Past

Details to be finalised

• as holomorphic functions on certain symplectic spaces
• as matrix coefficients of the Heisenberg (and metaplectic) representation,
• as sections of line bundles on abelian varieties.

Based on mathematical ecological models, this report reviews the impact of spatially heterogeneous environments on the persistence and distribution of biological populations. The report aims to elucidate the interplay between population density and key factors, including diffusion coefficients, resource availability, and habitat structure. The study first investigates the ecological consequences of dispersal strategies within environments characterized by uneven resource distribution, demonstrating the monotonic dependence of peak population densities on diffusion rates. Furthermore, analysis of a consumer-resource system indicates that under resource-limited conditions, the ecosystem converges to a globally asymptotically stable state of coexistence. Building on these findings, the report additionally explores the constraints imposed by domain geometry on the spatial patterning of populations.

Approximate subrings are subsets $A$ of a ring $R$ satisfying \[ A + A + AA \subset F + A \] for some finite $F \subset R$. They encode the failure of sum-product phenomena, much like approximate subgroups encode failure of growth in groups.

I will discuss how approximate subrings mirror approximate subgroups and how model-theoretic tools, such as a stabilizer lemma for approximate subrings due to Krupiński, lead to structural results implying a general, non-effective sum-product phenomenon in arbitrary rings: either sets grow rapidly under sum and product, or nilpotent ideals govern their structure. I will also outline related results for infinite approximate subrings and conjectures unifying known (effective) sum-product phenomena.

Based on joint work with Krzysztof Krupiński.

Professor Kaibo Hu will be talking about: 'Finite element form-valued forms'

Some of the most successful vector-valued finite elements in computational electromagnetics and fluid mechanics, such as the Nédélec and Raviart-Thomas elements, are recognized as special cases of Whitney’s discrete differential forms. Recent efforts aim to go beyond differential forms and establish canonical discretizations for more general tensors. An important class is that of form-valued forms, or double forms, which includes the metric tensor (symmetric (1,1)-forms) and the curvature tensor (symmetric (2,2)-forms). Like the differential structure of forms is encoded in the de Rham complex, that of double forms is encoded in the Bernstein–Gelfand–Gelfand (BGG) sequences and their cohomologies. Important examples include the Calabi complex in geometry and the Kröner complex in continuum mechanics.
These constructions aim to address the problem of discretizing tensor fields with general symmetries on a triangulation, with a particular focus on establishing discrete differential-geometric structures and compatible tensor decompositions in 2D, 3D, and higher dimensions.
 

We all sleep. But what determines when and for how long? In this talk I’ll describe some of the fundamental mechanisms that regulate sleep. I’ll introduce the nonsmooth coupled oscillator systems that form the basis of current mathematical models of sleep-wake regulation and discuss their dynamical behaviour. I will describe how we are using models to unravel environmental, societal and physiological factors that determine sleep timing and outline how constructing digital-twins could enable us to create personalised light interventions for sleep timing disorders.

The family of right-angled Artin groups (RAAGs) interpolates 
between free groups and free abelian groups. These groups are defined by 
a simplicial graph: the vertices correspond to generators, and two 
generators commute if and only if they are connected by an edge in the 
defining graph. A key feature of RAAGs is that many of their algebraic 
properties can be detected purely in terms of the combinatorics of the 
defining graph.
The family of outer automorphism groups of RAAGs similarly interpolates 
between Out(F_n) and GL(n, Z). While the l2-Betti numbers of GL(n, Z) 
are well understood, those of Out(F_n) remain largely mysterious. The 
aim of this talk is to introduce automorphism groups of RAAGs and to 
present a combinatorial criterion, expressed in terms of the defining 
graph, that characterizes when the first l2-Betti number of Out(RAAG) 
vanishes.
If time permits, we will also discuss higher l2-Betti numbers and 
algebraic fibring properties of these group

Koszul duality is a powerful mathematical construction.  In this talk, I will take a physical perspective to demonstrate one instance of this duality: an algebraic approach to coupling quantum field theories to a quantum mechanical system on a line.  I will explain how a Lagrangian coupling results in an algebraic object, called a Maurer-Cartan element, and show that there is a sense in which the Koszul dual to the algebra of local operators gives a “universal coupling”.  I will then describe what Koszul duality really “is”, and why many other mathematical constructions deserve the same name.

