# Junior Analysis Seminars

## Junior Analysis Seminar

• Friday, 26 October, Room: Huxley 140
2:00 - 3:00, Volker Schlue (Sorbonne (Paris)), Scattering from infinity for semilinear wave equations

Abstract: In this talk I will discuss the construction of global solutions from scattering data (at null infinity) for various semilinear wave equations on Minkowski space satisfying the (weak) null condition.  I will elaborate on the proof which relies, (i) on a fractional Morawetz estimate, and (ii) on the construction of suitable approximate solutions from the scattering data. Finally I will outline the application of these results to Einstein's equations in harmonic coordinates. This is joint work with Hans Lindblad.

• Friday, 26 October, Room: Huxley 140
3:30 - 4:30, Christoph Kehle (Cambridge), Uniform boundedness and continuity at the Cauchy horizon for linear waves on Reissner--Nordström--AdS black holes

Abstract: I will present a recent result on solutions to the massive linear wave equation $\Box_g \psi - \mu \psi =0$ on the interior of Reissner--Nordström--AdS black holes. This is motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS black holes with negative cosmological constant $\Lambda <0$. Our main result shows that linear waves arising from a spacelike hypersurface with Dirichlet (reflecting) boundary conditions imposed at infinity remain bounded in the interior and can be extended continuouslybeyond the Cauchy horizon. This result is surprising because in contrast to black hole backgrounds with non-negative cosmological constant, the decay of $\psi$ in the exterior region for asymptotically AdS black holes is only logarithmic (cf. polynomial ($\Lambda =0$) and exponential ($\Lambda >0$)).

• Friday, 9 November, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Stefano Marmi (Pisa), Quasianalyticity of spaces of generalized analytic functions and applications

Abstract: I will introduce what kind of generalized analytic functions we will deal with, explain a little bit the history of the subject (dating back to Borel) and prove a theorem of quasianaliticity of the space. Finally I will illustrate how these examples arise naturally in small divisor problems in dynamical systems, as first conjectured by Kolmogorov.

• Friday, 16 November, Room: Huxley 140
2:00 - 3:00, Dejan Gajic (Cambridge), Conservation laws and late-time asymptotics of waves on black hole spacetimes

Abstract: A precise understanding of the evolution and global behaviour of small perturbations of stationary black hole spacetimes is a long-standing open problem in general relativity. One aspect of this problem is the conjectured existence of so-called “polynomial tails” in the late-time asymptotics of metric perturbations. In this talk, I will discuss recent work in collaboration with Y. Angelopoulos and S. Aretakis that establishes rigorously the existence of polynomial late-time tails in the context of a toy model problem, the wave equation on fixed Schwarzschild black hole backgrounds. I will describe how polynomial tails emerge from simple conservation laws.

• Friday, 16 November, Room: Huxley 140
3:30 - 4:30, Maxime Van de Moortel (Cambridge), Non-linear interaction of three impulsive waves for the Einstein equations in U(1) polarized symmetry

Abstract: An impulsive gravitational wave is an idealization of gravitational waves produced by a strongly gravitating source. In the presence of multiple sources, the impulsive waves eventually interact, and it is interesting to study this interaction. From the perspectives of PDEs, impulsive gravitational waves are low regularity solutions of the Einstein equations, seen as a system of non-linear wave equations, thus even well-posedness of the initial value problem is not clear a priori. Tremendous progress has been made on this topic by Luk and Rodnianski in 2013, who proved local well-posedness for initial data featuring the interaction of two gravitational waves. One crucial idea is to exploit that the metric is very regular in two directions, those parallel to the intersection of the two singular wave-fronts. However, their method does not apply to the case where three or more impulsive waves interact transversally, since the space-time no longer admits two privileged directions. I will first review several related low regularity problems in General Relativity from the perspective of PDEs. Then I will present a new local existence result for Cauchy data featuring three impulsive gravitational waves of small amplitude propagating towards each other’s.
The talk will mainly emphasize PDE-related aspects and no prior exposure to General Relativity will be assumed. This is joint work with Jonathan Luk.

• Friday, 23 November, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Yulia Kuznetsova (University Bourgogne Franche Comte), Introduction to the theory of quantum (semi)groups

Abstract: Starting from scratch, I will speak of main ideas of the theory of quantum groups and semigroups, arriving at the end at some recent results.

• Friday, 30 November, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Diogo Oliveira e Silva (Birmingham), Maximal and Variational Fourier Restriction Theory

Abstract: Müller, Ricci, and Wright recently established the first "maximal restriction theorem" for the Fourier transform. As a direct consequence, they clarified certain subtle measure theoretic aspects underlying Fourier restriction theory. In the first half of this talk, we will give a brief introduction to the restriction problem, and illustrate its importance to modern analysis. We will then focus on the endpoint Tomas-Stein inequality in 3-dimensional Euclidean space, together with its maximal and variational variants, for which especially simple proofs are available. Finally, we will describe a recent generalisation, and present some open problems. This is partly based on joint work with Vjekoslav Kovač.

