Past

The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that this function often takes algebraic values that satisfy explicit reciprocity laws and that are related to derivatives of Hecke L-functions at s=0. Thus this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12^th problem. This is joint work with Nicolas Bergeron and Pierre Charollois.

Tensor decomposition express a tensor as a linear combination of elementary tensors. They have applications in chemometrics, computer science, machine learning, psychometrics, and signal processing. Their uniqueness properties render them suitable for data analysis tasks in which the elementary tensors are the quantities of interest. However, in applications, the idealized mathematical model is corrupted by measurement errors. For a robust interpretation of the data, it is therefore imperative to quantify how sensitive these elementary tensors are to perturbations of the whole tensor. I will give an overview of recent results on the condition number of tensor decompositions, established with my collaborators C. Beltran, P. Breiding, and N. Dewaele.

Metal forming involves permanently deforming metal into a required shape.  Many forms of metal forming are used in industry: rolling, stamping, pressing, drawing, etc; for example, 99% of steel produced globally is first rolled before any subsequent processing.  Most theoretical studies of metal forming use Finite Elements, which is not fast enough for real-time control of metal forming processes, and gives little extra insight.  As an example of how little is known, it is currently unknown whether a sheet of metal that is squashed between a large and a small roller should curve towards the larger roller, or towards the smaller roller.  In this talk, I will give a brief overview of metal forming, and then some progress my group have been making on some very simplified models of cold sheet rolling in particular.  The mathematics involved will include some modelling and asymptotics, multiple scales, and possibly a matrix Wiener-Hopf problem if time permits.

PhD_course_Roux_1.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

We will be joined by Philip Maini, Professor of Mathematical Biology and Ethnic Minorities Fellow at St John's College, to discuss his mathematical journey and experiences.

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades (subtrees) are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. This model turns out to have interesting properties. There is a canonical embedding into a continuous-time model ($\operatorname{CTCS}(n)$). There is an inductive construction of $\operatorname{CTCS}(n)$ as $n$ increases, analogous to the stick-breaking constructions of the uniform random tree and its limit continuum random tree. We study the heights of leaves and the limit fringe distribution relative to a random leaf. In addition to familiar probabilistic methods, there are analytic methods (developed by co-author Boris Pittel), based on explicit recurrences, which often give more precise results. So this model provides an interesting concrete setting in which to compare and contrast these methods. Many open problems remain.
Preprints at https://arxiv.org/abs/2302.05066 and https://arxiv.org/abs/2303.02529

More than 30 year ago Jerrum studied the planted clique problem and proved that under worst-case initialization Metropolis fails to recover planted cliques of size $\ll n^{1/2}$ in the Erdős-Rényi graph $G(n,1/2)$. This result is classically cited in the literature of the problem, as the "first evidence" that finding planted cliques of size much smaller than square root $n$ is "algorithmically hard". Cliques of size $\gg n^{1/2}$ are easy to find using simple algorithms. In a recent work we show that the Metropolis process actually fails to find planted cliques under worst-case initialization for cliques up to size almost linear in $n$. Thus the algorithm fails well beyond the $\sqrt{n}$ "conjectured algorithmic threshold". We also prove that, for a large parameter regime, that the Metropolis process fails also under "natural initialization". Our results resolve some open questions posed by Jerrum in 1992. Based on joint work with Zongchen Chen and Iias Zadik.

PhD_course_Roux_0.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

Generating n-tuples of a group G, or in other words epimorphisms Fₙ→G are usually studied up to the natural right action of Aut(Fₙ) on Epi(Fₙ,G); here Fₙ is the free group of n generators. The orbits are then called Nielsen classes. It is a classic result of Nielsen that for any n ≥ k there is exactly one Nielsen class of generating n-tuples of Fₖ. This result was generalized to surface groups by Louder.

In this talk the case of Fuchsian groups is discussed. It turns out that the situation is much more involved and interesting. While uniqueness does not hold in general one can show that each class is represented by some unique geometric object called an "almost orbifold covers". This can be thought of as a classification of Nielsen classes. This is joint work with Ederson Dutra.

TBA

A self-similar group is a group $G$ acting on a regular, infinite rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups, like Grigorchuk's 2-group of intermediate growth, are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims, and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk, we discuss a result of the speaker and Benjamin Steinberg characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

For a reductive group defined over a p-adic field, the wavefront set is an invariant of an admissible representations which roughly speaking measures the direction of the singularities of the character near the identity. Studied first by Roger Howe in the 70s, the wavefront set has important connections to Arthur packets, and has been the subject of thorough investigation in the intervening years. One of main lines of inquiry is to determine the relation between the wavefront set and the L-parameter of a representation. In this talk we present new results answering this question for unipotent representations with real infinitesimal character. The results are joint with Dan Ciubotaru and Lucas Mason-Brown.

