Past

In 1977 Mazur proved that the rational torsion subgroup of the Jacobian of the modular curve $X_0(N)$, $N > 5$ prime, is generated by the linear equivalence class of the difference of the two cusps. More generally, it is conjectured that for a general $N$, the rational torsion subgroup of the Jacobian of $X_0(N)$ is generated by cusps.  In this talk, we'll discuss a generalisation of this to other modular curves, namely certain covers of $X_0(N)$, indexed by subgroups of $(\mathbf{Z}/N\mathbf{Z})^\times$.

The ’t Hooft expansion is a powerful organizational framework for understanding QFTs as perturbations away from the large N limit and has deep connections to string theory and holography. In this talk, I will discuss categorical aspects of the ’t Hooft expansion, i.e. what one learns about topological defects from the ’t Hooft expansion and, correspondingly, topological strings and twisted holography. This talk is based off the paper arXiv:2411.00760 from last year as well as the more recent review paper arXiv:2511.19776.

In this talk, we'll consider the numerical approximation of singularly perturbed reaction-diffusion partial differential equations, by finite element methods (FEMs).

Solutions to such problems feature boundary layers, the width of which depends on the magnitude of the perturbation parameter. For many hears, some numerical analysts have been preoccupied with constructing methods that can resolve any layers present, and for which one can establish an error estimate that is  independent of the perturbation parameter. Such methods are called "parameter robust", or (in some norms) "uniformly convergent".

In this talk we'll begin with the simplest possible parameter robust FEM: a standard Galerkin finite element method (FEM) applied on a suitably constructed  mesh using a priori information. However, from a practical point of view, not very scalable. To resolve this issue we consider the application of sparse grid techniques. These methods have many variants, two of which we'll consider: the hierarchical basis approach (e.g., Zenger, 1991) and the
two-scale method (e.g., many papers by Aihui Zhou and co-authors). The former can be more efficient, while the latter is considered simpler in both theory and practice.

Our goal is to try to unify these two approaches (at least in two dimensions), and then extend to three-dimensional problems, and, moreover, to other FEMs.
 

In this talk, I will discuss an approach to free boundary minimal surfaces which comes out of recent work by Struwe on a non-local energy, called the half-energy. I will introduce the gradient flow of this functional and its theory in the already studied case of disc type domains, covering existence, uniqueness, regularity and singularity analysis and highlighting the striking parallels with the theory of the classical harmonic map flow. Then I will go on to present new work, joint with Melanie Rupflin and Michael Struwe, which extends this theory to all compact surfaces with boundary. This relies upon combining the above ideas with those of the Teichmüller harmonic map flow introduced by Rupflin and Topping.

In this talk, we employ a level-set method to define complex computational domains and propose a ghost nodal finite element strategy tailored for two distinct applications. In the first part, we introduce a model for a Poisson-Nernst-Planck system that accounts for the correlated motion of positive and negative ions through Coulomb interactions. For very short Debye lengths, one can adopt the so called Quasi-Neutral limit which drastically simplifies the system, reducing it to a diffusion equation for a single carriers with effective diffusion coefficient. This approach, while simplifying the mathematical model, can limit the scope of numerical simulations, as it may not capture the full range of behaviors near the Quasi-Neutral limit. Our goal is therefore to design an Asymptotic Preserving (AP)  to handle both regimes: the full system when the Debye length is small but non-negligible, and the Quasi-Neutral regime as the Debye length approaches zero. In the second part, we study the formation of biological transportation networks governed by a nonlinear elliptic equation for the pressure coupled with a reaction-diffusion parabolic equation for the conductivity tensor. We compute numerical solutions using the proposed ghost nodal finite element method, which shows that the network becomes highly intricate and its branches extend over large portions of the domain.

Wave energy has the theoretical potential to meet global electricity demand, but it remains less mature and less cost-competitive than wind or solar power. A key barrier is the absence of engineering convergence on an optimal wave energy converter (WEC) design. In this work, I demonstrate how geometry optimisation can deliver step-change improvements in WEC performance. I present methodology and results from optimisations of two types of WECs: an axisymmetric point-absorber WEC and a top-hinged WEC. I show how the two types need different optimisation frameworks due to the differing physics of how they make waves. For axisymmetric WECs, optimisation achieves a 69% reduction in surface area (a cost proxy) while preserving power capture and motion constraints. For top-hinged WECs, optimisation reduces the reaction moment (another cost proxy) by 35% with only a 12% decrease in power. These result show that geometry optimisation can substantially improve performance and reduce costs of WECs.