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 aimedl at PhD students and post-docs but everyone is welcome.

We give an introduction to a rigorous renormalization group analysis of the sine-Gordon model with a focus on deriving the lowest-order beta function.

This talk by Tim Austin, at the University of Warwick, will be an introduction to "almost periodic entropy".  This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra.  It is an analog of Lewis Bowen's "sofic entropy" from ergodic theory.  This analogy extends to many of its properties, but some important differences also emerge.  Tim will not assume any prior knowledge about sofic entropy.

After setting up the basic definition, Tim will focus on the special case of finitely generated free groups, about which the most is known.  For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations.  This, in turn, offers a new proof of the Collins—Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.

S-duality is an intriguing symmetry of (twisted) N=4 supersymmetric Yang-Mills theory on a four-manifold. When the four-manifold underlies a complex projective surface, it leads to the Vafa-Witten invariants defined by Tanaka-Thomas in 2017. I will discuss some developments related to Azumaya algebras, universality, Seiberg-Witten invariants, wall-crossing for Nakajima quiver varieties, the structure of S-duality, and modular curves (including relations to the Rogers-Ramanujan continued fraction and Klein quartic).

The theory of JSJ decomposition plays a key role in the classification of hyperbolic groups, in analogy with the case of 3-manifolds. While this theory can be extended to larger families of groups, the JSJ decomposition displays significant flexibility in general, making a complete understanding of its behaviour more challenging. In this talk, Dario Ascari explores this flexibility, with an emphasis on the case of generalized Baumslag-Solitar groups.

A temporal graph $G$ is a sequence of graphs $G_1, G_2, \ldots, G_t$ on the same vertex set. In this talk, we are interested in the analogue of the Travelling Salesman Problem for temporal graphs. It is referred to in the literature as the Temporal Exploration Problem, and asks for the minimum length of an exploration of the graph, that is, a sequence of vertices such that at each time step $t$, one either stays at the same vertex or moves along a single edge of $G_t$.

One natural and still open case is when each graph $G_t$ is connected and has bounded maximum degree. We present a short proof that any such graph admits an exploration in $O(n^{3/2}\sqrt{\log n})$ time steps. In fact, we deduce this result from a more general statement by introducing the notion of average temporal maximum degree. This more general statement improves the previous best bounds, under a unified approach, for several studied exploration problems.

This is based on joint work with Carla Groenland, Lukas Michel and Clément Rambaud.

Let g be the Lie algebra of a simple algebraic group over an algebraically closed field of characteristic p. When p=0 the celebrated Jacobson-Morozov Theorem promises that every non-zero nilpotent element of g is contained in a simple 3-dimensional subalgebra of g (an sl2). This has been extended to odd primes but what about p=2? There is still a unique 3-dimensional simple Lie algebra, known colloquially as fake sl2, but there are other very sensible candidates like sl2 and pgl2. In this talk, Adam Thomas from the University of Warwick will discuss recent joint work with David Stewart (Manchester) determining which nilpotent elements of g live in subalgebras isomorphic to one of these three Lie algebras. There will be an abundance of concrete examples, calculations with small matrices and even some combinatorics.

