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.

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 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.

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.

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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.

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.

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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.

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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.

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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.

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.

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.

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)

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.  

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.

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.

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.

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

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. 

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