Your chance to ask Mathematrix DPhil students about the process of applying to PhD programs, including written stages and interviews! 

Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model,  while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model.  We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.

In this talk, we will explore the emerging role of generative AI in mathematical research. Building on insights from the “Malliavin–Stein experiment”, carried out in collaboration with Charles-Philippe Diez and Luis Da Maia, we will discuss our experience and reflect on how AI might influence the way mathematics is conceived, proven, and created.

The simplex of traces of a unital C*-algebra has long been regarded as a central invariant in the theory. Likewise, from the group-theoretic perspective, the simplex of traces of a discrete group (namely, the simplex of traces of its maximal C*-algebra) is a fundamental object in harmonic analysis, and the study of this simplex led to many applications in recent years.

Itamar Vigdorovich , UCSD, will discuss several results describing the simplex of traces in concrete and significant cases. These include Property (T) groups and especially higher rank lattices, for which the simplex of traces is as tame as possible. In contrast, for free products, the simplex is typically as wild as possible, yet still admits a canonical and universal structure—the Poulsen simplex. In ongoing work, an analogous result is obtained for the space of traces on the fundamental group of a closed surface of genus g≥2.

Itamar presents these results, outlines the main ideas behind the proofs, and gives an overview of the central concepts. The talk is based on joint works with Gao, Ioana, Levit, Orovitz, Slutsky, and Spaas.

The weak noise theory (WNT) provides a framework for accessing large deviations in models of the Kardar-Parisi-Zhang (KPZ) universality class, probing the regime where randomness is small, fluctuations are rare, and atypical events dominate. Historically, two methods have been available: asymptotic analysis of Fredholm determinant formulas—applicable only for special initial data—and variational or saddle-point formulations leading to nonlinear evolution equations, which were mostly accessible perturbatively.

This talk explains how these approaches can be unified: the weak-noise saddle equations of KPZ-class models form classically integrable systems, admitting Lax pairs, conserved quantities, and an inverse scattering framework. In this setting, the large-deviation rate functions arise directly from the conserved charges of the associated integrable dynamics.

The discussion will focus on three examples:

1. The scalar Strict-Weak polymer ;
2. A matrix Strict-Weak polymer driven by Wishart noise ;
3. If time permits, the continuous-time q-TASEP.

In the 1990s, physicists introduced an ideal way to count curves inside a Calabi-Yau 3-fold, called the Gopakumar-Vafa (GV) theory. Building on several previous attempts, Maulik-Toda recently gave a mathematical rigorous definition of the GV invariants. We expect that the GV invariants and the Gromov-Witten (GW) invariants are related by an explicit formula, but this stands as a challenging open problem. In this talk, I will explain recent mathematical developments on the GV theory, especially for local curves, including the cohomological chi-independence theorem and the GV/GW correspondence in a special case.

An important invariant in the complex representation theory of reductive p-adic groups is the wavefront set, because it contains information about the character of such a representation. In this talk, Mick Gielen will introduce a new invariant called the canonical dimension, which can be said to measure the size of a representation and which has a close relation to the wavefront set.  He will then state some results he has obtained about the canonical dimensions of compactly induced representations and show how they teach us something new about the wavefront set. This illustrates a completely new approach to studying the wavefront set, because the methods used to obtain these results are very different from the ones usually used.

How many vectors are needed to simultaneously generate $m$ complete flags in $\mathbb{R}^d$, in the worst-case scenario?  A classical linear algebra fact, essentially equivalent to the Bruhat cell decomposition for $\text{GL}_d$, says that the answer is $d$ when $m=2$.  We obtain a precise answer for all values of $m$ and $d$.  Joint work with Federico Glaudo and Chayim Lowen.

TBA

Surfactants are chemicals that preferentially reside at interfaces. Once surfactant molecules have adsorbed to an interface, they reduce the surface tension between the two neighbouring fluids and may induce fluid flow. Surfactants have many household applications, such as in cleaning products and cosmetics, as well as industrial applications, like mineral processing and agriculture. Thus, understanding the dynamics of surfactant solutions is particularly important with regards to improving the efficacy of their applications as well as highlighting how they work. In this seminar, we will explore the spreading of a droplet over a substrate, in which there is constant injection of liquid and soluble surfactant through a slot in the substrate. Firstly, we will see how the inclusion of surfactant alters the spreading of the droplet. We will then investigate the early- and late-time behaviour of our model and compare this with numerical simulations. We shall conclude by briefly examining the effect of changing the geometry of the inflow slot.

