# Analysis Seminar 2014-15

### Term 3 2014/15 -

The seminars are held on Thursdays at 16:00 in Room B3.02 - Mathematics Institute unless noted otherwise

Organiser: Claude Warnick

23rd April **Jonathan Ben-Artzi** (Imperial)

**Title:** *How does one compute in infinite dimensions?
*

**Abstract:**It is often desirable to compute "infinite dimensional" quantities as the limit of finite dimensional (Galerkin) approximations. However, are such approximations guaranteed to converge? Even in the finite dimensional case this is not a simple problem. For instance, the problem of finding roots of polynomials of degree higher than three starting from some initial guess and then iterating was only solved in the 1980s (Newton's method isn't guaranteed to converge): Doyle and McMullen showed that this is only possible if one allows for multiple independent limits to be taken, not just one, and they called such objects "towers of algorithms". In this talk I will apply this idea to other problems (such as computational quantum mechanics, inverse problems, spectral analysis), show that towers of algorithms are a necessary tool, and introduce the Solvability Complexity Index. An important consequence is that some problems can never be calculated numerically. If time permits, I will mention connections with analogous notions in logic and theoretical computer science. Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).

30th April **Hao Yin** (Warwick)

**Title****:** *Ricci flow on surfaces with conical singularities
*

**Abstract:**In this talk, we will introduce the normalized Ricci flow on surfaces with conical singularities. We discuss the PDE problem related to this flow from an elementary point of view and also its applications (by other authors) in finding good metrics on such surfaces.

7th May **Alessandro Carlotto** (Imperial)

**Title:** *The complexity of a minimal subvariety: Morse index versus topology.
*

**Abstract:**There are several ways to quantify the "complexity" of a minimal subvariety: on the one hand we have analytic data (like the Morse index, the value of the first eigenvalue of the Jacobi operator etc...), on the other we have topological invariants (like the Betti numbers, the sigma invariant etc...). But what is the relation between these two pieces of information? I will give a broad overview of some recent results with special emphasis on two of them. I will first mention my recent construction in S^4 of minimal hyperspheres of arbitrarily large Morse index and uniformly bounded volume. I will then move to some joint work with Ambrozio and Sharp where we show that in various geometric settings (ellipsoids, complex projective spaces, real projective spaces, product of spheres...) the Morse index of a closed minimal hypersurface is bounded from below by an affine function of the first Betti number (with universal coefficients).

14th May **J****an Sbierski** (Cambridge)

**Title:** *The C^0 inextendibility of the Schwarzschild spacetime
*

**Abstract:**The strong cosmic censorship conjecture is one of the major open problems in general relativity. It states, that for generic asymptotically flat initial data for the vacuum Einstein equations, the maximal globally hyperbolic development is inextendible as a Lorentzian manifold with Christoffel symbols locally in L^2. This motivates the study of low regularity extensions of Lorentzian manifolds. So far, however, the few techniques available for showing inextendibility only yield inextendibility as a Lorentzian manifold with a C^2 regular metric. In this talk, I will present techniques for proving C^0 inextendibility. In particular, I outline the proof of the C^0 inextendibility of the Schwarzschild space-time.

21st May **Reto Mueller** (QMUL)

**Title:*** The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds
*

**Abstract:**A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy The Nguyen

28th May **Alexander Volkmann** (Potsdam)

**Title:** *Relative isoperimetric inequalities via mean curvature flow with Neumann boundary condition*

**Abstract:** In this talk, we present a flow approach to proving relative isoperimetric inequalities. After a brief introduction into the relative isoperimetric problem, we develop a theory of weak solutions for (nonlinear) mean curvature flow with Neumann boundary condition. We then outline the proof of an existence result for the weak level set flow under conceptually optimal conditions on the support surface. Finally, we explain how this flow can be used to obtain relative isoperimetric inequalities. In particular, we obtain a new proof of the relative isoperimetric inequality outside convex sets in R3 due to Choe-Ghomie-Ritore.

### Organiser: Peter Topping

### Term 2 2014/15 -

The seminars are held on Thursdays at 16:00 in Room B3.02 - Mathematics Institute unless noted otherwise

9th January 2pm** Jean Raimbault** (Toulouse)

15th January **Alix Deruelle** (Warwick)

Title: *Desingularizing positively curved metric cones by the Ricci flow*

Abstract : The Ricci flow is a non linear heat equation on the space of metrics and is therefore expected to smooth out instantaneously Riemannian metrics with singularities in some cases. We investigate here the possibility of desingularizing metric cones over a positively curved sphere by the Ricci flow : more specifically, we explain how this can be done with expanding self-similarities of the Ricci flow.

22nd January No Seminar

29th January **Eleonora Di Nezza** (Imperial)

Title:* Regularizing properties and uniqueness of the Kaehler-Ricci flow*

5th February **Martin de Borbon** (Imperial)

Title: *Asymptotically Conical Ricci Flat Kaehler metrics with cone singularities
*Abstract: I will state an existence result for Ricci flat metrics with cone singularities along a smooth complex curve in C^2. I am not going to talk about the context in which these metrics arise (as blow-up limits). I will discuss Holder spaces for metrics with cone singularities and the relevant a priori estimates needed to prove the existence result.

