Past

Constructing $G_2$ manifolds is a notoriously difficult task. Taking inspiration from physics, one can try to reverse engineer new $G_2$ manifolds by geometrising, via M-theory, the flux torus construction of 4d $\mathcal{N}=1$ quiver gauge theories. The resulting $G_2$ manifolds are described by highly non-trivial fibrations of an elliptically fibered Calabi-Yau threefold over a circle. In this talk I will describe the construction of a new infinite family of non-compact $G_2$ manifolds arising from flux torus compactifications of the rank 1 E-string theory. This talk will focus on the geometry and not the physics of this construction. Based on arXiv 2608.21238. 

Finitely presented multiparameter persistence modules are a natural target category for encoding homological invariants, but comparing such modules requires choosing a notion of distance. Our goal is to develop a zoo of distances on multiparameter persistence modules, capturing both the algebraic features of the representation category as well as the geometry of the parameter space. The parameters of this construction make the comparison flexible by capturing different notions of noise and similarity. This is joint work with Jens Agerberg and Wojciech Chachólski.

Make mathematical crafts and get to know other Mathematrix members! Materials provided.

This session will look at how you can get the most out of your lectures and tutorials. We’ll talk about how to prepare effectively, make lectures more productive, and understand what tutors expect from you during tutorials. You’ll leave with practical tips to help you study more confidently and make your learning time count.

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

Every population consists of individuals who vary in their traits, and each trait may, or may not, be associated with frailty or fitness. Variation in frailty and fitness traits makes population studies prone to selective depletion bias (SDB). The issue is widespread across scientific domains. When an ageing cohort exhibits declining mortality, is it individuals becoming healthier or selective depletion of the frail? In an epidemic, when growth in cumulative infections decelerates, is it individuals cautiously changing behaviour or selective depletion of the most susceptible? In microbial populations, when mutations increase population vulnerability to stress, it is individuals becoming more vulnerable or mutant populations having higher variance in fitness? In each case, the first explanation invokes individuals changing, while the second recognises that populations change due to selection on pre-existing variation. While the former are intuitive and widely adopted, explanations that rely on selective depletion are more neutral and less commonly considered due to cognitive biases and challenges in estimating all variation that matters. 

Remodelling selection (ReMS) is proposed as a general strategy of study design and analysis to address the SDB problem. I will show how the approach has been employed in specific case studies and how it has been formalised in general.

I will discuss an application to probability theory (more precisely, to the theory of exchangeable random variables) of recent developments in affine logic, the fragment of continuous logic in which only affine connectives are allowed.

An affine theory is simplicial if its type spaces, which are always compact convex sets, form Choquet simplices. A remarkable dichotomy holds: for a complete simplicial theory, either all type spaces are Bauer simplices (i.e., the set of extreme points is closed) or they are all Poulsen simplices (i.e., the extreme points are dense).

On the other hand, simplices of invariant measures arising in ergodic theory and probability theory can be represented as *quantifier-free* type spaces of appropriate universal affine theories. I will discuss a quantifier-free adaptation of the dichotomy result mentioned above and its application to the geometry of certain simplices of invariant measures arising in exchangeability theory. Relative property (T) for Polish groups will play an important role.

The established mod p local Langlands correspondence is a representation of GL2(Qp) that is associated to a local Galois representation and that occurs in the cohomology of the modular curve. It is believed that more general correspondences occur but, as of yet, this remains open. Examples of these generalisations would be representations of GLn(Qp) that are associated to higher dimensional Galois representations and that occur in the cohomology of suitable arithmetic manifolds. 

In this talk, I will discuss joint work with Shu Sasaki in which we construct a purely local framework for associating a mod p representation of GLn(Qp) to the data of a mod p Galois representation whose image lives in a Borel subgroup (a mod p, p-ordinary Galois representation). We will see how our construction is motivated by earlier work of Breuil and Herzig and why we hope to use our framework to generalise their mod p results to include non generic cases.  