Biomolecular condensates are membraneless assemblies of biomolecules (such as proteins or nucleic acids) formed through liquid-liquid phase separation. Many biomolecules are electrically charged, making condensates highly sensitive to the local electrochemical environment. In this talk, I will discuss our recent theoretical work on the dynamics of charged condensates and the role of salt concentration in their evolution toward equilibrium. Two-dimensional simulations of a thermodynamically consistent phase-field model reveal that salt can arrest coarsening by affecting the relative strength of interfacial energy, associated with the condensate surface, and electrostatic energy, arising from the formation of an electric double layer across liquid interfaces. At low salt concentrations, the electrostatic energy of the double layer becomes comparable to the interfacial energy, resulting in the emergence of multiple condensates with a fixed size. These results show that salt can act as a dynamic regulator of condensate size, with implications for both understanding biological organisation and modulating the behaviour of synthetic condensates.

Chirality, the property that an object cannot be superimposed on its mirror image, arises across all scientific disciplines, yet its ultimate origin remains one of the central open questions in Nature. Both fundamental and elusive, chirality plays a decisive role in shaping the structure and behaviour of natural systems. Starting from its classical geometric definition and the long-standing challenge of defining meaningful measures of chirality, this talk develops a natural extension of the concept to field theories by examining the physical response of chiral bodies immersed in fluid flows. This framework leads to a further novel concept in which chirality is attached not only to objects, but also to their smooth deformations. I will address the general problems of chirality, its quantification, and its transfer across scales, trace their historical development, and illustrate the theory through examples drawn from fluid mechanics, chemistry, and biology, revealing unifying principles with some surprising twists.

This talk will describe recent studies of how time-dependent, unsteady flow physics can be exploited to improve the performance of energy harvesting systems such as wind turbines. A theoretical analysis will revisit the seminal Betz derivation to identify the role of unsteady flow from first principles. Following will be a discussion of an experimental campaign to test the predictions of the theoretical model. Finally, a new line of research related to turbulence transition and inspired by the work of T. Brooke Benjamin will be introduced.

In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.

I will discuss recent and ongoing work (mostly with J. Zou).

Details to be finalised

In our first event of term, we have a fun quiz competition to help you get to know other Mathematrix members and a free pizza lunch from White Rabbit! What more could you want?

In this talk we present different modeling approaches to describe and analyse the dynamics of large pedestrian crowds. We start with the individual microscopic description and derive the respective partial differential equation (PDE) models for the crowd density. Hereby we are particularly interested in identifying the main driving forces, which relate to complex dynamics such as lane formation in bidirectional flows. We then analyse the time-dependent and stationary solutions to these models, and provide interesting insights into their behavior at bottlenecks. We conclude by discussing how the Bayesian framework can be used to estimate unknown parameters in PDE models using individual trajectory data.

We generalize the seminal polynomial partitioning theorems of Guth and Katz [1, 2] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n - k - r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemeredi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves.

These results, together with some of my other recent work (e.g., bounding the number of distinct distances on plane Pfaffian curves), are steps in a larger program - pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk.

This talk is based on multiple joint works with Saugata Basu, Antonio Lerario, Martin Lotz, Adam Sheffer, and Nicolai Vorobjov.

[1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge
Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.

[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of
mathematics (2015), 155–190.
 

Professor Nick Trefethen will speak about: 'Quadrature = rational approximation'

Whenever you see a string of quadrature nodes, you can consider it as a branch cut defined by the poles of a rational approximation to the Cauchy transform of a weight function.  The aim of this talk is to explain this strange statement and show how it opens the way to calculation of targeted quadrature formulas for all kinds of applications.  Gauss quadrature is an example, but it is just the starting point, and many more examples will be shown.  I hope this talk will change your understanding of quadrature formulas. 

This is joint work with Andrew Horning. 
 

I will review an elegant, theory-independent argument that proves the existence of exactly marginal operators in the presence of a conformal manifold. The proof relies on a few technical assumptions, which I will discuss in detail. The rest of the discussion will be phrased in terms of conformal interfaces separating two CFTs on the conformal manifold, which we take as an opportunity to discuss the fundamentals of defect CFTs. The overarching topic into which this result fits is that of proving certain (AdS) swampland conjectures from CFT principles.