12th February **Lu Wang** (Imperial)

Title: *Geometry of Two-dimensional Self-shrinkers*

19th February **Sebastian Schwarzacher**

Title: *On the time derivative of degenerated parabolic PDE*

26th February **Melanie Rupflin** (Leipzig)

Title: *Teichmueller harmonic map flow from cylinders*

5th March **Ulrich Menne** (Potsdam)

Title: *Weakly differentiable functions on 'singular submanifolds'*

12th March **Xavier Tolsa** (UA Barcelona)

Title: *The David-Semmes problem and related results
*Abstract: In this talk I plan to survey the last advances on the David-Semmes problem about the L^2 boundedness Riesz transforms and rectifiability. In particular, I will describe some ideas involved in the proof of the codimension 1 case of this problem by Nazarov, Volberg, and myself, and I will also explain other related results and open questions.

### Term 1 2014/15 -

The seminars are held on Thursdays at 16:00 in Room B3.02 - Mathematics Institute unless noted otherwise

Organiser: Claude Warnick

9th October: **Arick Shao** (Imperial)

**Title:** Unique continuation from infinity for linear waves

**Abstract: **We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects.

16th October: **Mahir Hadzic** (KCL)

**Title:** Stability problem in the dust-Einstein system with a positive cosmological constant

**Abstract: **The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstrom’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.

23rd October: **Alan Sola** (Cambridge)

**Title:** Cyclic polynomials in two variables.

**Abstract:** I will discuss recent work that has led to a characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk. It turns out that the cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary of the bidisk. The results will be illustrated by a number of examples. This reports on joint work with C. Beneteau, G. Knese, L. Kosinski, C. Liaw, and D. Seco.

30th October: **Jonathan Luk** (MIT/Cambridge)

**Title:** Stability of the Kerr Cauchy horizon and the strong cosmic censorship conjecture in general relativity

**Abstract:** The celebrated strong cosmic censorship conjecture in general relativity suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present recent result proving that the Cauchy horizon is C^{0} stable and discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

6th November: **Costante Bellettini (**Princeton/Cambridge)

**Title:** Regularity properties for semi-calibrated integral 2-cycles

**Abstract:** TBA

13th November: **Antoine Choffrut** (Edinburgh)

**Title: **Rayleigh-Benard convection: physically relevant a priori estimates

**Abstract: **A fluid contained between two horizontal plates is heated from below and cooled from above. Heat transfer is effected via two mechanisms: (1) thermal conduction, at the microscopic level; and (2) thermal convection, where lighter, warmer particles carry their internal energy to the top. The governing equations are those of the Boussinesq approximation. The average upward heat flux relative to pure conduction is measured by the Nusselt number (Nu). The temperature gradient is measured by the Rayleigh number (Ra). The relative strength of viscosity over inertia is measured by the Prandtl number (Pr).

In this talk I will present near optimal scaling laws for Nu as a function of Ra for two regimes of Pr. Previous work, pioneered by Constantin and Doering, with contributions from many others, assumed infinite Pr. The proof relies on maximal regularity estimates for the (linear) Stokes system in $L^\infty$- and $L^1$-type spaces, the latter with a borderline failing Muckenhoupt weight. This is joint work with Camilla Nobili and Felix Otto.

20th November: **Marc Troyanov** (EPFL)

**Title:** Recent Results in L_{qp} Cohomology

**Abstract:** I will recall the basic definition of (reduced and non reduced) L_{qp} Cohomology of a non compact Riemannian manifold, and survey relation with other notions of geometric analysis such the L_{p}-Hodge theorem, quasi-conformal structures or the topology at infinity. I will also present several notions of L_{qp} Cohomology on metric measure spaces and discuss conditions for invariance under quasi-isometry.

27th November: **Qian Wang** (Oxford)

**Title:** A geometric approach for sharp Local well-posedness of quasilinear wave equations

**Abstract:** The commuting vector fields approach for Strichartz estimates was employed for proving the local well-posedness in the Sobolev spaces Hs with for general quasi-linear wave equation in by Klainerman and Rodnianski.

Via this approach they obtained the local well-posedness in Hs with s>2 for (1+3) vacuum Einstein equations, by taking advantage of the vanishing Ricci curvature. The sharp, , local well-posedness result for general quasilinear wave equation was achieved by Smith and Tataru by constructing a parametrix using wave packets. Using the vector fields approach, one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations.

I will talk about my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.

4th December: **Laurent Hauswirth** (Marne-la-Vallée)

**Title: **Spectral curves, Harmonic map and minimal surfaces

**Abstract:** This talk concer the space moduli of periodic harmonic map into S(2) and S(3) parametrized by spectral data introduced by Hitchin in 1987. I will explain how obtain result of classification for minimal surfaces.

11th December: **Emil Wiedemann** (HCM Bonn) **NOTE UNUSUAL ROOM: B3.03**

**Title: **Renormalisation Defects for Transport Equations

**Abstract:** Linear transport equations can be uniquely solved by the method of characteristics as long as the transporting vectorfield has Lipschitz regularity. In a seminal paper from 1989, DiPerna and Lions treated the case of less regular vectorfields (in Sobolev spaces), thereby introducing the notion of renormalised solutions. On the other hand, individual examples of non-renormalised solutions of transport equations have been constructed when the transporting field is less regular than Sobolev. We present a convex integration approach which allows to recover and vastly generalise such counterexamples, showing that essentially any prescribed renormalisation defect can be realised. Joint work with G. Crippa, N. Gusev, and S. Spirito.