Past

Courserequirements: Basicmathematicalanalysis.

Examination and grading: The exam will consist in the presentation of some previously as- signed article or book chapter (of course the student must show a good knowledge of those issues taught during the course which are connected with the presentation.).

SSD: MAT/05 Mathematical Analysis
Aim: to make students aware of smooth and non-smooth controllability results and of some

applications in various fields of Mathematics and of technology as well.

Course contents:

Vector fields are basic ingredients in many classical issues of Mathematical Analysis and its applications, including Dynamical Systems, Control Theory, and PDE’s. Loosely speaking, controllability is the study of the points that can be reached from a given initial point through concatenations of trajectories of vector fields belonging to a given family. Classical results will be stated and proved, using coordinates but also underlying possible chart-independent interpretation. We will also discuss the non smooth case, including some issues which involve Lie brackets of nonsmooth vector vector fields, a subject of relatively recent interest.

Bibliography: Lecture notes written by the teacher.

In this talk, we give an overview of recent results on the fluctuation of the statistic $\sum_{i\neq j} f(L_N(\theta_i-\theta_j))$ for the Circular Beta Ensemble in the global, mesoscopic and local regimes. This work is morally related to Johansson's 1988 CLT for the linear statistic $\sum_i f(\theta_i)$ and Lambert's subsequent 2019 extension to the mesoscopic regime. The special case of the CUE ($\beta=2$) in the local regime $L_N=N$ is motivated by Montgomery's study of pair correlations of the rescaled zeros of the Riemann zeta function. Our techniques are of combinatorial nature for the CUE and analytical for $\beta\neq2$.

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nyström method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them. 

Data that have an intrinsic network structure are becoming increasingly common in various scientific applications. Compressing such data for storage or transmission is an important problem, especially since networks are increasingly large. From an information theoretic perspective, the limit to compression of a random graph is given by the Shannon entropy of its distribution. A relevant question is how much of the information content of a random graph pertains to its structure (i.e., the unlabelled version of the graph), and how much of it is contained in the labels attached to the structure. Furthermore, in applications in which one is interested only in structural properties of a graph (e.g., node degrees, connectedness, frequency of occurrence of certain motifs), the node labels are irrelevant, such that only the structure of the graph needs to be compressed, leading to a more compact representation. In this talk, I will consider the random geometric graph (RGG), where pairs of nodes are connected based on the distance between them in some latent space. This model captures well important characteristics of biological systems, information networks, social networks, or economic networks. Since determination of the entropy is extremely difficult for this model, I will present upper bounds we obtained for the entropy of the labelled RGG. Then, we will focus on the structural information in the one-dimensional RGG. I will show our latest results in terms of the number of structures in the considered model and bounds on the structural entropy, together with the asymptotic behaviour of the bounds for different regimes of the connection range. Finally, I will also present a simple encoding scheme for one-dimensional RGG structures that asymptotically achieves the obtained upper limit on the structural entropy.

arXiv link: https://arxiv.org/abs/2107.13495

Mass transfer in multicomponent systems occurs through convection and diffusion. For a viscous Newtonian flow, convection may be modelled using the Navier–Stokes equations, whereas the diffusion of multiple species within a common phase may be described by the generalised Onsager–Stefan–Maxwell equations. In this talk we present a novel finite element formulation which fully couples convection and diffusion with these equations. In the regime of vanishing Reynolds number, we use the principles of linear irreversible dynamics to formulate a saddle point system which leads to a stable formulation and a convergent discretisation. The wide scope of applications for this novel numerical method is illustrated by considering transport of oxygen through the lungs, gas separation processes, mixing of water and methanol and salt transport in electrolytes.

Independent sets in bipartite regular graphs have been studied extensively in combinatorics, probability, computer science and more. The problem of counting independent sets is particularly interesting in the d-dimensional hypercube $\{0,1\}^d$, motivated by the lattice gas hardcore model from statistical physics. Independent sets also turn out to be very interesting in the context of random graphs.

The number of independent sets in the hypercube $\{0,1\}^d$ was estimated precisely by Korshunov and Sapozhenko in the 1980s and recently refined by Jenssen and Perkins.

In this talk we will discuss new results on the number of independent sets in a random subgraph of the hypercube. The results extend to the hardcore model and rely on an analysis of the antiferromagnetic Ising model on the hypercube.

This talk is based on joint work with Yinon Spinka.

It the talk, various conditions of local regularity of axisymmetric suitable weak solutions, including the so-called slightly supercritical ones, will be discussed.

The moments of Hardy’s function have been of interest to number theorists since the early 20th century, and to random matrix theorists especially since the seminal work of Keating and Snaith, who were able to conjecture the leading order behaviour of all moments. Studying joint moments offers a unified approach to both moments and derivative moments. In his 2006 thesis, Hughes made a version of the Keating-Snaith conjecture for joint moments of Hardy’s function. Since then, people have been calculating the joint moments on the random matrix side. I will outline some recent progress in these calculations. This is joint work with Theo Assiotis, Benjamin Bedert, and Mustafa Alper Gunes.

