Past

Wireless communication systems with many antennas and users naturally give rise to large random channel matrices. In one-bit precoding, the transmitted signal is restricted to binary values and is designed using the channel matrix. It is then multiplied by the same channel matrix during transmission. This dependence, together with the entrywise sign nonlinearity, makes the performance difficult to characterize. Focusing on the standard i.i.d. Gaussian channel model, this talk studies the large-system behaviors of two classes of one-bit precoding schemes. For linear-quantized precoding, we use a recursive representation of Haar matrices, known as Householder Dice, to construct an asymptotic scalar signal-plus-independent-Gaussian-noise model.  We next consider nonlinear symbol-level precoding, focusing on a scheme based on convex relaxation followed by one-bit quantization. Using approximate message passing and state evolution, we characterize the limiting empirical laws of the relaxed solution and of the received signal–symbol pairs. These results provide explicit performance predictions and guidance for precoder design.

Any countable group G comes equipped with a canonical dynamical system, namely its conjugation action on its space of subgroups Sub(G). This 0-dimensional compact space is a central object in measured and geometric group theory, especially because it supports invariant and stationary random subgroups.

In general, describing this space is hard. In 2024, Carderi, Gaboriau, Le Maître, and Stalder initiated the study of the space of subgroups of non-amenable Baumslag-Solitar groups BS(m,n). They provided an explicit description of the perfect kernel of Sub(BS(m,n)). This is the largest closed subspace without isolated points, i.e. the space that remains after performing successive derivations that remove the isolated points.

In this talk, I will provide a complete classification of the spaces Sub(BS(m,n)) up to homeomorphism. I will prove that there exist exactly four homeomorphism types among the non-amenable ones. This relies on a detailed study of the Cantor-Bendixson erasing process, which depends on the arithmetical properties of the parameters m,n. This is based on a joint work with Damien Gaboriau, François Le Maître, and Yves Stalder.

Infinite-dimensional hyperbolic spaces serve as universal objects containing all finite-dimensional hyperbolic spaces at once and carry group actions with surprising properties. For example, the Mostow–Karpelevich theorem, which says that a semisimple Lie group acting on a symmetric space preserves a copy of its own symmetric space, fails there. Using kernels of hyperbolic type, Monod–Py and Monod obtain exotic actions on infinite-dimensional real and complex hyperbolic spaces. We explain these constructions and classify all such actions for any combination of real, complex and quaternionic hyperbolic spaces. The classification is partially based on joint work in progress with Bruno Duchesne and Gonzalo Ruiz Stolowicz.

The study of the maximum diameter of $d$-dimensional pseudomanifolds on $n$ vertices was initiated by Criado and Santos as an abstraction of the analogous problem for polytopes, in relation to the Polynomial Hirsch Conjecture.
A series of works by Santos, Criado, Newman and Bohman establish the asymptotics. Here we use a mixture of deterministic and random tools to 
determine the exact value for every large enough $n$ when $d=2$, and for a positive fraction of $n$ when $d\geq 3$.  

Our pseudomanifolds with maximum diameter crucially depend on a surprising connection to the cube of Euler trails in uniform hypergraphs. To this end, for every fixed uniformity $d$ and power $r \geq 2$, we show that satisfying the natural divisibility conditions implies the existence of the $r$th power of an Euler tour/trail in any large enough $d$-uniform hypergraph with large enough codegree.  The talk represents joint work with Stefan Glock, Olaf Parczyk, and Silas Rathke.

The local Langlands correspondence is a conjectural relation between the representations of a p-adic group and its L-parameters (which one can think of as representations of a `dual' group). Given a representation, it is natural to construct the hermitian dual representation. In this talk we present an explicit conjectural local Langlands correspondence, beyond the tempered case and including enhancements. We then show, assuming this conjecture, that constructing the hermitian dual corresponds to twisting its enhanced L-parameter by the compact real form on the dual group, generalising a result of Adams and Vogan. If time permits we will also show that standard desiderata of the local Langlands correspondence are in fact equivalent to this compatibility with hermitian duals.