We consider a one-dimensional compressible Navier-Stokes model for reacting gas mixtures with the same γ-law in dynamic combustion. The unknowns of the PDE system consist of the inverse density, velocity, temperature, and mass fraction of the reactant (Z). First, we show that the graph of Z cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any time, based on a Bernis-type inequality by M. Winkler (2012) and the recent works by T. Cieślak et al (2023). In addition, we establish the global well-posedness theory of small BV weak solutions for initial data that are small perturbations around the constant equilibrium state (1, 0, 1, 0) in the L¹(R)∩BV(R)-norm, via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier-Stokes and Navier-Stokes-Fourier systems developed by T. Liu and S.-H. Yu (2022).

*Joint with Prof. Haitao Wang and Miss Jianing Yang (SJTU)

Casey Garner will talk about; 'General Matrix Optimization'

Since our early days in mathematics, we have been aware of two important characteristics of a matrix, namely, its coordinates and its spectrum. We have also witnessed the growth of matrix optimization models from matrix completion to semidefinite programming; however, only recently has the question of solving matrix optimization problems with general spectral and coordinate constraints been studied. In this talk, we shall discuss recent work done to study these general matrix optimization models and how they relate to topics such as Riemannian optimization, approximation theory, and more.

I will present an overview of a range of interdisciplinary modelling challenges that I have been working on in collaboration with experimentalists and external partners. I will begin with mathematical modelling of calcium signalling in In-Vitro fertilization (IVF) and embryogenesis, illustrating how multiscale approaches can link molecular dynamics to cellular and developmental outcomes. I will then discuss our ongoing work on modelling viral transmission in indoor environments, carried out in collaboration with architects and policymakers, with the aim of informing evidence-based policy decisions for future epidemics.

In this ``journal club''-style advanced class, I will present some material from a recent paper of Tom Scanlon https://arxiv.org/abs/2508.17485 . Motivated by the question of decidability of the field C(t) of complex rational functions in one variable, he considers the structure $(\mathbb{C};+,\cdot,CM)$ of the complex field expanded by a predicate for the set CM of j-invariants of elliptic curves with complex multiplication (the "special points"). Analogous to Zilber's result from the 90s on stability of the expansion by a predicate for the roots of unity, Scanlon shows that Pila's solution to the André-Oort conjecture implies that this structure is stable, and moreover that effectivity in this conjecture due to Binyamini implies decidability. I aim to explain Scanlon's proof of this result in some detail.
 

I will report on ongoing joint work with Sam Fisher on showing that the mapping class group has a finite index subgroup whose group ring embeds in a division ring. Our methods involve p-adic analytic groups, but no prior knowledge of this will be assumed and much of the talk will be devoted to explaining some of the underlying theory. Time permitting, I will also discuss some consequences for the profinite topology for the mapping class group and potential extensions to Out(RAAG).

In this talk, Konstantin Recke, University of Oxford,  will report on some results pertaining to the interplay between geometric group theory, operator algebras and probability theory. Konstantin will introduce so-called invariant percolation models from probability theory and discuss their relation to geometric and analytic properties of groups such as amenability, the Haagerup property (a-T-menability), $L^p$-compression and Kazhdan's property (T). Based on joint work with Chiranjib Mukherjee (Münster).

Let $X_0 $ be a rational surface with a cyclic quotient singularity $(1,a)/r$.  Kawamata constructed a remarkable vector bundle  $F_0$  on $X_0$ such that the finite-dimensional algebra End$(F_0)$ "absorbs'' the singularity of $X_0$ in a categorical sense. If we deform over an irreducible component of the versal deformation space of $X_0$ (as described by Kollár and Shepherd-Barron), the vector bundle $F_0$ also deforms to a vector bundle $F$. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute End$(F)$ explicitly in a certain bulk-deformed Fukaya category. In the case of a $Q$-Gorenstein smoothing, this algebra End$(F)$ is a matrix order over $k[t]$ and "absorbs" the singularity of the corresponding terminal 3-fold singularity. This is based on joint work with Jenia Tevelev.