• Friday, 7 December, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Michele Coti Zelati (Imperial), Mixing and metastability in the Navier-Stokes/Euler equations of incompressible fluids

Abstract: We present recent developments in the study of the longtime dynamics of the incompressible Navier-Stokes equations and related scalar models. The main mechanism we analyse is mixing, which is a purely advective effect which causes a transfer of energy to high frequencies. In linear models, this effect can be understood in a dynamical system fashion by analyzing the decay of correlations of the associated flow, or in a more spectral theoretic way by studying the properties of the linear operators involved with the help of the RAGE theorem. It can be also associated to hypo-elliptic regularisation effects.  Mixing has important quantitative stability consequences for certain stationary solutions to the Euler equations, and it causes a relaxation effect called inviscid damping. When dissipation is present, mixing gives rise to what we refer to as enhanced dissipation: this can be understood by the identification of a time-scale faster than the purely diffusive one. This talk is based on recently obtained results: (1) a general quantitative criterion that links mixing rates to enhanced dissipation time-scales, with nice connections to the dynamics of contact Anosov flows, and (2) a precise identification of the enhanced dissipation time-scale for the Navier-Stokes equations linearised around the Poiseuille flow, along with metastability and nonlinear transition stability thresholds results, which show a direct link with 1D Schrodinger equations.

• Friday, 14 December, Room: Huxley 140
2:00 - 2:30, Esther Bou Dagher (Imperial), Logarithmic Sobolev Inequalities, Orlicz Imbeddings, and Supercontractivity

AbstractWe present a paper by R. A. Adams whose purpose is to obtain sharp L^p-logarithmic Sobolev inequalities for a wide class of measures and to consider some implications of such inequalities for the imbedding of Sobolev spaces.

• Friday, 14 December, Room: Huxley 140
2:45 - 3:15, Yifu Wang (Imperial), Logarithmic Sobolev Inequalities, Orlicz Imbeddings, and Supercontractivity

Abstract: We present a method by J. Rosen that allows us to get higher order logarithmic Sobolev inequalities and show how these are used to prove supercontractivity of the semigroup e^(-t del* . del) , t>0.

• Friday, 11 January, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Massimiliano Esposito (Imperial), Generalizations of Weyl quantisation: non linear \tau quantising functions on R^n and a possible approach of Weyl quantisation on Z^n

Abstract: In this talk we are going to introduce the Weyl quantisation as arose in the context of Quantum Mechanics, along with the Weyl calculus associated to this quantisation. Further we are going to present the Heisenberg group H^n and a possible definition of the Weyl quantisation on H^n. This latter has been the central motivation to extend the Weyl calculus on Z^n and to more general pseudo-differential calculi on R^n.

• Friday, 18 January, Room: Huxley 140
2:30 - 4:30, Joe Keir (Cambridge), TBA

Abstract: TBA

• Friday, 25 January, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Vasiliki Evdoridou (Open University), Simply connected wandering domains of entire functions

Abstract: Let f be a transcendental entire function and U be a connected component of the Fatou set of f. If U is not eventually periodic then it is called a wandering domain. Sullivan's celebrated result showed that rational functions have no wandering domains. Contrary to rational functions, transcendental entire functions can have wandering domains (simply or multiply connected), but they are not well understood. In this talk we will start with an introduction to the iteration of transcendental entire functions and give some background on wandering domains. We will then focus on simply connected wandering domains and give a trichotomy of simply connected wandering domains in terms of the hyperbolic distance of the orbits of pairs of points in the wandering domain. We will then present several different types of simply connected wandering domains. We will also discuss the result which allows us to construct such examples and is based on approximation theory. Finally, we will give details on how some of the examples have been constructed. This is joint work with A.-M. Benini, N. Fagella, P. Rippon and G. Stallard.

• Friday, 1 February, Room: Huxley 140
3:30 - 4:30, Andrew McLeod (UCL), Pyramid Ricci Flow

Abstract: Central to recent ground-breaking developments regarding the regularity of three-dimensional Ricci limit spaces has been the use of Ricci flow as a mollification tool. The main achievement has been the introduction of a weakened notion of Ricci flow, culminating in the so-called Pyramid Ricci flows of McLeod and Topping, that may be run in situations ill-suited to the classical theory; see the works of Hochard, Simon and Topping and McLeod and Topping. The aim of this talk is to introduce the classical theory of Ricci flow, explain why they are well-suited to be used as “smoothing tools”, before giving an overview of the classical limitations and how such issues are overcome by Pyramid Ricci flow in dimension three.

