Past

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.

TBC

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.

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.

In this talk, I will introduce the notion of $n$-morphisms between two $A_\infty$-algebras. These higher morphisms are such that 0-morphisms correspond to standard $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies. Their combinatorics are encoded by new families of polytopes,  which I call the $n$-multiplihedra and which generalize the standard multiplihedra.
Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of $A_\infty$-algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.

Random unitary circuits (RUCs) have served as important sources of insights in studying operator dynamics. While the simplicity of RUCs allows us to understand the nature of operator growth in a quantitative way, randomness of the dynamics in time prevents them to capture certain aspects of operator dynamics. To explore these aspects, in this talk, I consider the operator dynamics of a minimal Floquet many-body circuit whose time-evolution operator is fixed at each time step. In particular, I compute the partial spectral form factor of the model and show that it displays nontrivial universal physics due to operator dynamics. I then discuss the out-of-ordered correlator of the system, which turns out to capture the main feature of it in a generic chaotic many-body system, even in the infinite on-site Hilbert space dimension limit.

To tie in with mental health awareness week, in this session we'll give a brief overview of the mental health support available through the department and university, followed by a panel discussion on how we can look after our mental health as in an academic setting. We're pleased that several of our department Mental Health First Aiders will be panellists - come along for hints and tips on maintaining good mental health and supporting your colleagues and friends.

Dimensionality reduction is the machine learning problem of taking a data set whose elements are described with potentially many features (e.g., the pixels in an image), and computing representations which are as economical as possible (i.e., with few coordinates). In this talk, I will present a framework to leverage the topological structure of data (measured via persistent cohomology) and construct low dimensional coordinates in classifying spaces consistent with the underlying data topology.

The Omicron BA.1 variant of SARS-CoV-2 was more transmissible and less severe than the preceding Delta variant, including in hosts without previous infection or vaccination.  To investigate why this was the case, we conducted in vitro replication experiments in human nasal and lung cells, then constructed and fitted ODE models of varying levels of complexity to the data, using Markov chain Monte Carlo methods.  Our results fitting a simple model suggest that the basic reproduction number and growth rate are higher for Omicron in nasal cells, and higher for Delta in lung cells. As growth in nasal cells is thought to correspond to transmissibility and growth in lung cells is thought to correspond to severity, these results are consistent with epidemiological and clinical observations.  We then fitted a more complex model, including different virus entry pathways and the immune response, to the data, to understand the mechanisms leading to higher infectivity for Omicron in nasal cells. This work paves the way for using within-host mathematical models to analyse experimental data and understand the transmission potential of future variants. 

While presenting the results of this study, I will use them to open a wider discussion on common problems in mathematical biology, such as the situations in which complex models are preferable to simpler models; when it is appropriate to fix model parameters; and how to present results which are contingent on unidentifiable parameters.

The Iwasawa main conjecture, first developed in the 1960s and later generalised to a modular forms setting, is the prediction that algebraic and analytic constructions of a p-adic L-function agree. This has applications towards the Birch—Swinnerton-Dyer conjecture and many similar problems. This was proved by Kato (’04) and Skinner—Urban (’06) for ordinary modular forms. Progress in the non-ordinary setting is much more recent, requiring tools from p-adic Hodge theory and rigid analytic geometry. I aim to give an overview of this and discuss a new approach in the setting of unitary groups where even more things go wrong.

The Zilber-Pink conjecture predicts that if V is a proper subvariety of an arithmetic variety S (e.g. abelian variety, Shimura variety, others) not contained in a proper special subvariety of V, then the “unlikely intersections” of V with the proper special subvarieties of S are not Zariski dense in V. In this talk I will present a strong counterpart to the Zilber-Pink conjecture, namely that under some natural conditions, likely intersections are in fact Euclidean dense in V.  This is joint work with Tom Scanlon.

The main object of study of the talk is the balanced triple product p-adic L-function; this is a p-adic L-function associated with a triple of families of (quaternionic) modular forms. The first instances of these functions appear in the works of Darmon-Lauder-Rotger, Hsieh, and Greenberg-Seveso. They have proved to be effective tools in studying cases of the p-adic equivariant Birch & Swinnerton-Dyer conjecture. With this aim in mind, we discuss the construction of a new p-adic L-function, extending Hsieh's construction, and allowing classical weight one modular forms in the chosen families. Such improvement does not come for free, as it coincides with the increased dimension of certain Hecke-eigenspaces of quaternionic modular forms with non-Eichler level structure; we discuss how to deal with the problems arising in this more general setting. One of the key ingredients of the construction is a p-adic extension of the Jacquet-Langlands correspondence addressing these more general quaternionic modular forms. This is joint work in progress with Aleksander Horawa.

On supercomputers that exist today, achieving even close to the peak performance is incredibly difficult if not impossible for many applications. Techniques designed to improve the performance of matrix computations - making computations less expensive by reorganizing an algorithm, making intentional approximations, and using lower precision - all introduce what we can generally call ``inexactness''. The questions to ask are then:

1. With all these various sources of inexactness involved, does a given algorithm still get close enough to the right answer?
2. Given a user constraint on required accuracy, how can we best exploit and balance different types of inexactness to improve performance?

Studying the combination of different sources of inexactness can thus reveal not only limitations, but also new opportunities for developing algorithms for matrix computations that are both fast and provably accurate. We present few recent results toward this goal, icluding mixed precision randomized decompositions and mixed precision sparse approximate inverse preconditioners.

We propose a unified theory of brain function called ‘Thermodynamics of Mind’ which provides a natural, parsimonious way to explain the underlying computational mechanisms. The theory uses tools from non-equilibrium thermodynamics to describe the hierarchical dynamics of brain states over time. Crucially, the theory combines correlative (model-free) measures with causal generative models to provide solid causal inference for the underlying brain mechanisms. The model-based framework is a powerful way to use regional neural dynamics within the hierarchical anatomical brain connectivity to understand the underlying mechanisms for shaping the temporal unfolding of whole-brain dynamics in brain states. As such this model-based framework fitted to empirical data can be exhaustively investigated to provide objectively strong causal evidence of the underlying brain mechanisms orchestrating brain states.