Estimating the correlation $\sum_{n \le x} d_k(n)d(n+h)$ is a central problem in analytic number theory. In this talk, I will present a method to obtain an asymptotic formula for a smoothed version of this sum. A key feature of the result is a power-saving error term whose exponent does not depend on $k$, improving earlier bounds where the quality of the saving deteriorates with $k$. The argument relies on balancing three distinct bounds for the remainder term according to the sizes of the factors of $n$.

With the classification theory of simple and nuclear C*-algebras of real rank zero advanced to a level which may very well be final, it is natural to wonder what happens when one allows ideals, but not too many of them. Contrasting the simple case, the K-theoretical classification theory for real rank zero C*-algebras with finitely many ideals is only satisfactorily developed in subcases, and in many settings it is even unclear and/or disputed which flavor of K-theory to use.

Restricting throughout to the setting of real rank zero, Søren Eilers will compare what is known of the classification of graph C*-algebras and of approximately subhomogeneous C*-algebras, with an emphasis on what kind of conclusion can be extracted from restrictions on the complexity of the ideal lattice. The results presented are either more than a decade old or joint with An, Liu and Gong.

I will discuss a uniform waist inequality in codimension 2 for the family of finite covers of a Riemannian manifold whose fundamental group has Kazhdan‘s property T. I will describe a general strategy to prove waist inequalities based on a higher property T for Banach spaces. The general strategy can be implemented in codimension 2 but is conjectural in higher codimension. We speculate about the situation for lattices in semisimple Lie groups. Based on joint work with Uri Bader

As discovered by Perelman, the study of ancient Ricci flows which are $\kappa$-noncollapsed is a crucial prerequisite to understanding the singularity behaviour of more general Ricci flows. In dimension three, these so-called "$\kappa$-solutions" have been fully classified through the groundbreaking work of Brendle, Daskalopoulos, and Šešum. Their classification result can be extended to higher dimensions, but only for those Ricci flows that have uniformly positive isotropic curvature (PIC), as well as weakly-positive isotropic curvature of the second type (PIC2); it appears the classification result fails with only minor modifications to the curvature assumption. Indeed, with the alternative assumption of non-negative curvature operator, a rich variety of new examples emerge, as recently constructed by Buttsworth, Lai, and Haslhofer; Haslhofer himself has conjectured that this list of non-negatively curved $\kappa$-solutions is now exhaustive in dimension four. In this talk, we will discuss some recent progress towards resolving Haslhofer's conjecture, including a compactness result for non-negatively curved $\kappa$-solutions in dimension four, and a symmetry improvement result for bubble-sheet regions. This is joint work with Anusha Krishnan and Timothy Buttsworth. 

 Hyper-Kähler manifolds are rigid geometric structures. They have three different symplectic and complex structures, in direct analogy with the quaternions. Being Ricci-flat, they solve the vacuum Einstein equations, and so there has been considerable interest among physicists to explicitly construct such spaces. We will discuss in detail the examples arising from the Gibbons-Hawking ansatz. These give concrete descriptions of the metric, giving many examples to work with. They also lead to the generalised classification as hyper-Kähler quotients by P.B. Kronheimer, with one such space for each finite subgroup of SU(2). Finally, we will look at the McKay correspondence, relating the finite subgroups of SU(2) with the simple Lie algebras of type A,D,E.

Merge trees are a topological descriptor of a filtered space that enriches the degree zero barcode with its merge structure. The space of merge trees comes equipped with an interleaving distance d_I , which prompts a naive question: is the interleaving distance between two merge trees equal to the bottleneck distance between their corresponding barcodes? As the map from merge trees to barcodes is not injective, the answer as posed is no, but as proposed by Gasparovic et al., we explore intrinsic metrics d_I and d_B realized by infinitesimal path length in merge tree space, which do indeed coincide. This result suggests that in some special cases the bottleneck distance (which can be computed quickly) can be substituted for the interleaving distance (in general, NP-hard).

