Junior Analysis seminar

Schedule of 2018-2019 academic year:

• 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 
  
  Abstract: We 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), TBA 
  
  Abstract: TBA

• Friday, 15 February, Room: Huxley 140
  2:00 - 3:00, Felicity Eperon (Cambridge), TBA 
  
  Abstract: TBA

• Friday, 15 February, Room: Huxley 140
  3:30 - 4:30, Shrish Parmeshwar (KCL), TBA 
  
  Abstract: TBA

• Friday, 22 February, Room: Huxley 140
  2:00 - 3:00, 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, 22 February, Room: Huxley 140
  3:30 - 4:30, Stergios Antonakoudis (Cambridge), On totally geodesic submanifolds of Teichm\"uller space
  
  Abstract. We will discuss recent results and progress on the study of
  totally geodesic submanifolds of Teichm\"uller space of Riemann surfaces.