# Seminars

There are three series of seminars: **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 - Spring term programme

♠ Anders Hansen(Cambridge), On non-computable problems in computer assisted proofs - Why foundations of computations may interest pure mathematicians

**Thursday 10 January**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar**Abstract: **Computer assisted proofs have become increasingly popular over the last decades and turn out to be instrumental when proving many long standing conjectures. The recent computer assisted proof, led by T. Hales, of the more than four century old conjecture of Kepler (Hilbert’s 18th problem) on optimal packing of 3-spheres, is a striking example. Another fascinating case is the computer assisted proof of the Dirac-Schwinger conjecture, by C. Fefferman and L. Seco, on the asymptotic behaviour of the ground-state energy of certain Schrodinger operators. What may be surprising is that the computational problems used in these proofs are non-computable according to Turing. In this talk we will discuss this paradoxical phenomenon: Not only can non-computable problems be used in computer assisted proofs, they are crucial for proving important conjectures. A key tool for understanding this phenomenon is the Solvability Complexity Index (SCI) hierarchy, which allows for a classification theory for all types of computational problems. This classification theory may be of use to pure mathematicians for determining which computational problems that may be used in computer assisted proofs. In particular, there are non-computable problems that can be used and there are non-computable problems that are so difficult that they can never be used in computer assisted proofs. The question is: which ones are safe to utilise? Examples from mathematical physics and spectral theory will be highlighted.

♣ * Leonid Parnovski* (UCL), ** Floating mats and sloping beaches: spectral asymptotics **

** Wednesday 23 January **, 1.30 - 2.30pm, King's College London, King's Building, room K3.11, London Analysis and Probability seminar,

**Abstract: **I will discuss the asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.

♣ *Valery Smyshlyaev* (UCL), **TBA**

**Wednesday 23 January**, 3.00 - 4.00pm, King's College London, King's Building, room K3.11, London Analysis and Probability seminar,

♣ *Dmitry Jakobson* (McGill), **TBA **

**Wednesday 23 January**, 4.30 - 5.30pm, King's College London, King's Building, room K3.11, London Analysis and Probability seminar,

♠ Alisa Knizel (Columbia), TBA

**Thursday 31 January**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

♣ * Paul Bourgade* (Courant), ** TBA**

** Thursday 7 February 2019 **, 3-4pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

♠ TBA;

**Thursday 14 February**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

♣ * Martin Hairer* (Imperial), ** TBA**

** Thursday 21 February**, 3-4pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

♣ *Vlad Vysotsky* (Sussex), **TBA**

**Thursday 21 February**, 4:30-5.30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

♠ TBA;

**Thursday 28 February**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

♠ TBA;

**Thursday 7 March**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

♣ * TBA* ,

**Thursday 14 March 2019 **, 3-5.30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

♠ TBA;

**Thursday 21 March**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

♦ TBA;

**Friday 29 March**, All day, Kings College London, The Paris-London Analysis Seminar.

## 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), ** Toeplitz determinants with Fisher-Hartwig singularities**

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

**Abstract:** We consider the large n asymptotics of n dimensional Toeplitz determinants with Fisher-Hartwig singularities, uniformly as the location of the singularities are allowed to merge with each other. We discuss applications to moments of averages of the characteristic polynomials of the Circular Unitary Ensemble.

♣ *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), **Delocalisation of waves under scaling limits**

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

**Abstract:** We establish quantitative quantum ergodicity type delocalisation theorem for waves on hyperbolic surfaces of large genus. In the compact setting our assumptions hold for random surfaces in the sense of Weil-Petersson volume in the TeichmÃ¼ller space due to the work of Mirzakhani and in non-compact setting for arithmetic surfaces coming from congruence covers of the modular surface. The methods are based on Benjamini-Schramm scaling limits of metric measure spaces and Stein type harmonic analysis ergodic theorems, and are inspired by similar results on graphs. We plan to give a gentle introduction to the field before going to our results. Joint work with Etienne Le Masson (Cergy-Pontoise University, France).

♠ Matthew Jacques (Open University), ** Semigroups of hyperbolic isometries and their parameter spaces **

**13 December **, 3:00-4:00, Imperial College London, (Huxley 140), Pure analysis and PDE seminar**Abstract:** Let M denote the collection of orientation-preserving isometries of the hyperbolic plane. Given an n-tuple x=(x_1, ..., x_n) in M^n, let S(x) denote the semigroup generated by the ordinates of x under composition. In a 2010 paper of Avila, Bochi and Yoccoz the authors define the hyperbolic locus, H, as the set of points in M^n whose ordinates generate composition sequences that grow exponentially. They also define the elliptic locus, E, as the set of those x in M^n for which S(x) contains an elliptic isometry. They show that both H and E are open, and that the closure of E is equal to the complement of H. Motivated by this work and by the theory of discrete (Fuchsian) subgroups contained in M, we introduce the term semidiscrete to describe a semigroup that does not contain the identity within its closure. The semidiscreteness property on semigroups appears to be a good analogue of the discreteness property on groups, and we give theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we find that every semigroup is one of four standard types: elementary, exceptional, semidiscrete, or dense in M. We use these ideas to characterise the set H in terms of the semidiscreteness property. Finally, we give an example of a point on the boundary of E but not on the boundary of H, and an example of a point on the boundary of H that does not lie on the boundary of any of its connected components, answering two questions posed by Avila, Bochi and Yoccoz.

♠ 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.