Past

Boundary layers control the transport of momentum, heat, solutes and other quantities between walls and the bulk of a flow. The Prandtl-Blasius boundary layer was the first quantitative example of a flow profile near a wall and could be derived by an asymptotic expansion of the Navier-Stokes equation. For higher flow speeds we have scaling arguments and models, but no derivation from the Navier-Stokes equation. The analysis of exact coherent structures in plane Couette flow reveals ingredients of such a more rigorous description of boundary layers. I will describe how exact coherent structures can be scaled to obtain self-similar structures on ever smaller scales as the Reynolds number increases.

A quasilinear approximation allows to combine the structures self-consistently to form boundary layers. Going beyond the quasilinear approximation will then open up new approaches for controlling and manipulating boundary layers.

Abstract: Akshay Venkatesh and his coauthors (Galatius, Harris, Prasanna) have recently introduced a derived Hecke algebra and a derived Galois deformation ring acting on the homology of an arithmetic group, say with p-adic coefficients. These actions account for the presence of the same system of eigenvalues simultaneously in various degrees. They have also formulated a conjecture describing a finer action of a motivic group which should preserve the rational structure $H^i(\Gamma,\Q)$. In this lecture we focus in the setting of classical modular forms of weight one, where the same systems of eigenvalues appear both in degree 0 and 1 of coherent cohomology of a modular curve, and the motivic group referred to above is generated by a Stark unit. In joint work with Darmon, Harris and Venkatesh, we exploit the Theta correspondence and higher Eisenstein elements to prove the conjecture for dihedral forms.

In many applications requiring the solution of a linear system Ax=b, the matrix A has been shown to have a low-rank property: its off-diagonal blocks have low numerical rank, i.e., they can be well approximated by matrices of small rank. Several matrix formats have been proposed to exploit this property depending on how the block partitioning of the matrix is computed.
In this talk, I will discuss the block low-rank (BLR) format, which partitions the matrix with a simple, flat 2D blocking. I will present the main characteristics of BLR matrices, in particular in terms of asymptotic complexity and parallel performance. I will then discuss some recent advances and ongoing research on BLR matrices: their multilevel extension, their use as preconditioners for iterative solvers, the error analysis of their factorization, and finally the use of fast matrix arithmetic to accelerate BLR matrix operations.

I will prove the 2d Biot-Savart law for the vorticity being an unbounded measure $\mu$, i.e. such that $\mu(\mathbb{R}^2)=\infty$, and show how can one infer some useful information concerning Kaden's spirals using it. Vorticities being unbounded measures appear naturally in the engineering literature as self-similar approximations of 2d Euler flows, see for instance Kaden's or Prandtl's spirals. Mathematicians are interested in such objects since they seem to be related to the questions of well-posedness of Delort's solutions of the 2d vortex sheet problem for the Euler equation. My talk is based on a common paper with K.Oleszkiewicz, M. Preisner and M. Szumanska.

Polycyclic groups either have polynomial growth, in which case they are virtually nilpotent, or exponential growth. I will give two interesting examples of "small" polycyclic groups which are extensions of $\mathbb{R}^2$ and the Heisenberg group by the integers, and attempt to justify the claim that they are small by sketching an argument that every exponential growth polycyclic group contains one of these.

Nets of lines are line arrangements satisfying very strict intersection conditions. We will see that nets can be defined in a very natural way in algebraic geometry, and, thanks to the strict intersection properties they satisfy, we will see that a lot can be said about classifying them over the complex numbers. Despite this, there are still basic unanswered questions about nets, which we will discuss. 
 

A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.

In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).

One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November).

All graduate students, post-docs faculty and visitors are welcome to come to the lectures. If you aren't a member of the CDT please email @email to confirm that you will be attending.

During the talk I will present some new computational technique based on excursion theory for Markov processes. Some new results for classical processes like Bessel processes and reflected Brownian Motion will be shown. The most important point of presented applications will be the new insight into Hartman-Watson (HW) distributions. It turns out that excursion theory will enable us to deduce the simple connections of HW with a hyperbolic cosine of Brownian Motion.

I will explain how complex varieties which have asymptotically large intersections with finite grids can be seen to correspond to projective geometries, exploiting ideas of Hrushovski. I will describe how this leads to a precise characterisation of such varieties. Time permitting, I will discuss consequences for generalised sum-product estimates and connections to diophantine problems. This is joint work with Emmanuel Breuillard.

The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

This work determines an aim point selection strategy for players in order to improve their chances of winning at the classic darts game of 501. Although many previous studies have considered the problem of aim point selection in order to maximise the expected score a player can achieve, few have considered the more general strategical question of minimising the expected number of turns required for a player to finish. By casting the problem as a Markov decision process, a framework is derived for the identification of the optimal aim point for a player in an arbitrary game scenario.

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the “large” sizes) are of the form $2^{k-1} + 2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot2^{k-6}$ . We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the “small” values is missing.

Polyfree groups are defined as groups having a series of normal
subgroups such that each sucessive quotient is free. This property
imples locally indicability and therefore also right orderability. Right
angled Artin groups are known to be polyfree (a result shown
independently by Duchamp-Krob, Howie and Hermiller-Sunic). Here we show
that Artin FC-groups for which all the defining relation are of even
type  are also polyfree. This is a joint work with Ruven Blasco and Luis
Paris.

The talk introduces the problem of completing a partially observed matrix whose columns obey a nonlinear structure. This is an extension of classical low-rank matrix completion where the structure is linear. Such matrices are in general full rank, but it is often possible to exhibit a low rank structure when the data is lifted to a higher dimensional space of features. The presence of a nonlinear lifting makes it impossible to write the problem using common low-rank matrix completion formulations. We investigate formulations as a nonconvex optimisation problem and optimisation on Riemannian manifolds.

