Conley complex is the optimal chain-homotopic reduction of the initial complex that respects the poset grading.  In this work, we give a purely algebraic derivation of the Morse-Conley complex using homological perturbation theory. Unlike Forman’s discrete Morse theory for cellular complexes, our algebraic formulation does not require the computation of acyclic partial matchings of cells.  We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex.  This talk features joint work with Álvaro Torras Casas and Ulrich Pennig in "Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory" (arXiv:2503.09301). 
 

The question can be formulated as a statistical hypothesis asserting that the distribution of the shapes of closed curves representing outlines of cell nuclei in a spatial domain is independent of the distribution of their locations. The key challenge in developing a procedure to test the hypothesis from a sample of spatially indexed curves (e.g. from an image) lies in how symmetries in the data are accounted for: shape of a curve is a property that is invariant to similarity transformations and reparameterization, and the shape space is thus an infinite-dimensional quotient space. Starting with a convenient geometry for the shape space developed over the last few years, I will discuss dependence measures and their estimates for spatial point processes with shape-valued marks, and demonstrate their use in testing for spatial independence of marks in a breast cancer application.  

I will sketch a proof of the statement in the title and outline how it is related to Ehrenfeucht–Fraïssé games on C*-algebras. I will provide the relevant background on C*-algebras (and descriptive set theory) and explain how to construct a standard Borel category X that can play a role of their `moduli'. The theorem from the title is an application of the compactness theorem, for a suitable first-order theory whose models correspond to functors from X. If time permits, I will mention some related problems and connections with conceptual completeness for infinitary logic. This talk is based on several discussions with Ehud Hrushovski, Jennifer Pi, Mira Tartarotti, and Stuart White after a reading group on the paper "Games on AF-algebras" by Ben De Bondt, Andrea Vaccaro, Boban Velickovic and Alessandro Vignati.

The study of periods of automorphic forms is a key theme in the Langlands program and has become an important tool to tackle various problems in Number Theory and Arithmetic Geometry.  For instance, Waldspurger formula and its generalisations have created a fertile ground for numerous arithmetic applications. In recent years, the conjectures of Sakellaridis and Venkatesh (and then Ben-Zvi, Sakellaridis, and Venkatesh) in the context of spherical varieties has led to a deeper understanding of automorphic periods and their relation to special values of $L$-functions. In this talk, I present work in progress aimed at looking at certain non-spherical cases. Precisely, I will describe a new integral representation of the degree 12 "exterior square x standard" $L$-function on generic cusp forms on $\mathrm{GU}(2,2) \times \mathrm{GL}(2)$ (or $\mathrm{GL}(4) \times \mathrm{GL}(2)$) and how it can be used to relate the non-vanishing of its central value to a certain cohomological period.  If time permits, I will describe how the same strategy applies to the case of $\mathrm{GSp}(6) \times \mathrm{GL}(2)$. This is joint work with Armando Gutierrez Terradillos.

Many matrices that arise in scientific computing and in data science have internal structure that can be exploited to accelerate computations. The focus in this talk will be on matrices that are either of low rank, or can be tessellated into a collection of subblocks that are either of low rank or are of small size. We will describe how matrices of this nature arise in the context of fast algorithms for solving PDEs and integral equations, and also in handling "kernel matrices" from computational statistics. A particular focus will be on randomized algorithms for obtaining data sparse representations of such matrices.

At the end of the talk, we will explore an unorthodox technique for discretizing elliptic PDEs that was designed specifically to play well with fast algorithms for dense structured matrices.

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.

Multicomponent fluids are mixtures of distinct chemical species (i.e. components) that interact through complex physical processes such as cross-diffusion and chemical reactions. Additional physical phenomena often must be accounted for when modelling these fluids; examples include momentum transport, thermality and (for charged species) electrical effects. Despite the ubiquity of chemical mixtures in nature and engineering, multicomponent fluids have received almost no attention from the finite element community, with many important applications remaining out of reach from numerical methods currently available in the literature. This is in spite of the fact that, in engineering applications, these fluids often reside in complicated spatial regions -- a situation where finite elements are extremely useful! In this talk, we present a novel class of high-order finite element methods for simulating cross-diffusion and momentum transport (i.e. convection) in multicomponent fluids. Our model can also incorporate local electroneutrality when the species carry electrical charge, making the numerical methods particularly desirable for simulating liquid electrolytes in electrochemical applications. We discuss challenges that arise when discretising the partial differential equations of multicomponent flow, as well as some salient theoretical properties of our numerical schemes. Finally, we present numerical simulations involving (i) the microfluidic non-ideal mixing of hydrocarbons and (ii) the transient evolution of a lithium-ion battery electrolyte in a Hull cell electrode.