Dr Peter Braam is going to talk about; 'State time geometry: causal performance profiles and optimal data transport in parallel execution'

The increasing complexity of parallel architectures and heterogeneous microarchitectures makes predicting and optimising program performance notoriously difficult. For Optimal Data Transport, we present a discrete variant of the Wasserstein–Fisher–Rao metric that quantifies the true cost of data layout transformations and movement across memory hierarchies. For Causal Performance Profiles, we introduce the Lyons–Gregg Signature, which combines the ideas of Terry Lyons' rough path signatures with hardware performance counters (eBPF) to capture cross-correlated, causal bottlenecks in execution streams. Both arose from State Time Geometry (STG), a model for stateful program execution on computing infrastructure, first modelled as a dynamical system governing state-values over the space of memory addresses. The address space generalises to geometric objects defining infrastructure and leads to the metric. The state transitions of parallel executions become a Grothendieck quantum field theory over the infrastructure and carry the statistical model for the Lyons-Gregg Signature. The central theme is that an intuitive faithful model is not doomed by complexity but forms a geometric domain in which both theoretical and engineering perspectives are simplified.

(In a companion lecture in the Computing Laboratory at 11:00 on Nov 13, we will discuss STG's underlying categorical and geometric structure and its relationship to programming languages and formal methods)

Chiral holography constructs a gravitational dual of N=4 self-dual Yang-Mills (SDYM), which provides a holographic duality at zero 't Hooft coupling. The top-down construction begins by considering the twistor string and its D-branes, whose world-volume theory is the twistor uplift of SDYM.  In this talk, I will review twistor theory, the holographic setup, and how local spacetime operators manifest on twistor space. This talk is based on https://arxiv.org/pdf/2512.04152.

Zheng Tang (Mathematical Institute, University of Oxford) is going to talk about: "Nyström is better"

Abstract: Subspace methods are widely used for computing a few eigenvalues and eigenvectors of large symmetric matrices. These methods first construct a low-dimensional subspace and then extract approximate eigenpairs from it. The Rayleigh–Ritz method is the standard approach for the extraction step and has several classical optimality properties. For positive semidefinite matrices, however, the Nyström method, widely used for low-rank approximation of kernel matrices in machine learning, provides an alternative that can achieve significantly higher accuracy at the same asymptotic computational cost.

In this talk, we compare the eigenvalue extraction errors of different subspace projection methods, including Rayleigh-Ritz and Nyström, and investigate the potential accuracy improvement. We show that the improvement can be significant under certain spectral conditions. We also discuss how our results apply when the low-dimensional search subspace is constructed by subspace iteration or Krylov subspace methods.

Underneath large glaciers and ice sheets, water flows through a permeable network of cavities and channels, held open by melting the ice above balancing the downwards flow of ice. Dissipation within the water flow is a significant enough source of heat that instabilities can develop if the flow rate is high enough, eroding large channels that rapidly drain water from the glacier bed. I will present a model for the system, and discuss the linear problem, observational evidence for the stability criterion, and the non-linear interactions that rapidly become the dominant control on channel spacing. Recently, there has been some discussion of slowing glaciers down by pumping water out from under them - I will consider the viability of this strategy in view of the results in this talk.

We show that the conjugator length spectrum of finitely presented groups contains every rational number in [1, ∞︎).  In particular, this shows there is no gap in the conjugator length spectrum, which is in contrast to isoperimetric spectra.  Our construction relies on upgrading distortion of cyclic subgroups to conjugator length functions, which we apply to snowflake groups.

We study 5d N=1 theories engineered by (p,q) five-brane webs and their 3d N=4 magnetic quivers, focusing on the N + SU(N)^(k-1) + N gauge theory. We establish an exact parameter dictionary mapping the 5d flavour masses and instanton couplings to the FI parameters of the strong-coupling magnetic quiver. We test this dictionary against mass deformations, weak-coupling limits, and moment map VEVs, implemented via the 5d superconformal index and the 3d squashed-sphere partition function. We analyze these deformations on the magnetic-quiver side using a new framework that exactly tracks Higgsing patterns, decoupled sectors, and infrared theories. This promotes magnetic quivers to quantitative tools for studying how the 5d superconformal index responds to physical deformations
We show that large 5d moment map VEVs render the superconformal index distributional, mirroring bad theories in lower dimensions. We also develop a brane-web interpretation of these results, identifying the Hanany–Witten moves associated with the cascade of VEVs and relating inequivalent frames to distinct tessellations of the generalized toric diagram. Furthermore, we demonstrate that magnetic quivers remain powerful predictive tools even when the theory enters the bad regime.