Modern atmospheric and ocean science require sophisticated geophysical fluid dynamics models. Among them, stochastic partial

differential equations (SPDEs) have become increasingly relevant. The stochasticity in such models can account for the effect

of the unresolved scales (stochastic parametrizations), model uncertainty, unspecified boundary condition, etc. Whilst there is an

extensive SPDE literature, most of it covers models with unrealistic noise terms, making them un-applicable to

geophysical fluid dynamics modelling. There are nevertheless notable exceptions: a number of individual SPDEs with specific forms

and noise structure have been introduced and analysed, each of which with bespoke methodology and painstakingly hard arguments.

In this talk I will present a criterion for the existence of a unique maximal strong solution for nonlinear SPDEs. The work

is inspired by the abstract criterion of Kato and Lai [1984] valid for nonlinear PDEs. The criterion is designed to fit viscous fluid

dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in Holm [2015]. As an immediate application, I show that 

the incompressible SALT 3D Navier-Stokes equation on a bounded domain has a unique maximal solution.

This is joint work with Oana Lang, Daniel Goodair and Romeo Mensah and it is partially supported by European Research Council (ERC)

Synergy project Stochastic Transport in the Upper Ocean Dynamics (https://www.imperial.ac.uk/ocean-dynamics-synergy/

Let $G$ be a compact connected Lie group and $k \in H^4(BG,\mathbb{Z})$ a cohomology class. The String 2-group $G_k$ is the central extension of $G$ by the smooth 2-group $BU(1)$ classified by $k$. It has a close relationship to the level $k$ extension of the loop group $LG$.
We will introduce smooth 2-groups and the associated notion of centre. We then compute this centre for the String 2-groups, leveraging the power of maximal tori familiar from classical Lie theory.
The centre turns out to recover the invertible positive energy representations of $LG$ at level $k$ (as long as we exclude factors of $E_8$ at level 2).

 

When a reductive group G acts on a complex projective variety
X, there exist different methods for finding an open G-invariant subset
of X with a geometric quotient (the 'stable locus'), which is a
quasi-projective variety and has a projective completion X//G. Mumford's
geometric invariant theory (GIT) developed in the 1960s provides one way
to do this, given a lift of the action to an ample line bundle on X,
though with no guarantee that the stable locus is not empty. An
alternative approach due to Kapranov and others in the 1990s is to use
Chow varieties to define a 'Chow quotient' X//G. The aim of this talk is
to review the relationship between these constructions for reductive
groups, and to discuss the situation when G is not reductive.

There have been several proposals of scale-separated AdS vacua in the literature. All known examples arise from the effective field theory of flux compactifications with low supersymmetry, and there are often doubts about their consistency as 10 or 11d backgrounds in string theory. These issues can often be tackled in the bulk theory, or by analysis of the dual CFT via holography. I will review the most common issues, and focus the analysis on the recently constructed family of 3d scale-separated AdS vacua, which is dual to a two-dimensional CFT, emphasizing the discrete symmetry structure of the model in comparison to DGKT. Finally, I will comment on the tantalizing observation of integer operator dimensions in DGKT-like vacua, and comment on possible places to look for consistency issues in these models.

This event will be hybrid and will take place in L1 and on Teams. A link will be available 30 minutes before the session begins.

Pascal Heid
Title: Adaptive iterative linearised Galerkin methods for nonlinear PDEs

Abstract: A wide variety of iterative methods for the solution of nonlinear equations exist. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable linear preconditioning operator to the original nonlinear equation. Based on this observation, we will derive an abstract linearisation framework which recovers some prominent iteration schemes. Subsequently, in order to cast this unified iteration procedure into a computational scheme, we will consider the discretisation by means of finite dimensional subspaces. We may then obtain an effective numerical algorithm by an instantaneous interplay of the iterative linearisation and an (optimally convergent) adaptive discretisation method. This will be demonstrated by a numerical experiment for a quasilinear elliptic PDE in divergence form.   

Ilyas Khan
Title: Geometric Analysis: Curvature and Applications

Abstract: Often, one will want to find a geometric structure on some given manifold satisfying certain properties. For example, one might want to find a minimal embedding of one manifold into another, or a metric on a manifold with constant scalar curvature, to name some well known examples of this sort of problem. In general, these problems can be seen as equivalent to solving a system of PDEs: differential relations on coordinate patches that can be assembled compatibly over the whole manifold to give a globally defined geometric equation.

In this talk, we will present the theories of minimal surfaces and mean curvature flow as representative examples of the techniques and philosophy that geometric analysis employs to solve problems in geometry of the aforementioned type. The description of the theory will be accompanied by a number of examples and applications to other fields, including physics, topology, and dynamics. 