Join us on Friday Week 7 for some chill board games! Meet in N4.01 at 12pm for a Taylors sandwich lunch and positive end-of-term vibes.

The universal infinitesimal symmetry of a holomorphic field theory is the Lie algebra of holomorphic vector fields. We introduce the higher-dimensional Virasoro algebra and prove a local, universal, form of the Riemann–Roch theorem using Feynman diagrams. We use the concept of a (Jouanoulou) higher-dimensional chiral algebra as developed recently with Gui and Wang. We will remark on applications to superconformal field theory. This project is joint work with Zhengping Gui.

This session will explore practical ways to manage your time effectively as a student. We’ll discuss how to find the right balance between revising and working on problem sheets, tools and strategies to help you plan your workload, and how to set realistic priorities. We’ll also talk about what kind of study balance makes sense over the Christmas break. Come along to pick up useful tips for staying organised, focused, and on top of your studies.

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

Microbial communities contain many evolving and interacting bacteria, which makes them complex systems that are difficult to understand and predict. We use theory – including game theory, agent-based modelling, ecological network theory and metabolic modelling - and combine this with experimental work to understand what it takes for bacteria to succeed in diverse communities. One way is to actively kill and inhibit competitors and we study the strategies that bacteria use in toxin-mediated warfare. We are now also using our approaches to understand the human gut microbiome and its key properties including ecological stability and the ability to resist invasion by pathogens (colonization resistance). Our ultimate goal is to both stabilise microbiome communities and remove problem species without the use of antibiotics.

Pfaffian functions, and by extension Pfaffian and semi-Pfaffian sets, play a crucial role in various areas of mathematics, including o-minimal theory. Incidence combinatorics has recently experienced a surge of activity, fuelled by the introduction of the polynomial partitioning method of Guth and Katz. While traditionally restricted to simple geometric objects such as points and lines, focus has shifted towards incidence questions involving higher dimensional algebraic or semi-algebraic sets. We present a generalization of the polynomial partitioning method to semi-Pfaffian sets and illustrate how this leads to Pfaffian generalizations of classic results in incidence geometry, such as the Szemerédi-Trotter Theorem. Finally, we outline an application of semi-Pfaffian geometry and Khovanskii's bound to the robustness of neural networks.

We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. To account for liquidity dynamics, we introduce a stochastic volume effect governing the recovery of the deviation process, which represents the difference between the impacted and unaffected price. Additionally, we incorporate stochastic liquidity variations through a regime-switching Markov chain to capture abrupt shifts in market conditions. We study this singular control problem, where the trader optimally determines the timing and rate of purchases to minimize execution costs. The associated value function to this optimization problem is shown to satisfy a system of variational Hamilton–Jacobi–Bellman inequalities. Moreover, we establish that it is the unique viscosity solution to this HJB system and study the analytical properties of the free boundary separating the execution and continuation regions. To illustrate our results, we present numerical examples under different limit-order book configurations, highlighting the interplay between price impact, resilience dynamics, and stochastic liquidity regimes in shaping the optimal execution strategy.

It is well-known that one can attach Galois representations to modular forms. In the case of cusp forms, the corresponding l-adic Galois representations are irreducible for every prime l, while in the case of Eisenstein series, the corresponding Galois representations are reducible. The Langlands correspondence is expected to generalise this picture, with cuspidal automorphic representations always giving rise to irreducible Galois representations. In the cuspidal, polarized, regular algebraic setting over a CM field, a construction of Galois representations is known, but their irreducibility is still an open problem in general. I will discuss recent joint work with Zachary Feng establishing new instances of irreducibility, and outline how our methods extend some previous approaches to this problem.

Linear discrete inverse problems arise in many areas of science and engineering, from medical imaging and geophysics to atmospheric modelling. Their numerical solution often relies on iterative algorithms, particularly Krylov subspace methods, that can efficiently handle large-scale, ill-posed systems. In many practical settings, however, exact computations of matrix–vector products, preconditioners, or right-hand sides are either infeasible or unnecessary, leading to inexact iterations. This talk explores the interplay between inexactness and the regularizing behaviour of Krylov subspace methods for inverse problems. We discuss how approximate computations influence the regularization effect inherent in early iterations, as well as  semiconvergence, and how controlled inexactness may be exploited to improve computational efficiency. The aim is to provide a broad perspective on recent insights and open questions at the interface of inverse problems, iterative solvers, and computational inexactness.