I will talk about preliminaries in Arakelov geometry. Also, a historical overview will be provided. This talk will be the basis of a later talk about the theory of globally valued fields.

Property (T) is a rigidity property for group representations. It is generally very difficult to determine whether an infinite group has property (T) or not. It has long been known that a discrete group with a finite symmetric generating set has property (T) if and only if the group Laplacian is a positive element in the maximal group C*-algebra. However, this characterization has not been useful in addressing the question for automorphism groups of (non-abelian) free groups. In his 2016 paper, Ozawa proved that the phenomenon of 'positivity' of the group Laplacian is observed in the real group algebra, meaning that the Laplacian can be decomposed into a 'sum of squares'. This result transformed checking property (T) into a finite-dimensional condition that can be performed with the assistance of computers. In this talk, we will introduce property (T) and discuss Ozawa's result in detail.

In this talk, I will describe a family of observables for 3D quantum Yang-Mills theory based on regularising connections with the YM heat flow. I will describe how these observables can be used to show that there is a unique renormalisation of the stochastic quantisation equation of YM in 3D that preserves gauge symmetries. This complements a recent result on the existence of such a renormalisation. Based on joint work with Hao Shen.

In this talk, I will discuss a model mixture of active (self-propelled) and passive (diffusive) particles with non-reciprocal effective interactions (or forces that violate Newton’s third law). We derive the hydrodynamic PDE limit for the particle densities, which is not a Wasserstein gradient flow of any free energy, consistent with the microscopic model having non-equilibrium steady states. We study the emergence of collective behaviour, which includes phase separation and dynamical (travelling) steady states.

Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension for an étale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra.  To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite.  I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph,  if the C*-algebra is stably finite, then its nuclear dimension is at most 1.  This is joint work with Dana Williams. 

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations, focusing on applications such as the canonical bundle formula and recent results toward the boundedness of Fano foliations.

Fundamental to the study of hyperbolic groups is their Gromov boundaries. The classical Cannon--Thurston map for a closed fibered hyperbolic 3-manifolds relates two such boundaries: it gives a continuous surjection from the boundary of the surface group (a circle) to the boundary of the 3-manifold group (a 2-sphere). Mj (Mitra) generalized this to all hyperbolic groups with hyperbolic normal subgroups. A generalization of the Gromov boundary to all finitely generated groups is called the Morse boundary. It collects all the "hyperbolic-like" rays in a group. In this talk we will discuss Cannon--Thurston maps for Morse boundaries. This is joint work with Ruth Charney, Antoine Goldsborough, Alessandro Sisto and Stefanie Zbinden.

Graphs (and structures) which have a linear ordering of their vertices with given local properties have a rich spectrum of complexities. Some have full power of class NP (and thus no dichotomy) but for biconnected patterns we get dichotomy. This also displays the importance of Sparse Incomparability Lemma. This is a joint work with Gabor Kun (Budapest).

The mod-$p$ cohomology of uniform pro-$p$ groups has been calculated by Lazard in the 1960s. Motivated by recent considerations in the mod-$p$ Langlands program, we consider the problem of extending his results to the case of compact $p$-adic Lie groups $G$ that are $p$-saturable but not necessarily uniform pro-$p$: when $F$ is a finite extension of $\mathbb{Q}_p$ and $p$ is sufficiently large, this class of groups includes the so-called pro-$p$ Iwahori subgroups of $SL_n(F)$. In general, there is a spectral sequence due to Serre and Lazard that relates the mod-$p$ cohomology of $G$ to the cohomology of its associated graded mod-$p$ Lie algebra $\mathfrak{g}$. We will discuss certain sufficient conditions on $p$ and $G$ that ensure that this spectral sequence collapses. When these conditions hold, it follows that the mod-$p$ cohomology of $G$ is isomorphic to the cohomology of the Lie algebra $\mathfrak{g}$.