# Seminars

There are three series of semianrs: **Imperial pure analysis and PDE**, **London analysis and probability**, and **Paris-London analysis seminar**. The first two altenate weekly, and are listed in the green and blue boxes below, the Paris-London series meets four times per year.

**Our PhD students also jointly organise the Junior Analysis seminar; which consists of informal talks by students and visitors. **

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## 2018-2019 - Autumn term programme

♦ *Sandrine Grellier* (Orléans), **Generic colourful tori and inverse spectral transform for Hankel operators**

**5 October**, 10:30-11:20, UCL (Room 706), Paris-London Analysis seminar

♦ *Tom Korner* (Cambridge), **Can we characterise sets of strong uniqueness**

**5 October**, 11:30-12:20, UCL (Room 706), London Analysis and Probability seminar

♦ *Emmanuel Fricain* (Lille), **Multipliers between sub-Hardy Hilbert spaces**

**5 October**, 14:00-14:50, UCL (Room 706), Paris-London Analysis seminar

♦ Tom Sanders (Oxford),**The Erdös Moser sum-free set problem**

**5 October**, 15:20-16:10, UCL (Room 706), London Analysis and Probability seminar,

For abstracts of the talks please visit here

♠ Andrzej Zuk (CNRS- Paris), From PDEs to groups

**12 October**, 3:00-4:00, Imperial College London, Huxley 140), Pure analysis and PDE seminar**Abstract: **We present a construction which associates to a KdV equation the lamplighter group. At crucial steps of it appear automata and random walks on ultra discrete limits. It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy invariants of closed manifolds.

♣ *Jürg Fröhlich* (ETH), ** The Arrow of Time - Images of Irreversible Behavior**

**18 October**, 3:00-4:00, UCL (Room 706), London Analysis and Probability seminar,

**Abstract:**I sketch various examples of physical systems with time-reversal invariant dynamics exhibiting irreversible behavior. I start with deriving the Second Law of Thermodynamics in the formulation of Clausius from the existence of quantum-mechanical heat baths and then derive the Carnot bound for the degree of efficiency of heat engines. I continue with the analysis of a quantum-mechanical model with unitary time evolution describing a particle that exhibits diffusive motion when coupled to a suitably chosen (non-interacting) heat bath. A classical model with a Hamiltonian time evolution describing a particle coupled to a wave medium exhibiting friction is sketched next. I conclude with an attempt to draw the attention of the audience to the fact that the dynamics of isolated, open quantum systems featuring events is fundamentally irreversible.

♣ *Thomas Spencer* (IAS), **Edge reinforced random walk as a toy model of localization**

**18 October**, 4:30-5:30, UCL (Room 707), London Analysis and Probability seminar,

**Abstract: **I will present some results and conjectures about edge reinforced random walk (ERRW). This is a history dependent walk which favors edges it has visited in the past. In three dimensions the walk has a phase transition as the reinforcement is varied. The relation of ERRW to a toy model of quantum localization will also be discussed.

♠ *Jan Sbierski* (Oxford), **Uniqueness & non-uniqueness results for wave equations**

**25 October**, 3:00-4:00, Imperial College (Huxley 140), Pure analysis and PDE seminar

**Abstract:**A well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, time evolution of globally hyperbolic solutions is unique. This talk investigates whether the same result holds for quasilinear wave equations defined on a fixed background. After recalling the notion of global hyperbolicity, we first present an example of a quasilinear wave equation for which unique time evolution in fact fails and contrast this with the Einstein equations. We then proceed by presenting conditions on quasilinear wave equations which ensure uniqueness. This talk is based on joint work with Harvey Reall and Felicity Eperon.

♣ *Thierry Lévy* (Paris 6), **Quantum spanning forests**

**1 November**, 3:00-4:00, UCL (Room 706), London Analysis and Probability seminar,

**Abstract:** I will report on a work in progress with Adrien Kassel (ENS Lyon) about an extension of Kirchhoffâ€™s matrix-tree theorem and determinantal point processes, to the framework of vector bundles over graphs. While trying to understand in combinatorial terms the determinant of the covariant Laplacian on the space of sections of a vector bundle over a graph endowed with a connection, we were led to the definition of a family of probability measures on the Grassmannian of a Euclidean or Hermitian space, associated with an orthogonal splitting of this space and a self-adjoint contraction on it. This family of measures contains and extends the family of determinantal point processes.

♣ Thomas Bothner (KCL), **When J. Ginibre met E. Schrödinger**

**1 November**, 4:30-5:30, UCL (Room 706), London Analysis and Probability seminar,

**Abstract: **The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibe matrix follows a different limiting law for purely real eigenvalues than for non-real ones. Building on previous work by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1+1 dimensions which are solvable by the inverse scattering method, for instance the nonlinear Schro ̈dinger equation. The results of this talk are based on the recent preprint arXiv:1808.02419, joint with Jinho Baik.

♠ *Vedran Sohinger* (Warwick) **Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimension d <= 3**

**8 November**, 3:00-4:00, Imperial College (Huxley 140), Pure Analysis and PDE seminar

**Abstract:** Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. We consider bounded defocusing interaction potentials and work either on the d-dimensional torus or on R^d with a confining potential. The analogous problem for d=1 and in higher dimensions with smooth non translation-invariant interactions was previously studied by Lewin, Nam, and Rougerie by means of variational techniques.