The study of closed geodesics on hyperbolic surfaces is a subject that shares deep connections with many areas of mathematics. We survey classic and recent results in the subject, emphasizing the relations between three different perspectives: topology, geometry, and arithmetic. In particular, we discuss recently discovered connections with mapping class groups and Teichmüller dynamics. The talk assumes no previous knowledge on the subject.

In this talk, we study the numerical approximation of the stochastic heat equation on the sphere driven by an additive Lévy process. For this, we first prove new regularity results for the solution of the stochastic heat equation under different regularity assumptions on the initial value and driving Lévy process. In these settings, we perform a spectral approximation based on the truncation of the series expansion with respect to the real-valued spherical harmonic functions. Further, we apply a forward resp. backward Euler-Maruyama scheme for the temporal approximation. We prove strong and weak convergence rates for the introduced approximation scheme and present numerical simulations that confirm our theoretical results.

This is joint work with Annika Lang and Andrea Papini.

Hypertoric varieties are conical symplectic singularities equipped with a hamiltonian action of a torus of maximal possible dimension. They behave analogously to toric varieties in many ways, with the hamiltonian torus replacing the dense torus. Examples include: (crepant partial resolutions of) rational surface singularities of type A, the cotangent bundle of projective space, and Nakajima quiver varieties with the `all 1s' dimension vector.

Arbo and Proudfoot conjectured a combinatorial classification of hypertoric varieties by zonotopal tilings. In this talk I will discuss a proof of the conjecture.

Ian Osband from DeepMind is going to talk about: 'The Price of Learning'

Policy gradient sits at the heart of modern RL, including LLM post-training. Yet on image classification, where the gradient of accuracy is exact, it loses to cross-entropy, even on accuracy. Our explanation is that learning is a sequential resource allocation problem. Data, compute and capacity are scarce, and each update sets where the next one starts. Policy gradient ignores this and spends greedily. This talk presents simple methods that price learning, and shows how they improve robustness, compute efficiency and accuracy.

 

TBA

Multiparameter persistent homology (MPH) associates to a vector-valued filter function on a simplicial complex a multiparameter persistence module. In this talk, I will present recent joint work with Heather Harrington and Ulrike Tillmann on the inverse problem for MPH: given a persistence module, what can we say about the data that generated it? To study the geometry of the fibers of MPH, we equip both the space of filters and the moduli space of essentially finite persistence modules with natural stratifications that are preserved by the map. Over each stratum in its image, MPH restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We also bound the dimension of MPH fibers in terms of multigraded Betti numbers, recovering the one-parameter bound of Leygonie and Tillmann as a special case.

Come along for free pizza and to hear about the Mathematrix events this term.

Junior Algebra & Representation Theory starts with a week 1 social. Join us in the Quillen Room!

In this talk, we introduce deep learning schemes for forward–backward systems whose backward component is a backward stochastic Volterra integral equation (BSVIE). BSVIEs generalize classical BSDEs by introducing a second time variable, and arise naturally in recursive utilities with memory, time-inconsistent stochastic control, and dynamic risk measures.

Building on deep solvers for BSDEs, we develop backward learning schemes adapted to the two-time structure of BSVIEs. When the forward process is Markovian, the solution is given by deterministic functions of the two time variables and the forward state, which we approximate with neural networks. We prove the algorithm's convergence by decomposing the error into a time-discretization error and a neural network approximation error, and validate it numerically.

When the forward process is itself of Volterra type, the Markovian structure is lost, and the solution depends on the entire past trajectory. Using a path-dependent PDE representation, we approximate the solution with neural networks taking path signatures as inputs, and test the method on examples with explicit solutions.

Based on joint works with Nacira Agram, and with Nacira Agram and Mounir Zebbar.

Ruth Misener will talk about: 'Optimizing over graphs: Challenges, Formulations, and Applications'

Applications involving optimization over graphs include molecular design, graph neural network verification, neural architecture search, etc. This talk discusses formulating graph spaces using mixed-integer optimization and incorporating application-specific constraints. We discuss computational challenges with these mixed-integer optimization formulations and zoom in on the practical implications for these applications. We mention what has been done (by both ourselves and others) and what other research still needs to be done.