• Friday, 15 February, Room: Huxley 140
2:00 - 3:00, Shrish Parmeshwar (KCL), Global Solutions of the Compressible Euler Equations

Abstract: We discuss recent results on global solutions to the compressible Euler equations with vacuum free boundary. Previously, in the presence of a free boundary, global solutions called affine flows have been found by relying on reducing the problem of constructing the fluid flow to that of solving a finite dimensional dynamical system, via a separation of variables argument. We show that if our initial density is small enough, and our initial velocity expands in a suitable manner, then global solutions can be constructed using a scaling property inherent in the compressible Euler system, and such solutions need not exhibit the same symmetries that are apparent in the affine flows.

• Friday, 22 February, Room: Huxley 140
3:30 - 4:30, Stergios Antonakoudis (Cambridge), Uniformization of Riemann Surfaces and Royden's theorem

Abstract. We will discuss the problem of uniformization of Riemann Surfaces by polygons in the plane; explain and present a new proof
of Royden's theorem
.

• Friday, 1 March, Room: Huxley 140
2:00 - 3:00, Harris Maroudas (Imperial), Gap probabilities in random matrix theory

Abstract
We will present a series of known asymptotic formulae describing the probability that certain large intervals of the real line are free from eigenvalues of the Gaussian unitary ensemble. These will highlight contributions in this area and motivate the presentation of a new such formula. Subject to time we will discuss the Riemann-Hilbert steepest descent method of Deift and Zhou employed in the proof.

This is ongoing work under the supervision of I. Krasovsky.

• Friday, 8 March, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:00, Matias Delgadino (Imperial), Capillarity and Alexandrov's theorem

Abstract: We will introduce the associated energy for capillarity model of stationary droplets. We will show how Alexandrov's theorem, which states that every smooth constant mean curvature surface is a sphere, implies the sphericity of small droplets. During the talk, we will introduce the notion of sets of finite perimeter and show the Alexandrov theorem in an utmost level of generality.

• Friday, 15 March, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Benjamin Fahs (Imperial), Toeplitz determinants with Fisher-Hartwig singularities

AbstractWe discuss Toeplitz determinants, their connection to orthogonal polynomials, and their application to studying the statistics of eigenvalues of the Circular Unitary Ensemble. I will present some known and some new results on asymptotics of Toeplitz determinants with Fisher-Hartwig singularities of dimension n for large n, and discuss some methods involved in proving such results.

• Friday, 22 March, Room: Huxley 140
2:00 - 3:00, Megan Griffin-Pickering (Cambridge), A Particle Approximation for the Kinetic Incompressible Euler Equation

Abstract: The Kinetic Incompressible Euler equation is a model for plasma. This is the formal limit of the classical Vlasov-Poisson system in the quasineutral’ limit where the Debye length tends to zero. The Vlasov-Poisson system can itself be derived formally from a system of interacting particles, in the limit as the number of particles tends to infinity. The rigorous justification of this mean field’ limit remains a major open problem. However, in recent years, researchers have derived the Vlasov-Poisson equation rigorously from various regularised microscopic systems. In this talk, I will present a joint work with Mikaela Iacobelli, in which we give a rigorous derivation of the Kinetic Incompressible Euler equation from a regularised particle system, using a combined mean field and quasi-neutral limit.

• Friday, 22 March, Room: Huxley 140
3:30 - 4:30, Rishabh Gvalani (Imperial), The McKean–Vlasov equation on the torus: Stationary solutions, phase transitions, and mountain passes

Abstract: We study the McKean–Vlasov equation on the torus which is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box.  We focus our attention on the stationary problem - under certain assumptions on the interaction potential, we show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations.  Finally, we attempt to classify continuous and discontinuous phase transitions for this system and show that at the point of discontinuous transition the free energy possesses a mountain pass point. Joint work with José A. Carrillo, Greg Pavliotis, and André Schlichting.

• Friday, 3 May, Room: Huxley 140
2:00 - 3:00, 3:30 - 4:30, Thomas Johnson (Imperial), The black hole stability problem in general relativity

Abstract: A major open problem in the field of geometric hyperbolic PDE is whether the subextremal Kerr family of black holes are non-linearly stable on their exterior regions as solutions to the Einstein vacuum equations of general relativity. In the first part of the talk we shall discuss both the set up of this problem in the context of the less elaborate Schwarzschild subfamily of the Kerr family and work of Dafermos—Rodnianski on establishing decay for solutions to the wave equation on Schwarzschild. The second part of the talk then concerns a discussion of my own recent work regarding the linear stability of the Schwarzschild family for which the earlier work of Dafermos—Rodnianski proved fundamental.