Topological quantum field theories are one of the few cases of QFT that can be treated mathematically.  The goal of this talk will be to recall a mathematician's definition of TFT, and demonstrate how it recovers features of QFT that are familiar to physicists.  I will then introduce the notion of extended TFT, and explain how dualizability properties can be used to classify TFTs.

In 2014, Glasner and Weiss introduced the theory of uniformly recurrent subgroups (URSs) as a topological analogue of the theory of invariant random subgroups. Later, Matte Bon and Todor showed that every URS arises from a minimal topological action of the group on a compact space. Subsequently, in 2020, Kennedy established a complete characterization of C ∗ -simplicity in terms of URSs, proving that a group is C ∗ -simple if and only if it admits no amenable URS. 

More recently, Bray and Kennedy proved that the reduced crossed product associated with a minimal action X is simple if and only if the set of points in X whose stabilizers are C ∗ -simple is comeager in X. Their proof relies on the theory of URSs, the existence of a minimal amenable URS, established by Le Boudec and Matte Bon, and a finiteness condition related to C ∗ -simplicity. 

In this talk, I will introduce the theory of uniformly recurrent subgroups and present a proof of the aforementioned result of Bray and Kennedy.

For subcritical singular SPDEs, solutions are typically exhibited as the limit of solutions of regularised and renormalised PDEs as the regularisation scale goes to 0. Existing general solution theories typically yield a very small exponent for the rate of convergence. This suggests the question of identifying the sharpest possible rate of convergence that can be achieved in some space of distributions. 

In this talk, I will discuss recent work with Máté Gerencsér and Yueh-Sheng Hsu in which we identify the sharp rate of convergence in the case of the KPZ equation.

Quantum cellular automata describe discrete-time dynamics of quantum lattice systems which have a causal light-cone. They can be mathematically characterized as the group of bounded-spread automorphisms of the quasi-local C*-algebra associated to the lattice system. Invertible phases are a class of topological phases of matter with no bulk anyon content, and are mathematically described by a class of states on the quasi-local algebra C*-algebra which have a stable inverse under stacking, which form a group. We will show that in any spatial dimension, if the local Hilbert spaces of the lattice system are infinite dimensional, the group of stable equivalence classes of quantum cellular automata and the group of invertible phases are isomorphic . We will then explain the construction of a large class of non-trivial examples on the 2D lattice from holomorphic conformal nets. This talk is geared toward operator algebraists and will require no knowledge of physics. Based on joint work Nikita Sopenko and Ryan Thorngren.

TBA

The classical Hochschild-Kostant-Rosenberg (HKR) theorem supplies an identification of Hochschild homology groups in terms of differential forms. In the setting of derived geometry, Ben Zvi and Nadler and Toën and Vezzosi supplied a geometric interpretation of the HKR theorem in terms of an equivalence between the derived loop space and the shifted tangent bundle of a scheme. There is however a nontrivial problem in globalising this equivalence from schemes to stacks, and it is well known that derived loops and shifted tangent bundle do not agree already on Deligne-Mumford stacks. Ben Zvi and Nadler provided an infinitesimal version that holds for Artin stacks ; on the other hand, Arinkin Caldararu and Hablicsek [ACH] provided a global formula for smooth global quotient Deligne-Mumford stacks. In joint work with Fu, Scherotzke and Sibilla we extended this global formula to general derived Deligne-Mumford stacks, establishing a coordinate-free version of the previous result of [ACH]. In this talk I will explain the main ideas as well as some more recent developments.