Co-authors: Shiqiang Zhang, Yilin Xie, Christopher Hojny, Juan Campos, Jixiang Qing, Christian Feldmann, David Walz, Frederik Sandfort, Miriam Mathea, Calvin Tsay

This talk is hosted by Rutherford Appleton Laboratory, Harwell Campus

I will present my recent results on global regularity for a canonical shell model of the Navier-Stokes equations, covering the unforced equation and certain classes of forcing. I will discuss the distinction between the forced and unforced regularity problems in this model, with the corresponding Navier-Stokes questions as motivation.

Internal gravity waves (IGWs) are oscillatory modes of motion that exist in the interior of a stably stratified fluid with gravity as the restoring force. In Earth's oceans, these waves play a fundamental role in the transport and mixing of energy, momentum, heat, carbon and nutrients. Much like ocean surface waves, IGWs become unstable and break when their amplitude is large and density interfaces become steeply inclined. This talk will explore the evolution and fate of such waves in two different scenarios.

I will first focus on IGWs that propagate into horizontally sheared currents, explaining how the fate of these waves depends both on their geometry and the strength of the shear flow, and why the total energy dissipated by ensuing turbulence is unexpectedly large. Second, I will discuss the dynamics of internal bores – long, large-amplitude solitary waves that propagate along sharp density interfaces. I will outline the mathematical links between internal bores and gravity currents and, by studying observations off the coast of California, demonstrate their relevance to understanding transport and dissipation on the continental shelf.

David Niederkofler (TU Wien) is going to talk about: "Optimal Time-Adaptivity for Parabolic Problems"

Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with Radau IIA methods of any order larger than one to obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error.

We show that if a group admits an action with no algebraicity, then all of its mixed identities are singular. This generalises a theorem of Abért showing such groups are lawless. Our result confirms a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups, and it also applies to several groups arising in topological contexts. I'll also discuss how ideas from independence relations in model theory arise in the proof and yield more general results, also applying to infinite-dimensional general (and projective) linear groups over arbitrary fields, recovering results of Bradford, Schneider, and Thom.

This is joint work with Michael Pinsker.

In this talk I will discuss homological and homotopical isoperimetric inequalities, including those of higher rank.  I will emphasise what is known in the non-positive curvature case.

The anyons of a (2+1)d topological order are described mathematically by a modular tensor category (MTC). Given an MTC, any n-component link becomes an n-tensor that stores some information about this MTC. For example, the 1-component unknot (twisted once) gives a vector "T" that stores the topological spins of all anyons; the 2-component Hopf link evaluates to matrix "S", from which the Verlinde formula recovers the fusion rules of the anyons. Do T and S store all the data? The answer is no: there are different MTCs with the same S and T. So the question becomes what other links to include in this list of tensors to tell MTCs apart? 
I will start by telling you how to turn any link you can drawn on a piece of paper (or a whiteboard) in to a tensor, then review the T and S story and then discuss some recent work where people considered adding the Whitehead link and the Borromean rings. There will be lots of drawings!

Kesten's theorem characterises amenability in terms of spectral radii: a symmetric random walk on a group G has spectral radius 1 if and only if it generates an amenable subgroup.

We prove a quantitative "1%" version of this result for G a free group, or a fully residually free group: for every ε > 0, there is δ > 0 such that for every probability measure μ on G with convolution operator norm at least ε, there is a co-set of an amenable subgroup of G of mass > δ under μ.

This answers a question of Emmanuel Breuillard. We believe the same result to hold in a broader setting of linear groups, with δ depending on the dimension. The proof uses uniform exactness and a generalisation of the Tits alternative.

Joint work with Emmanuel Breuillard.
 

Riemannian manifolds whose holonomy group lies inside the exceptional Lie group G_2 are called G_2-manifolds. These manifolds have several interesting properties (they are Ricci-flat) and are of interest in geometric analysis as well as in mathematical physics and other fields. In this seminar, we will give an introduction to the theory of G_2-manifolds and discuss an associated non-linear Laplacian-type operator whose kernel essentially determines whether a compact manifold is G_2. We will present uniqueness and existence results for the Poisson’s equation of this operator on homogeneous and cohomogeneity-one manifolds.