• Friday, 10 May, Room: Huxley 140
2:00 - 3:00, Matteo Capoferri (UCL), Global hyperbolic propagators and spectral asymptotics

Abstract: In my talk I will present a global, invariant and explicit construction of propagators of hyperbolic PDEs on closed Riemannian manifolds. This can be achieved by representing the propagator as a single Fourier integral operator with distinguished complex-valued phase function.  The knowledge of the propagator allows one, in turn, to recover asymptotic spectral properties of the operators at hand. As a prototypical example, I will analyse the construction of the wave propagator. Time permitting, I will discuss similarities and fundamental differences between scalar equations and first order systems.

• Friday, 10 May, Room: Huxley 140
3:30 - 4:30, Michele Dolce (GSSI), Linear stability of 2D isothermal compressible Couette flow

Abstract: The interest in studying stability of shear flows dates back to the origin of fluid dynamics.
In the talk, we will briefly introduce the problem in the incompressible setting. Then we consider the 2D isothermal compressible Euler equations in T×R, where a steady solution is given by the Couette flow, i.e. a shear flow with a linear velocity profile, with constant density. For this problem, in the physics literature, the linearization around the Couette flow was considered by Chagelishvili et al. By some heuristic argument they claim that there is a linear growth for a sort of linearized energy. Our result regards the characterization on the frequency space of the solution of the linearization around the Couette flow. Thanks to standard Fourier transform techniques, by proper weighting, we are able to reduce the system to a 2D non-autonomous dynamical system, where we study the dynamics. As a consequence we are able to prove in a rigorous way the claim about the linear growth, being more precise on the dependece of compressible and incompressible effects. In the end, if time permits, we discuss also about the stability of monotone shear flows "close" to Couette.

• Friday, 17 May, Room: Huxley 311
2:00 - 3:00, 3:30 - 4:30, Sabine Boegli (Imperial), Spectral approximation for non-selfadjoint linear operators

Abstract: We study spectral convergence for sequences of unbounded linear operators that converge to some operator T in strong resolvent sense. It is well known that, even in the case of purely discrete spectra, the eigenvalues of the approximating operators may accumulate at a point that is not an eigenvalue of T. In addition to the occurrence of such spurious eigenvalues, for non-selfadjoint operators not every eigenvalue of T may be approximated. In the first part of the talk I will give an introduction to spectral approximation, with focus on the projection method (truncating infinite matrices to finite sections) and the domain truncation method (truncating the domain of a singular differential operator). Usually spectral convergence fails in the presence of essential spectrum (non-discrete parts of the spectrum) of the operator T. In the second part of the talk I will introduce the essential numerical range, which is a useful tool to describe all possible spurious eigenvalues produced by projection methods.

• Friday, 31 May, Room: Huxley 140
2:00 - 3:00, Owain Salter Fitz-Gibbon (Cambridge), The Klein-Gordon equation on the hyperboloidal anti-de Sitter Schwarzschild black hole

Abstract: In this talk I aim to establish energy decay for solutions to the Klein-Gordon equation on the positive mass hyperboloidal anti-de Sitter Schwarzschild black hole, subject to Dirichlet, Neumann and Robin boundary conditions at infinity, for a range of the (negative) mass squared parameter. To do so we use vector field methods with a renormalised energy to avoid divergences that would otherwise appear in the energy integrals. For another region of the parameter space, we use the existence of negative energy solutions to demonstrate linear instability.

• Friday, 31 May, Room: Huxley 140
3:30 - 4:30, Nikolaos Athanasiou (Oxford), Formation of singularities for the Relativistic Euler equations

Abstract: An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.

• Tuesday, 9 July, Room: Huxley 340
2:00 - 3:00, John Anderson (Princeton), Global stability for nonlinear wave equations with multi-localized initial data: part I

Abstract: We will discuss the global stability of the trivial solution to systems of quasilinear wave equations satisfying the null condition where the initial data is not assumed to be localized around a single point. We shall first review previous global stability results and describe why having data that is not localized around a single point causes difficulties. We shall then describe the main strategy for overcoming these difficulties. Finally, we shall provide details of the main parts of the argument in a simplified setting.

• Tuesday, 9 July, Room: Huxley 340
3:15 - 4:15, Federico Pasqualotto (Princeton), Global stability for nonlinear wave equations with multi-localized initial data: part II

Abstract: We will discuss the global stability of the trivial solution to systems of quasilinear wave equations satisfying the null condition where the initial data is not assumed to be localized around a single point. We shall first review previous global stability results and describe why having data that is not localized around a single point causes difficulties. We shall then describe the main strategy for overcoming these difficulties. Finally, we shall provide details of the main parts of the argument in a simplified setting.