Models of opinion dynamics describe how opinions are shaped in various environments. While these models are able to replicate general opinion distributions observed in real-world scenarios, their capacity to align with data at the user level remains mostly untested. In this talk, I will demonstrate that the MultiState Voter model with Zealots (MSVZ) can capture individual opinions in a fine-grained Twitter dataset collected during the 2017 French presidential elections. I will first lay out the theoretical background, with formulas to evaluate opinion fluctuation at the node level. I will also present exact equations to compute the probability of discord between nodes, that is, the probability that any two nodes hold different opinions. In particular, I will highlight how discord levels between two nodes hinge heavily on the statistical dependency between their states, which can be seen as a topological property of the underlying network. I will briefly illustrate how discord levels fluctuate in synthetic examples. I will then focus on the Twitter dataset, consisting of 19K accounts with pre-identified political leanings. Fitting the MSVZ on the retweet network with accounts of political parties as zealots, I will reveal a strong correspondence between individual opinion distributions in the equilibrium state and ground-truth political leanings. Additionally, I will demonstrate how discord probabilities can accurately identify pairs of like-minded users. These results depend heavily on taking into account the directionality and/or the weight of edges. Overall, this contribution advances our theoretical knowledge of the Voter Model, emphasizes its ability to identify individual characteristics in real-world datasets, and advocates for further empirical evaluations of opinion dynamics models at the user level.

Given a complicated geometric object, we often want to simplify it without losing information about the structure we are interested in. In this talk, I will describe a method for doing this by combining a random sparse set with a deterministic one, gaining some of the advantages of both. We will then see how this idea can be used to show that torsion in the first homology of higher-rank lattices grows subexponentially, answering a question of Abért, Gelander, and Nikolov and confirming the degree-one case of several conjectures. We will also use the same principle to give a new general criterion for a property known as fixed price one, which applies to a broad class of groups.

The cutoff phenomenon is a sharp transition in the convergence of high-dimensional Markov chains to equilibrium: the total variation distance remains close to 1 for a long time and then rapidly decreases to almost 0 over a much shorter time window. It was initially discovered in the context of card shuffling by Diaconis and Shahshahani, and it has since then been observed in a variety of different models. In spite of its ubiquity, it is still largely unexplained, and most proofs are model-specific. In this talk, we discuss a general approach to establishing cutoff based on transport inequalities, and we illustrate it with a few examples. Based on joint work with Justin Salez.

Finite non-invertible symmetries have been extensively studied in both (1+1)-dimensional conformal field theories and quantum lattice systems. The kinematics of these settings look very different. We will show that nevertheless, for a class of rational CFT symmetries preserving a common rational chiral algebra, the continuum and lattice descriptions are exactly equivalent in a precise sense. The main idea is to encode symmetry through the inclusion of symmetry-invariant observables into the full observable algebra. After adjoining symmetry-trivial infinite-dimensional degrees of freedom to an anyon chain, these inclusions can be identified by an isomorphism of algebras that preserves locality up to a uniformly bounded enlargement of regions in both directions. Starting from familiar examples of CFT defects and anyon chains, we will explain this equivalence, the resulting classification of local symmetry structures by fusion categories, and its implications for transporting states and dynamics between the continuum and the lattice.

In seminal work, Harper showed that the partial sums of random multiplicative functions enjoy "better than square-root cancellation". This surprising phenomenon gives rise to many questions; for example, recent research has focused on whether certain subsums possess this property. In this talk, I will introduce smooth numbers, discuss their "anatomy", and relate this to recent progress on random multiplicative functions over smooth numbers. This is based on joint work with Max Wenqiang Xu.

Diffusion models and flow-based methods have achieved strong results in image generation, but often rely on costly iterative sampling. We introduce W-Flow, a framework that compresses a Wasserstein gradient flow into a one-step neural generator. By minimizing the Sinkhorn divergence, W-Flow transports a reference distribution toward the data distribution through efficient optimal-transport updates that capture global distributional discrepancies. We prove that, under suitable assumptions, the finite-sample training dynamics converge to the continuous-time distributional dynamics. Empirically, W-Flow sets a new state of the art in one-step ImageNet generation, with improved mode coverage and domain transfer. We will also discuss extensions using alternative energy functionals and applications to post-training.

In this talk we will discuss $G_2$-instantons and their relation to the heterotic $G_2$-system on 7-dimensional 2-step nilpotent Lie groups. We will first consider left-invariant coclosed $G_2$-structures and present necessary and sufficient conditions for their characteristic connection to be a $G_2$-instanton, expressed in terms of the torsion of the $G_2$-structure, the torsion of the connection, and the Lie group structure. We will then explain how these conditions imply that the corresponding left-invariant Riemannian metrics define naturally reductive structures.

