# 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 - Summer term programme

♠ *Mira Shamis* (Queen Mary), Applications of the Ky Fan inequality to random (and almost periodic) operators

**Thursday 16 May**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar**Abstract: **We shall discuss the Ky Fan inequality for the eigenvalues of the sum of two Hermitian matrices. As an application, we shall derive a sharp version of a recent result of Hislop and Marx pertaining to the dependence of the integrated density of states of random Schroedinger operators on the distribution of the potential. Time permitting, we shall also discuss an application to quasiperiodic operators.

♠ *Jonathan Fraser* (St. Andrews), Approximate arithmetic structure in large sets of integers

**Thursday 23 May**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar**Abstract: **The celebrated Erdos conjecture on arithmetic progressions states that sets of positive integers whose reciprocals form a divergent series should contain arbitrarily long arithmetic progressions. I will discuss recent joint work with Han Yu, where we prove that such sets must at least approximate arithmetic progressions in a quantifiable sense.

## 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), **Two-scale homogenisation for a general class of high-contrast PDE systems**

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

**Abstract: **I present recent work with Ilia Kamotski [1] on two-scale homogenisation of general PDE systems with periodic coefficients with a critically scaled high contrast, which reflects certain interesting effects due to underlying ``micro-resonancesâ€™â€™. It appears that a strong two-scale resolvent convergence of associated high-contrast elliptic operators holds under a rather generic decomposition assumption. This implies in particular (two-scale) convergence of parabolic and hyperbolic semigroups with applications to a wide class of initial value problems. In the end I briefly discuss most recent stronger results with operator-type error estimates for high-contrast problems (with Shane Cooper and I. Kamotski), as well as situations where the micro-resonances display certain randomness. In simplest cases, the resulting two-scale limit behaviour appears to be rather explicit and the macroscopic equations display a form of wave trapping by the micro-resonances due to their randomness.

[1] I.V. Kamotski, V.P. Smyshlyaev, Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients, {\it Applicable Analysis} 98 (1-2), 64--90 (2019).

♣ *Dmitry Jakobson* (McGill), **Quantum ergodicity for ray-splitting (branching) billiards**

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

**Abstract: **This is joint work with Yu. Safarov and A. Strohmaier.

After giving an overview of Quantum Ergodicity results on compact Riemannian manifolds with ergodic geodesic flow (due to Shnirelman, Zelditch, Colin de Verdiere and others), we discuss joint work with Yuri Safarov and Alex Strohmaier, which concerns the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. We prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics. If time permits, we outline an example (provided by Y. Colin de Verdiere) of a system where the ergodicity assumption holds for the discontinuous system.

♠ Alisa Knizel (Columbia), **Log-gases on a quadratic lattice via discrete loop equations**

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

**Abstract: **We study a general class of log-gas ensembles on a quadratic lattice.

Using a variational principle we prove that the corresponding

empirical measures satisfy a law of large numbers and that their

global fluctuations are Gaussian with a universal covariance. We apply

our general results to analyze the asymptotic behavior of a q-boxed

plane partition model introduced by Borodin, Gorin and Rains. In

particular, we show that the global fluctuations of the height

function on a fixed slice are described by a one-dimensional section

of a pullback of the two-dimensional Gaussian free field.

Our approach is based on a q-analogue of the Schwinger-Dyson (or

loop) equations, which originate in the work of Nekrasov and his

collaborators, and extends the methods developed by Borodin, Gorin and

Guionnet to a quadratic lattice.

The talk is based on a joint work with Evgeni Dimitrov (Columbia University).

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

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

♣ *Mikhail Menshikov* (Durham), **Non-homogeneous random walks in critical regimes**

**Thursday 7 February 2019 **, 4:30-5:30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar.

**Abstract: http://www.london-analysis-seminar.org.uk/abstracts/Menshikov_070219.pdf**

♠ *Christopher Kauffman *(Imperial), **Global Stability for Charged Scalar Fields in Spacetimes close to Minkowski**** Thursday 14 February**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

**Abstract:**We prove global stability for the Charge-Scalar Field system on a background metric close to 1+3-dimensional Minkowski space. In particular, we consider a class of background metrics which satisfy certain estimates consistent with small-data solutions to Einstein's Vacuum Equations in harmonic coordinates. Our results are analogous to results obtained in Minkowski space, with slightly higher restrictions on the decay of initial data. The proof relies on a single-parameter modification of the standard Lorentz fields which depends on the initial ADM mass"

♣ * Martin Hairer* (Imperial), **Reconciling Ito and Stratonovich**

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

♣ *Vlad Vysotsky* (Sussex), **Large deviations of convex hulls of planar random walks**

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

♠ *Disheng Xu * (University of Chicago),** Lyapunov exponents and spectral theory of Schrodinger operator.****Thursday 28 February**, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

**Abstract:**In this talk we define Lyapunov exponents (LE) over ergodic systems, and explain why LE is useful in the study of dynamical systems and the spectral theory of Schrodinger operator. We present a brief introduction to Kotani theory and Avila's density result on positive LE, and our recent results for higher dimensional case. This will be an introductory talk.

♣ *Bassam Fayad* (Paris), **On the stability of quasi-periodic motion for analytic Hamiltonian systems**

**Thursday 14 March**, 3-4pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

**Abstract: **The stability of an elliptic fixed point of a Hamiltonian system is a central question in mathematical physics and is one of the founding problems of dynamical systems. We explore this question from three points of view : topological stability (Lyapunov stability), statistical stability (KAM theory), and effective stability (finite time stability). We introduce in particular new diffusion mechanisms that give the first examples of real analytic Hamiltonians (in three or more degrees of freedom) with unstable elliptic fixed points, or unstable invariant quasi-periodic tori. We produce examples with arbitrary (including Diophantine) frequency vectors and with divergent or convergent (in the case of tori) Birkhoff Normal Forms. We also give examples of analytic Hamiltonians that are integrable on half of the phase space and diffusive on the other half.

♣ Eero Saksman (Helsinki), **Decompositions of log-correlated fields with applications**

**Thursday 14 March**, 4:30-5.30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar.

**Abstract: **We consider a simple idea to decompose of log-correlated Gaussian fields into two-parts, both of which behave well in suitable sense. Applications include Onsager type inequalities in all dimensions, analytic dependence and existence of critical chaos measures for a large class of log-correlated fields. Talk is based on Joint work with Janne Junnila (EPFL) and Christian Webb (Aalto University).

♠ *Diane Holcomb *(KTH Stockholm); **Random matrices through differential operators**

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

**Abstract:** In 1984 Trotter described a tridiagonal random matrix model that has the same eigenvalues as the Gaussian Orthogonal Ensemble. This model and a later generalization share many structural properties with discrete Schrödinger operators. This led to the conjecture that the largest eigenvalues of the ensemble converged to the the eigenvalues a certain random differential operator. We will give an overview of where the conjecture comes from and a bit of the proof. We will then look at a process that appears when looking at eigenvalues of submatrices of the tridiagonal model. This process is notably different than the one that appears when considering submatrices of the full matrix model. This talk will survey work by Dumitriu-Edelman, Edelman-Sutton, and Ramirez-Rider-Virag. It finishes up with work that is joint A. Gonzalez.

♦ Program: http://www.london-analysis-seminar.org.uk/Paris-London/;

**Friday 22 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.