In our work, we apply a perturbative expansion of the interaction, motivated by ideas from field theory. The terms of the expansion are analysed using a diagrammatic representation and their sum is controlled using Borel resummation techniques. When d=2,3, we apply a Wick ordering renormalisation procedure. Moreover, in the one-dimensional setting our methods allow us to obtain a microscopic derivation of time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

♠ *Edward Crane* (Bristol), **Circle Packing and Uniformizations**

**15 November**, 3:00-4:00, Imperial College (Huxley 140), Pure analysis and PDE seminar

**Abstract:** Koebe discovered his circle packing theorem in the 1930s as a limiting case of his uniformization theorem for multiply-connected plane domains. After Thurston had interpreted circle packing as a discretization of conformal structure, Rodin and Sullivan showed how one could deduce the Riemann mapping theorem as a limiting case of the circle packing theorem. I will explain conformal welding and its circle packing analogue. I will show how this technique can be used to approximate an unusual uniformization of multiply-connected domains, in which each complementary component is a disc in the hyperbolic metric associated to the complement of all the other complementary components.

♣ *Jonathan Bennett* (Birmingham) **The nonlinear Brascamp-Lieb inequality and applications**

**22 November**, 3:00-4:00, UCL (Room tba), London Analysis and Probability seminar,

**Abstract:** The Brascamp--Lieb inequality is a broad generalisation of many well-known multilinear inequalities in analysis, including the multilinear H\"older, Loomis--Whitney and sharp Young convolution inequalities. There is by now a rich theory surrounding this inequality, along with diverse applications in convex geometry, partial differential equations, number theory and beyond. Of particular importance is Lieb's Theorem (1990), which states that the best constant in this inequality is exhausted by centred gaussian functions. In this talk we present a recent "nonlinear" variant of the Brascamp--Lieb inequality, and describe some of its applications in harmonic analysis and PDE. A key ingredient in our proof is a certain effective version of Lieb's theorem, providing information about the shapes of gaussian near-extremisers for the classical Brascamp--Lieb inequality. This is joint work with Stefan Buschenhenke, Neal Bez, Michael Cowling and Taryn Flock.

♣ Herbert Koch (Bonn), A continuous family of conserved energies for the Gross-Pitaevskii equation,

**22 November**, 4:30-5:30, UCL (Room tba), London Analysis and Probability seminar,

**Abstract:** The Gross-Pitaevskii equation is the defocusing cubic nonlinear Schrödinger equation with the boundary conditions |u(t,x)| -> 1 at infinity. A difficulty in the study of the Gross-Pitaevskii equation is that the state space is nonlinear. In joint work with Xian Liao we study the equation in one space dimension, equip it with a new metric, and construct a continuous family of conserved energies.

♠ Benjamin Fahs (Imperial College), **title of the talk**

**29 November**, 3:00-4:00, Imperial College (Huxley 140), Pure analysis and PDE seminar

**Abstract:**

♣ *Horst Knörrer* (ETH) **Construction of oscillatory singular homogenuous space times**

**6 December**, 3:00-4:00, UCL (Room tba), London Analysis and Probability seminar,

**Abstract: **The vacuum Einstein equations for Bianchi space times (that is space times that can be foliated into three dimensional space like slices that are all homogenuous spaces) reduce to a system of ordinary differential equations. The conjectures of Belinskii, Khalatnikov and Lifshitz predict that for almost all initial data the solutions of these differential equation behave like trajectories of a billiard in a Farey triangle in the hyperbolic plane, that is, a triangle whose three vertices are ideal points. In joint work with M.Reiterer and E.Trubowitz we show that, for a set of initial data that has positive measure, this is indeed the case. We use ideas inspired by scattering theory for approximations of the system. The fact that billiard in a Farey triangle is chaotic leads us to small divisor problems similiar to those of KAM theory in Hamiltonian dynamics.

♣ *Tuomas Sahsten* (Manchester), **title TBA**

**6 December**, 4:30-5:30, UCL (Room tba), London Analysis and Probability seminar,

**Abstract:**

♠ Matthew Jacques (Open University), ** TBA **

**13 December **, 3:00-4:00, Imperial College London, (Huxley 140), Pure analysis and PDE seminar**Abstract:**

♠ Tom Claeys (Uni Louvain-la-Neuve), ** TBA **

**13 December **, 4:30-5:30, Imperial College London, (Huxley 140), Pure analysis and PDE seminar**Abstract:**

♠ Ivan Gentil (Lyon), ** Analytic point of view of the Schrödinger problem : a review on the subject. **

**20 December **, 3:00-4:00, Imperial College London, (Huxley 140), Pure analysis and PDE seminar**Abstract:**We are going to describe the Schrödinger problem as a minimisation of a cost along paths. This point of view allows us to simplify the problem and to see how the Schrödinger problem approaches the optimal transportation problem and also dual formulation. This is a joint work with C. Léonard and L. Ripani.