As a continuation, we will consider solutions to the heterotic $G_2$-system on 2-step nilmanifolds. We will describe constructions of solutions with non-abelian gauge groups and discuss how the geometry of the underlying $G_2$-structure and the choice of gauge connection enter the equations. In particular, we will present families of solutions arising from the $G_2$-instantons described in the first part.

This is joint work with Andrew Clarke, Andrés J. Moreno, and Udhav Fowdar.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards.

We show that the group cohomology of simple connected Lie groups, and consequently their lattices and symmetric spaces, with L^p-coefficients vanishes below the rank. This confirms a conjecture of Gromov in the special case of the coefficient module L^p(G). We relate the vanishing of L^1-cohomology to a fixed-point theorem and confirm a conjecture of Farb. Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
In this talk, I will give a short introduction to group cohomology and discuss ideas from the proofs of both results. This talk is based on joint work with Saar Bader, Uri Bader, and Roman Sauer.

From contours of 2D shapes to knotted proteins, many areas of statistics and physical sciences are concerned with data in the form of geometric loops - embeddings of the circle into a Euclidean space. It is desirable to find a representation of the geometry of these loops that is invariant under Euclidean symmetries, and indifferent to how the loops are parametrised. In this joint work with Ka Man Yim (Oxford) and Otto Sumray (MPI Dresden), we propose a novel representation of a loop’s geometry by rep resenting the distance matrix of the loop as a Morse function on a Möbius band - the second finite subset space of the circle. By considering the persistent homology of this function, we obtain a representation of the loop’s geometry that has the desired symmetry invariance. We show that this feature map is stable with respect to perturbations, give a geometric interpretation of the features obtained and explore analogues involving higher order finite subset spaces.

In this talk, I will discuss the challenges of solving string theory through the two-dimensional quantum field theory on the string worldsheet. Focusing on the spectrum of superstrings in AdS3, I will explain how conformal field theory techniques yield exact results for special backgrounds, and how methods from integrable models allow us to go beyond these cases.

Mapping class groups arise in an elementary way in the study of surfaces, and their properties are used in many areas of group theory and low-dimensional topology. This talk will be an introduction to the study of mapping class groups, in particular via a geometric space known as the curve complex. This approach is classical and yields strong properties and a deep understanding of mapping class groups. In the second half of the talk, we will discuss a broadening of this construction to a family of groups known as Artin groups, whose properties are still very mysterious and are the subject of an active area of research.

Renewal equations are one of the most popular ways to model outbreaks of infectious diseases and estimate key control indicators for decision-makers. This talk will focus on new mathematical methods that address where traditional renewal equations often fail to capture the human, biological, and environmental complexities of epidemics. These methods cover both directly transmitted and vector-borne diseases, describe epidemics on complex human mobility networks, and enable more targeted interventions across space and time.

Control policies optimized in simulation can perform poorly in the real system when the simulator's parameters are estimated from limited data but the resulting parameter uncertainty is not represented inside the simulation. A common way to incorporate such ambiguity is to simulate each trajectory under a randomly drawn and unobservable parameter value. This forces the policy to act robustly at the start of the control episode. Over time, however, the policy can often infer the parameter value from its observations and specialize its actions accordingly, which is undesirable in systems where latent factors are expected to shift. In financial markets, for example, a policy hedging a derivative payoff should remain robust to changes in the volatility regime. To induce such continual robustness, we propose training policies in simulators where ambiguity can vary with the system's state but does not systematically vanish over time. We formalize this as stationary ambiguity: the simulator induces a stationary filter process over the latent state. We show how to construct such simulators and demonstrate, on hedging problems, that policies trained under stationary ambiguity maintain robustness to latent factors over time, leading to strong performance on real market data. As a modeling principle, stationary ambiguity informs which models make realistic simulators, how their parameters should be randomized, and how simulator and policy should be initialized. While our experiments focus on hedging, stationary ambiguity may also be useful for other control problems driven by exogenous stochastic processes with shifting latent structure.