Maximum likelihood estimation is a ubiquitous task in statistics and its applications. The task is: given some observations of a random variable, find the distribution(s) in your statistical model which best explains these observations. A modern perspective on this classical problem is to study the "likelihood geometry" of a statistical model. By focusing on models which have a polynomial parametrization, i.e., lie on an algebraic variety, this perspective brings in tools, algorithms and invariants from algebraic geometry and combinatorics.

In this talk, I will explain some of the key ideas in likelihood geometry and discuss its recent application to the study of likelihood asymptotics, i.e., understanding likelihood estimation for very large or very small observation counts. Agostini et al. showed that these asymptotics can be modeled and understood using tools from tropical geometry, and they used this to completely describe the asymptotics for linear models. In our work, we use the same approach to treat the class of log-linear models (also known as Gibbs distributions or maximum entropy models) and give a complete and combinatorial description of the likelihood asymptotics under some conditions.

This talk is based on joint work with Emma Boniface (UC Berkeley) and Serkan Hoşten (San Francisco SU), available at: https://epubs.siam.org/doi/full/10.1137/24M1656839

An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the pth-order tensor, based on lower-order derivatives. Between each recalculation of the pth-order derivative approximation, a high-order secant equation can be used to update the pth-order tensor as proposed in (Welzel 2022) or the approximation can be kept constant in a lazy manner. When refreshing the pth-order tensor approximation after m steps, an exact evaluation of the tensor or a finite difference approximation can be used with an explicit discretization stepsize. For all the newly adaptive regularization variants, we retrieve standard complexity bound to reach a second-order stationary point.  Discussions on the number of oracle calls for each introduced variant are also provided. When p = 2, we obtain a second-order method that uses quasi-Newton approximations with optimal number of iterations bound. 

Elekes and Szabó established non-trivial incidence bounds for binary algebraic relations in characteristic 0, generalizing the Szemerédi-Trotter theorem for point-line-incidence. This was later generalized to binary relations defined in reducts of so-called distal structures in a result of Chernikov, Peterzil and Starchenko. For fields of positive characteristic, such bounds fail to hold in general. Bays and Martin apply the bounds for distal structures in the context of valued fields to derive incidence bounds in the sense of Szemerédi-Trotter in fields admitting valuations with finite residue field, such as F_p(t). We show that this result can be made uniform in the size of the finite residue field, by making precise in some sense the intuition that ACVF is distal relative to the residue field. In this talk, I will introduce the relevant notions from incidence combinatorics and distality, before outlining a proof of the uniform-in-p result.

A d-dimensional TQFT is a topological invariant which assigns (d-1)-dimensional manifolds to vector spaces and d-dimensional cobordisms to linear maps. In the early 90s, Reshetikhin and Turaev constructed examples of these in the case d=3, using the data of certain types of linear categories. In this talk, I will provide an overview of this construction, and then explore how this might be meaningfully extended downwards to assign 1-manifolds to "2-vector spaces". Minimal knowledge of category theory assumed!

Witten’s seminal 1988 work revealed the connection between 3-dimensional Chern-Simons theory and knot invariants. In this talk, I will provide a physically motivated overview and explain how skein relations manifest from a path-integral/partition-function perspective on 3-manifolds with Wilson lines inserted. There will also be some fun topological brain-twisters for the audience. If time permits, I will comment on recent developments involving factorization homology and its relation to correlators for logarithmic CFTs.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

We present an adjoint method for optimization of the spatially inhomogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a "discretize then optimize" approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian.  The boundary conditions that are included in this analysis include spectral reflection, thermal reflection, and inflow boundary conditions. For thermal reflection, a "score function" is included as a statistical regularization. This is joint work with Yunan Yang (Cornell). This special seminar is jointly held with the Keble Complexity Research Cluster.

I'll discuss some of the history of the use of random matrices for studying the structure of operator algebras, starting with Voiculescu's notion free independence. We'll see that the original notions of convergence of random matrix models to certain infinite-dimensional operators is actually fairly weak, and discuss the more recent "strong convergence" phenomenon and its applications to C*-algebras. Finally, I'll touch upon some ongoing work, joint with A. Shiner and S. White, for continuing to use random matrix tools to prove structural properties of C*-algebras.

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.