Recently, it has been more clearly understood that the N=4 superconformal index leads to the microstate counting of the BPS black holes in AdS_5 X S^5. The leading N^2 behavior of the free energy was shown in various ways to match that of the known BPS black hole in the gravity side, but this correspondence has not been realized at the level of the saddle point analysis of the full matrix model for the N=4 index. Here, I will try to clarify how such saddles corresponding to the BPS black holes arise as 'areal' distributions. The talk is based on https://arxiv.org/abs/2111.10720 with Sunjin Choi, Seok Kim, and Eunwoo Lee; https://arxiv.org/abs/2103.01401 with Sunjin Choi and Seok Kim.

We use Cech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain nine homology and six homotopy theories of closure spaces. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Cech complexes to closure spaces and prove that their persistent homology is stable.

This is joint work with Nikola Milicevic.

Stably density stratified shear flows arise widely in geophysical settings. Instabilities of these flows occur on scales that are too small to be directly resolved in numerical simulations, e.g., of the oceans and atmosphere, yet drive diabatic mixing events that often exert a controlling influence on much larger-scale processes. In the limit of strong stratification, the flows are characterized by the emergence of highly anisotropic layer-like structures with much larger horizontal than vertical scales. Owing to their relative horizontal motion, these structures are susceptible to stratified shear instabilities that drive spectrally non-local energy transfers. To efficiently describe the dynamics of this ``layered anisotropic stratified turbulence'' regime, a multiple-scales asymptotic analysis of the non-rotating Boussinesq equations is performed. The resulting asymptotically-reduced equations are shown to have a generalized quasi-linear (GQL) form that captures the essential physics of strongly stratified shear turbulence. The model is used to investigate the mixing efficiency of certain exact coherent states (ECS) arising in strongly stratified Kolmogorov flow. The ECS are computed using a new methodology for numerically integrating slow--fast GQL systems that obviates the need to explicitly resolve the fast dynamics associated with the stratified shear instabilities by exploiting an emergent marginal stability constraint.

In this talk, I will talk about the category of coherent sheaves on the affine Flag variety of a simply-connected reductive group over $\mathbb{C}$. In particular, I'll explain how the convolution product naturally leads to a construction of the Iwahori-Hecke algebra, and present some combinatorics related to computing duals in this category.

In collective navigation a population travels as a group from an origin to a destination. Famous examples include the migrations of birds and whales, between winter and summer grounds, but collective movements also extend down to microorganisms and cell populations. Collective navigation is believed to improve the efficiency of migration, for example through the presence of more knowledgeable individuals that guide naive members ("leader-follower behaviour") or through the averaging out of individual uncertainty ("many wrongs"). In this talk I will describe both individual and continuous approaches for modelling collective navigation. We investigate the point at which group information becomes beneficial to migration and how it can help a population navigate through areas with poor guidance information. We also explore the effectiveness of different modes through which a leader can herd a group of naïve followers. As an application we will consider the impact of noise pollution on the migration of whales through the North Sea.

Let X be a smooth projective curve over a finite field equipped with an action of a finite group G. I’ll first briefly introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result will be a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic if the projection X → X/G is at most weakly ramified. This generalises a theorem of Chinburg for the tamely ramified case. I’ll end with some speculations in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and with Adriano Marmora.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Polynomial preconditioning for linear solvers is well-known but not frequently used in current scientific applications.  Furthermore, polynomial preconditioning has long been touted as well-suited for parallel computing; does this claim still hold in our new world of GPU-dominated machines?  We give details of the GMRES polynomial preconditioner and discuss its simple implementation, its properties such as eigenvalue remapping, and choices such as the starting vector and added roots.  We compare polynomial preconditioned GMRES to related methods such as FGMRES and full GMRES without restarting. We conclude with initial evaluations of the polynomial preconditioner for parallel and GPU computing, further discussing how polynomial preconditioning can become useful to real-word applications.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

This is a preparatory study for our ultimate goal of understanding the various instabilities associated with an electrodes-coated dielectric membrane that is subject to mechanical stretching and electric loading. Leaving out electric loading for the moment, we consider bifurcations from the homogeneous solution of a circular or square hyperelastic sheet that is subjected to equibiaxial stretching under either force- or displacement-controlled edge conditions. We derive the condition for axisymmetric necking and show, for the class of strain-energy functions considered, that the critical stretch for necking is greater than the critical stretch for the Treloar-Kearsley (TK) instability and less than the critical stretch for the limiting-point instability. Abaqus simulations are conducted to verify the bifurcation conditions and the expectation that the TK instability should occur first under force control, but when the edge displacement is controlled the TK instability is suppressed, and it is the necking instability that will be observed. It is also demonstrated that axisymmetric necking follows a growth/propagation process typical of all such localization problems.