This is a joint work with Amira Akkari, Ben Wood, and Lukas Gonon.

Let K be a degree d number field. A famous theorem of Faltings states that a "nice" curve C of genus at least 2 defined over K has only finitely many points defined over K. However, C can of course admit infinitely algebraic points of degree d. For example, a hyperelliptic curve defined over the rationals admits infinitely many quadratic points. In this talk, we will see many examples of curves which have infinitely many points of degree 2 or 3 and, moreover, we shall see the number fields arising from these points are relatively rare.

Professor Sara Shashaani is going to talk about: 'Adaptive Sampling and Regularization for Stochastic Trust-region Methods'

Trust-region methods have proven highly effective for unconstrained nonconvex stochastic optimization problems where objective and gradient information are available only through noisy stochastic oracles. ASTRO is a class of adaptive sampling trust-region methods that dynamically determine sampling effort while constructing local quadratic models from noisy function and gradient observations. By exploiting dependence among samples and the stochastic structure of the problem, ASTRO achieves strong convergence and complexity guarantees. Its derivative-free variant, ASTRO-DF, relies solely on noisy function evaluations and also enjoys almost-sure convergence guarantees.

Tim van Beeck (University of Göttingen) is going to talk about: "An Embedded Trefftz DG Approach to Least-Squares Finite Element Methods"

Least-squares finite element methods (LSFEM) minimise the residual of the PDE directly. This gives discrete problems that are stable by construction, have symmetric positive definite system matrices, and come with a built-in reliable and efficient error estimator. In this talk, we present an embedded Trefftz DG approach to LSFEM. Trefftz methods use basis functions that locally lie in the kernel of the differential operator, which greatly reduces the number of degrees of freedom. The embedded Trefftz DG method obtains these spaces through an element-local embedding into a standard DG space, so no explicit Trefftz basis is needed. We apply the method to a range of linear second-order problems, including Helmholtz-type equations and PDEs in non-divergence form, and show that it keeps the properties above under suitable mesh resolution conditions, while requiring far fewer degrees of freedom than standard DG or first-order least-squares formulations. We conclude with an outlook on nonlinear problems such as the Hamilton-Jacobi-Bellman equation.

Self-propelled particles such as swimming micro-organisms often accumulate at boundaries. Active Brownian particles (ABPs), that is, particles moving at constant speed with an orientation undergoing rotational diffusion, provide a simple model. I will discuss the distribution of ABPs inside a channel. 

Stallings core graphs are a powerful graphical representation of the finitely generated subgroups of a free group. They bring geometric insight to the study of free groups, make many properties and algorithmic tasks transparent, and are a ubiquitous tool in combinatorial and geometric group theory.
Free group algebras are strikingly analogous to free groups. Most notably, just as the Nielsen-Schreier theorem says that every subgroup of a free group is free, the Cohn-Lewin theorem says that every right (or left) ideal of a free group algebra is free. Yet free group algebras have had no counterpart to core graphs.
In this talk I will present one. We define the core of a module over a free group algebra, and show that it is unique whenever the module is algebraic (also known as bound). Many of the concepts of Stallings' theory, such as folding and pullbacks, generalize to these new objects, and this yields simple proofs and algorithms for many natural problems about free group algebras. For example, I will present an algorithm for testing primitivity in free modules which generalizes (Wilton's simplification of) Whitehead's classical algorithm for testing primitivity in free groups.
This is joint work with Doron Puder.
 

Scale and conformal symmetries play a central role in various areas of theoretical physics, ranging from statistical mechanics and quantum field theory to string theory. A natural question is under what conditions scale invariance implies the larger symmetry of conformal invariance.  In this talk, I will discuss non-perturbative arguments for this enhancement in two and four dimensions, highlighting the assumptions under which they hold. I will first explain how Zamolodchikov's $c$-theorem, under suitable assumptions, can be used to establish the enhancement of scale invariance to local conformal invariance in two dimensions. I will then turn to four dimensions and discuss the argument of Dymarsky, Komargodski, Schwimmer, and Theisen based on dilaton scattering amplitudes and unitarity. I will conclude by commenting on possible generalisations to other dimensions and, time permitting, recent progress on the question.