In gas-liquid two-phase pipe flows, flow regime transition is associated with changes in the micro-scale geometry of the flow. In particular, the bubbly-slug transition is associated with the coalescence and break-up of bubbles in a turbulent pipe flow. We consider a sequence of models designed to facilitate an understanding of this process. The simplest such model is a classical coalescence model in one spatial dimension. This is formulated as a stochastic process involving nucleation and subsequent growth of ‘seeds’, which coalesce as they grow. We study the evolution of the bubble size distribution both analytically and numerically. We also present some ideas concerning ways in which the model can be extended to more realistic two- and three-dimensional geometries.

Quantum Yang-Mills theory is an important part of the Standard model built
by physicists to describe elementary particles and their interactions. One
approach to this theory consists in constructing a probability measure on an
infinite-dimensional space of connections on a principal bundle over
space-time. However, in the physically realistic 4-dimensional situation,
the construction of this measure is still an open mathematical problem. The
subject of this talk will be the physically less realistic 2-dimensional
situation, in which the construction of the measure is possible, and fairly
well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the
distribution of a stochastic process with values in a compact Lie group (for

example the unitary group U(N)) indexed by the set of continuous closed
curves with finite length on a compact surface (for example a disk, a sphere
or a torus) on which one can measure areas. It can be seen as a Brownian
motion (or a Brownian bridge) on the chosen compact Lie group indexed by
closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills
measure is constructed, and describe it without assuming any prior
familiarity with the subject. I will then present a set of results obtained
in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel,
Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N
tends to infinity of the Yang-Mills measure constructed with the unitary
group U(N).

Mechanical percolation is a phenomenon in materials processing wherein ‘filler’ rod-like particles are incorporated into polymeric materials to enhance the composite’s mechanical properties. Experiments have well-characterized a nonlinear phase transition from floppy to rigid behavior at a threshold filler concentration, but the underlying mechanism is not well understood. We develop and utilize an iterative graph compression algorithm to demonstrate that this experimental phenomenon coincides with the formation of a spatially extending set of mutually rigid rods (‘rigidity percolation’). First, we verify the efficacy of this method in two-dimensional fiber systems (intersecting line segments), then moving to the more interesting and mechanically representative problem of three-dimensional fiber systems (cylinders). We show that, when the fibers are uniformly distributed both spatially and orientationally, the onset of rigidity percolation appears to co-occur with a mean field prediction that is applicable across a wide range of aspect ratios.

A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.

In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).

One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November)

All graduate students, post-docs faculty and visitors are welcome to come to the lectures. If you aren't a member of the CDT please email @email to confirm that you will be attending.

Two curves in a closed hyperbolic surface of genus g are of the same type if they differ by a mapping class. Mirzakhani studied the number of curves of given type and of hyperbolic length bounded by L, showing that as L grows, it is asymptotic to a constant times L^{6g-6}. In this talk I will discuss a generalization of this result, allowing for other notions of length. For example, the same asymptotics hold if we put any (singular) Riemannian metric on the surface. The main ingredient in this generalization is to study measures on the space of geodesic currents.

Solutions to Rough Differential Equations (RDE) may be constructed by several means. Beyond the fixed point argument, several approaches rely on using approximations of solutions over short times (Davie, Friz & Victoir, Bailleul, ...). In this talk, we present a generic, unifying framework to consider approximations of flows, called almost flows, and flows through the non-linear sewing lemma. This framework unifies the approaches mentioned above and decouples the analytical part from the algebraic part (manipulation of iterated integrals) when studying RDE. Beyond this, flows are objects with their own properties.New results, such as existence of measurable flows when several solutions of the corresponding RDE exist, will also be presented.

From a joint work with Antoine Brault (U. Toulouse III, France).

In the context of randomly fluctuating surfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang and the Edwards-Wilkinson. Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively. Starting from a modification of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account. We will describe how it arises, briefly discuss its connections to KPZ and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. 

Introduced by Konno, hyperpolygon spaces are examples of Nakajima quiver varieties.  The simplest of these is a noncompact complex surface admitting the structure of a gravitational instanton, and therefore fits nicely into the Kronheimer-Nakajima classification of complete ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2).  For more general hyperpolygon spaces, we can speculate on how
this classification might be extended by studying the geometry of hyperpolygons at "infinity". This is ongoing work with Hartmut Weiss.

We are very excited to have another session with invited speakers joining us for the lunch next week. Annika Heckel, Frances Kirwan and Marc Lackenby will all be joining us for a panel discussion on balancing family with academia. All are welcome to join us and to ask questions. 

We hope to see many of you at the lunch - Monday 1-2pm Quillen Room (N3.12).

We test various conjectures on quantum gravity for general 6d string compactifications in the framework of F-theory. Starting with a gauge theory coupled to gravity, we first analyze the limit in Kähler moduli space where the gauge coupling tends to zero while gravity is kept dynamical. A key observation is made about the appearance of a tensionless string in such a limit. For a more quantitative analysis, we focus on a U(1) gauge symmetry and determine the elliptic genus of this string in terms of certain meromorphic weak Jacobi forms, of which modular properties allow us to determine the charge-to-mass ratios of certain string excitations. A tower of these asymptotically massless charged states are then confirmed to satisfy the (sub-)Lattice Weak Gravity Conjecture, the Completeness Conjecture, and the Swampland Distance Conjecture. If time permits, we interpret their charge-to-mass ratios in two a priori independent perspectives. All of this is then generalized to theories with multiple U(1)s.