In 1989, Benjamin Weiss proved that for a countable group G, there exists a global minimal G-flow which is a topological model for every essentially free p.m.p. Borel G-action. In this talk, I will present our vast generalization, asserting that any essentially free Borel measured action (without the p.m.p. assumption), admits a minimal compact model. Moreover, we show that given a random walk on the group, there exists a global minimal topological model for all essentially free stationary Borel actions.

Based on joint work with Yair Hartman.

In their recent ICM address, Dan Halpern-Leistner and Jarod Alper posed the question: does the stack of canonically polarized curves admit a theta-stratification whose semistable locus is the moduli space of stable curves? In other, perhaps more familiar, language: is there some notion for curves that understands instability the way Harder-Narasimhan type does for vector bundles/coherent sheaves?

I will explain some joint work with Dave Swinarski that solves the analogous problem for certain rational curves, and then sketch out an argument that I think should solve the problem in general.

This talk presents a geometric machine learning framework for resolving algebraic decision problems, with a focus on the Word Problem and post-quantum cryptography. Unreduced words in the Baumslag-Solitar group BS(1,2) and Artin groups are mapped into graphs constructed by their specific generators and defining relations.

To construct a continuous representation of the underlying group, a Graph Neural Network is trained using contrastive triplets (W,P,N). In this setup, W represents a base word, P is a positive example (a word algebraically equivalent to W), and N acts as a negative decoy (a non-equivalent word). The network embeds these graphs into a 128-dimensional unit sphere, forcing algebraically identical elements to cluster together while separating distinct elements.

This learned geometric space offers direct computational solutions to classical group-theoretic problems. It is used to successfully launch a cryptanalytic attack on the Wagner-Magyarik cryptosystem. Furthermore, by directly linking the continuous embedding back to the discrete metric space of the group, a variant network architecture accurately predicts the reduced geodesic length of randomly generated sequences.

The framework is subsequently applied to Right-Angled Artin Groups (RAAGs) and random Artin groups to evaluate the Nielsen equivalence of subgroup generating sets. The presentation concludes by outlining how these structural graph representations can be adapted to estimate Gromov hyperbolicity and extract other coarse geometric invariants from infinite groups.
 

Conlon and Gowers in 2016 described a general approach to proving sparse random analogues of extremal results in combinatorics, such as bounding the minimum and maximum number of triangles in any subgraph of G(n,p) with a given number of edges. The general part of this approach is a functional-analytic statement which, given a sparse setting, constructs a 'dense model'. However there is a condition which must be shown to hold with high probability to apply the dense model theorem. In Conlon and Gowers' work, there is a technical difficulty with the probabilistic part which leads to a rather involved proof, which applies only in a restricted setting (for example, they can handle triangles but not triangles with a pendant edge), and with quite poor bounds on 'high probability'.

Around the same time Schacht, with a very different method, was able to prove a related result, which is applicable in a much more general setting and which has optimal bounds on 'high probability': but Schacht's result does not provide sharp lower bounds on the number of triangles, and does not provide any upper bounds. What it does do is identify the number of edges in a subgraph of G(n,p) which guarantee that triangles will appear; subsequently the 'hypergraph container method' provides another approach to proving this kind of result, but again does not provide sharp lower bounds or any upper bounds.

We revisit Conlon and Gowers' approach, and show how to avoid their technical problem, giving a simpler proof of their counting result which applies in the general setting and with optimal probability bounds. As a corollary, we prove the 'Counting KLR' theorem of Conlon, Gowers, Samotij and Schacht, but for general hypergraphs and with optimal probability bounds. This is joint work with Julia Boettcher, Joanna Lada and Domenico Mergoni.

I will describe a method that deforms analytic cycles to algebraic cycles. This will be done using global derived analytic geometry and analytic (Efimov) K-theory. I will conjecture how this is related to the Tate and Hodge conjectures. If time permits, I will speculate about a global arithmetic site. This is joint work in progress with Federico Bambozzi, Jack Kelly, and Devarshi Mukherjee.