General Relativity and Quantum Theory are the two main achievements of physics in the 20th century. Even though they have greatly enlarged the physical understanding of our universe, there are still situations which are completely inaccessible to us, most notably the Big Bang and the inside of black holes: These circumstances require a theory of Quantum Gravity — the unification of General Relativity with Quantum Theory. The most natural approach for that would be the application of the astonishingly successful methods of perturbative Quantum Field Theory to the graviton field, defined as the deviation of the metric with respect to a fixed background metric. Unfortunately, this approach seemed impossible due to the non-renormalizable nature of General Relativity. In this talk, I aim to give a pedagogical introduction to this topic, in particular to the Lagrange density, the Feynman graph expansion and the renormalization problem of their associated Feynman integrals. Finally, I will explain how this renormalization problem could be overcome by an infinite tower of gravitational Ward identities, as was established in my dissertation and the articles it is based upon, cf. arXiv:2210.17510 [hep-th].

We talk about the Hausdorff measure of level sets of the fields, say length of level lines of a planar field. Given two coupled stationary fields  $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using the Kac-Rice type formula as the main tool in the analysis. 

Reaction–diffusion equations are widely used to model spatial propagation, and constant-speed "traveling waves" play a central role in their dynamics. These waves are well understood in "essentially 1D" domains like cylinders, but much less is known about waves with noncompact transverse structure. In this direction, we will consider traveling waves of the KPP reaction–diffusion equation in the Dirichlet half-space. We will see that minimal-speed waves are unique (unlike faster waves) and exhibit curious asymptotics. The arguments rest on potential theory, the maximum principle, and a powerful connection with the probabilistic system known as branching Brownian motion.

This is joint work with Julien Berestycki, Yujin H. Kim, and Bastien Mallein.

It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.

Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over $\mathbb{R}P^3$ that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a  smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds. Joint work with Tristan Ozuch.

Accurately predicting turbulent fluid mechanics remains a significant challenge in engineering and applied science. Reynolds-Averaged Navier–Stokes (RANS) simulations and Large-Eddy Simulation (LES) are generally accurate, though non-Boussinesq turbulence and/or unresolved multiphysical phenomena can preclude predictive accuracy in certain regimes. In turbulent combustion, flame–turbulence interactions lead to inverse-cascade energy transfer, which violates the assumptions of many RANS and LES closures. We survey the regime dependence of these effects using a series of high-resolution Direct Numerical Simulations (DNS) of turbulent jet flames, from which an intermediate regime of heat-release effects, associated with the hypothesis of an “active cascade,” is apparent, with severe implications for physics- and data-driven closure models. We apply adjoint-based data assimilation method to augment the RANS and LES equations using trusted (though not necessarily high-fidelity) data. This uses a Python-native flow solver that leverages differentiable-programming techniques, automatic construction of adjoint equations, and solver-in-the-loop optimization. Applications to canonical turbulence, shock-dominated flows, aerodynamics, and flow control are presented, and opportunities for reacting flow modeling are discussed.

There has been recent advances in understanding the microscopic origin of the Bekenstein-Hawking entropy of supersymmetric ant de Sitter (AdS) black holes using holography and localization applied to the dual quantum field theory. In this talk, after a brief overview of the general picture, I will propose a BPS partition function -- based on gluing elementary objects called gravitational blocks -- for known AdS black holes with arbitrary rotation and generic magnetic and electric charges. I will then show that the attractor equations and the Bekenstein-Hawking entropy can be obtained from an extremization principle.

Come to this session to learn how to get started in consultancy from Dawn Gordon at Oxford University Innovation (OUI). After an introduction to what consultancy is, we'll explore case studies of consultancy work performed by mathematicians and statisticians within the university. This session will also include practical advice on how you can explore consultancy opportunities alongside your research work, from finding potential clients to the support that OUI can offer.

In this talk, I will present the notion of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). I will explain how the fibered barcode is a particular instance of projected barcodes and how the ISM and the SCD provide lower bounds for the convolution distance. 

Furthermore, in the case where the persistence module considered is the sublevel-sets persistence modules of a function f : X -> R^n, we will explain how, under mild conditions, the projected barcode of this module by a linear map u : R^n \to R is the collection of sublevel-sets barcodes of the composition uf . In particular, it can be computed using software dedicated to one-parameter persistence modules. This is joint work with Nicolas Berkouk.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves is special if and only if it contains a Zariski-dense set of special points. I'll discuss joint work with Gal Binyamini, Harry Schmidt, and Margaret Thomas in which we use pfaffian methods to obtain an effective uniform version of Manin-Mumford for products of CM elliptic curves. Using this we then prove an effective version of Habegger's result.