Calibrated fibrations are expected to play an important role in exceptional holonomy, much as special Lagrangian fibrations do in the SYZ picture for Calabi--Yau manifolds. Singular fibres are expected to be essential, and a natural question is whether they are forced by the geometry. In his PhD thesis, Baraglia showed that coassociative fibrations of compact full-holonomy G_2-manifolds must have singular fibres. The analogous problem for Cayley fibrations remained open for many years and turns out to be substantially more difficult, with its resolution relying on deep results from 4-manifold topology.

The proof takes some unexpected twists and turns, involving a Diophantine equation arising from the Spin(7)-structure, families Seiberg--Witten theory and parametrized homotopy theory. After a short crash course on Spin(7) geometry, I will explain how these pieces fit together. This is joint work with Jianfeng Lin and Viktor Majewski.

Professor Lorenzo Pareschi is going to talk about; 'Bridging high-order numerics and machine learning for kinetic plasma simulation'

Reliable uncertainty quantification is a central challenge in kinetic plasma simulation, where high dimensionality, multiple physical scales, and sensitivity to uncertain inputs make repeated high-fidelity computations prohibitively expensive. This is particularly relevant in fusion-oriented applications, for which accurate predictions require sophisticated numerical solvers but direct sampling is often out of reach.

In this talk, I will present a multifidelity framework for the Vlasov–Poisson–Landau system designed to combine, rather than replace, high-order numerical simulation with machine learning. At the high-fidelity level, asymptotic-preserving and structure-aware solvers provide accurate kinetic descriptions across different regimes. These are coupled with reduced plasma models and tensor neural surrogates constructed through a micro–macro decomposition, so that the dominant physical structure is treated analytically and numerically, while learning is used only for the lower-complexity kinetic correction.  The resulting hierarchy produces inexpensive low-fidelity samples that remain strongly correlated with the high-fidelity kinetic solution.  When used as control variates, these models yield substantial variance reduction and computational savings while retaining the high-order solver as the reference description.

Beyond the specific plasma application, the main message is that classical numerical analysis and machine learning need not be competing approaches. High-order solvers can provide structure, reliability, and asymptotic consistency, while learned models provide efficient approximations that can be exploited within rigorous multifidelity estimators. This interaction offers a general route toward trustworthy machine learning for computational science.
 

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from (for example) kites and darts, squares and triangles, rhombi or shield-shaped tiles and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal (12-fold rotation symmetry) type, and many can be described in terms of square and equilateral triangular tiles. Here, we explore what contributes to the thermodynamic stability of soft-matter quasicrystals, both in two dimensions and in three, and how the details of how soft-matter particles interact leads to different kinds of aperiodic tilings. Although dodecagonal quasicrystals are the most common, this work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.

Erion Morina is going to talk about: 'Learning PDE-based models from data: An analysis-driven perspective on identifiability, consistency, and interpretability'

Learning governing equations from data is a central problem in scientific machine learning. Given noisy and incomplete observations of a physical process, the goal is to recover the underlying law. This is often approached by fitting a neural network or a dictionary of candidate terms to the observed dynamics. A good fit, however, does not determine identifiability of the law, its approximability by what is computed, physical consistency, or interpretability as a formula. These depend largely on how the learning problem is posed.

This talk formulates the learning task as a regularized inverse problem in function space and studies how these properties can be established. Over a parameterized class of candidate laws, one minimizes a model residual, a data misfit, and a regularizer, with the class and the regularizer as the design choices. With a sufficiently expressive class and suitable regularization, one obtains a regularization-based notion of identifiability, under which the regularization-minimizing law consistent with the data is unique. As the approximation scale grows and the regularization parameters are chosen accordingly, minimizers of the parameterized problem converge to that law. For classes carrying the structural constraints of the underlying physical model, the learned models are physically consistent and well-posed by construction at every finite approximation scale. For symbolic networks built from rational building blocks, the recovered law is a readable formula. The emphasis of this talk is analytical, and first numerical experiments illustrate the recovery in practice. 

Overall, the results show that identifiability, consistency, and interpretability are not competing objectives, but can be unified in one analysis-driven framework.