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems.  This is a report on joint work with Marco Sangiovanni Vincentelli.

Given a non-orientable Lagrangian surface L in a symplectic 4-manifold, how far
can the cohomology class of the symplectic form be deformed before there is no
longer a Lagrangian isotopic to L? I will properly introduce this and a
related question, both of which are less interesting for orientable
Lagrangians due to topological conditions. The majority of this talk will be
an exposition on Evans' 2020 work in which he solves this deformation
question for a particular Klein bottle. The proof employs the heavy machinery
of symplectic field theory and more classical pseudoholomorphic
curve theory to severely constrain the topology and intersection properties of
the limits of certain pseudoholomorphic curves under a process called
neck-stretching. The treatment of SFT-related material will be light and focus
mainly on how one can use the compactness theorem to prove interesting things.

Data-driven reduced-order modeling aims at constructing models describing the underlying dynamics of unknown systems from measurements. This has become an increasingly preeminent discipline in the last few years. It is an essential tool in situations when explicit models in the form of state space formulations are not available, yet abundant input/output data are, motivating the need for data-driven modeling. Depending on the underlying physics, dynamical systems can inherit differential structures leading to specific physical interpretations. In this work, we concentrate on systems that are described by differential equations and possess linear dynamics. Extensions to more complicated, nonlinear dynamics are also possible and will be briefly covered here if time permits.

The methods developed in our study use rational approximation based on Loewner matrices. Starting with the approach by Antoulas and Anderson in '86, and moving forward to the one by Mayo and Antoulas in '07, the Loewner framework (LF) has become an established methodology in the model reduction and reduced-order modeling community. It is a data-driven approach in the sense that what is needed to compute the reduced models is solely data, i.e., samples of the system's transfer function. As opposed to conventional intrusive methods that require an actual large-scale model to reduce (described by many differential equations), the LF only needs measurements in compressed format. In the former category of approaches, we mention balanced truncation (BT), arguably one of the most prevalent model reduction methods. Introduced in the early 80s, this method constructs reduced-order models (ROMs) by using balancing and truncating steps (with respect to classical system theory concepts such as controllability and observability). We show that BT can be reinterpreted as a data-driven approach, by using again the Loewner matrix as a central ingredient. By making use of quadrature approximations of certain system theoretical quantities (infinite Gramian matrices), a novel method called QuadBT (quadrature-based BT) is introduced by G., Gugercin, and Beattie in '22. We show parallels with the LF and, if time permits, certain recent extensions of QuadBT. Finally, all theoretical considerations are validated on various numerical test cases.

A growing plant is a fascinating system involving multiple fields. Biologically, it is a multi-cellular system controlled by bio-chemical networks. Physically, it is an example of an "active solid" whose element (cells) are active, performing mechanical work to drive the evolving geometry. Computationally, it is a distributed system, processing a multitude of local inputs into a coordinated developmental response. In this talk I will discuss how plants, a living information-processing organism, uses physical laws and biological mechanisms to alter its own shape, and negotiate its environment. Here I will focus on two examples reflecting the computational and mechanical aspects: (i) probing temporal integration in gravitropic responses reveals plants sum and subtract signals, (ii) the interplay between active growth-driven processes and passive mechanics.

A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, and show that the class group with planar Cayley graphs is QI-rigid.

Mathematics has always been part of the fabric of culture. References to mathematics in literature go back at least as far as Aristophanes, and encompass everyone from Dostoevsky to Oscar Wilde. In this talk I’ll explore some of the ways that literature has engaged with mathematical ideas, from the 17^th and 18^th century obsession with the cycloid (the “Helen of Geometry”) to the 19^th century love of the fourth dimension.

Spectral properties of Schrödinger operators are studied a lot in mathematical physics. They can give the description of trapped fermionic particles. This presentation will focus on the non-interacting case. I will explain why it is relevant to estimate L^p bounds of orthonormal families of eigenfuntions at the semiclassical regime and then, give the main ideas of the proof.

Lie groupoids are closely connected to pseudo-differential calculus. On a vector bundle considered as a `commutative Lie groupoid' (i.e. as a family of commutative Lie groups), they can be treated using the Fourier transform. In this talk, we explore the extension of this idea to the noncommutative space by employing the tubular neighborhood construction and subsequently adopting a global approach through the introduction of deformation to the normal cone (groupoid). By utilizing this groupoid, we can construct the analytic index of pseudo-differential operators without relying on pseudo-differential calculus.

Furthermore, through the canonical construction of the space of functions with Schwartz decay, pseudo-differential operators on a manifold can be represented as an integral associated with smooth functions on the deformation to the normal cone. This perspective provides a geometric characterization that allows for the direct proof of fundamental properties of pseudo-differential operators.

COURSE_PROPOSAL (12)_2.pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

The study of the fundamental group of an affine manifold has a long history that goes back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact Euclidian affine manifold has a subgroup of a finite index such that every element of this subgroup is translation. The motivation was the study of the symmetry groups of crys- talline structures which are of fundamental importance in the science of crystallography. A natural way to generalize the classical problem is to broaden the class of allowed mo- tions and consider groups of affine transformations. In 1964, L. Auslander in his paper ”The structure of complete locally affine manifolds” stated the following conjecture, now known as the Auslander conjecture: The fundamental group of a compact complete locally flat affine manifold is virtually solvable.

In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”, J.Milnor asked if a free group can be the fundamental group of complete affine flat mani- fold.
The purpose of the talk is to recall the old and to talk about new results, methods and conjectures which are important in the light of these questions .

The talk is aimed at a wide audience and all notions will be explained 1

The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.

Magnant and Martin conjectured that every $d$-regular graph on $n$ vertices can be covered by $n/(d+1)$ vertex-disjoint paths. Gruslys and Letzter verified this conjecture in the dense case, even for cycles rather than paths. We prove the analogous result for directed graphs and oriented graphs, that is, for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles. Moreover if $G$ is an oriented graph, then $n/(2d+1)$ cycles suffice. This also establishes Jackson's long standing conjecture for large $n$ that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian.
This is joint work with Viresh Patel and  Mehmet Akif Yıldız.

TBC

Let K be an algebraically closed field. Given three elements a Lie algebra over K, we say that these elements form an sl_2-triple if they generate a subalgebra which is a homomorphic image of sl_2(K). In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of sl_2-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic p, and discuss results for both the classical and exceptional Lie algebras. 

We describe a network model for the progression of Alzheimer's disease based on the underlying relationship to toxic proteins. From human patient data we construct a network of a typical brain, and simulate the concentration and build-up of toxic proteins, as well as the clearance, using reaction--diffusion equations. Our results suggest clearance plays an important role in delaying the onset of Alzheimer's disease, and provide a theoretical framework for the growing body of clinical results.

The usual approach to study 2d CFT relies on the Virasoro algebra and its representation theory. Moving away from the criticality, this infinite dimensional symmetry is lost so it is useful to have a look at 2d CFTs from the point of view of more general framework of quantum integrability. Every 2d conformal field theory has a natural infinite dimensional family of commuting higher spin conserved quantities that can be constructed out of Virasoro generators. Perhaps surprisingly two different sets of Bethe ansatz equations are known that diagonalise these. The first one is of Gaudin/Calogero type and was discovered by Bazhanov–Lukyanov–Zamolodchikov in the context of ODE/IM correspondence. The second set is a very natural generalisation of the Bethe ansatz for the Heisenberg XXX spin chain and was found more recently by Litvinov. I will discuss these constructions as well as their relation to W-algebras and the affine Yangian.

We build a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both environmentally concerned and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors' allocations. We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of environmentally concerned investors on the market decarbonization dynamics and share prices. We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions. However, if future climate policies are uncertain, even a large fraction of environmentally concerned investors is unable to bring down the emission curve: clear and predictable climate policies are an essential ingredient to allow green investors to decarbonize the economy.

Joint work with Pierre Lavigne

PhD_course_Roux.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.

The subject of “geometric” fluid dynamics flourished following the seminal work of VI.
Arnold in the 1960s. A famous paper was published in 1970 by David Ebin and Jerrold
Marsden, who used the manifold structure of certain groups of diffeomorphisms to obtain
sharp existence and uniqueness results for the classical equations of fluid dynamics. Of
particular importance are the fixed points of the underlying dynamical system and the
“accessibility” of these Euler equilibria. In 1985 Keith Moffatt introduced a mechanism
for reaching these equilibria not through the Euler vortex dynamics itself but via a
topology-preserving diffusion process called “Magnetic Relaxation”. In this talk, we will
discuss some recent results for Moffatt’s MR equations which are mathematically
challenging not only because they are active vector equations but also because they have
a cubic nonlinearity.

This is joint work with Rajendra Beckie, Adam Larios, and Vlad Vicol.

Beginning with the foundational work of Daniel Quillen, an understanding of aspects of the cohomology of finite groups evolved into a study of representations of finite groups using geometric methods of support theory. Over decades, this approach expanded to the study of representations of a vast array of finite dimensional Hopf algebras. I will discuss how related geometric and categorical techniques can be applied to linear algebra groups.