Title: Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution.

Abstract: We will start by recalling the weak cosmic censorship conjecture. Then we will review Christodoulou’s construction of naked singularities for the spherically symmetric Einstein-scalar field system. Finally, we will discuss joint work with Igor Rodnianski which constructs spacetimes corresponding to the exterior region of a naked singularity for the Einstein vacuum equations.

]]>Title: Two dimensional gravity waves at low regularity.

Abstract: In this talk, we will consider the low regularity well-posedness problem for the two dimensional gravity water waves. This quasilinear dispersive system admits an interesting structure which we exploit to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier energy estimates of Hunter-Ifrim-Tataru. These results allow us to significantly lower the regularity threshold for local well-posedness, even without using dispersive properties. This is joint work with Mihaela Ifrim and Daniel Tataru.

]]>Title: Strong cosmic censorship and generic mass inflation for charged black holes in spherical symmetry.

Abstract: In this talk, I will first review a previous work with J. Luk, in which the C2-formulation of the strong cosmic censorship is proved for the Einstein-Maxwell-(real)-Scalar Field system in spherical symmetry for two-ended asymptotically flat data. More precisely, it was shown that a “generic” class of data for this model gives rise to maximal future developments which are future C2-inextendible. In the second part of the talk, I will present a new, complementary theorem (also joint with J. Luk) that for a further “generic” subclass of such data, the Hawking mass blows up identically along the Cauchy horizon. This result confirms, rigorously and unconditionally, the mass inflation scenario of Poisson-Israel and Dafermos for the model at hand.

]]>Title: Multisolitons and their stability in 1-d cubic NLS.

Abstract: The aim of this talk is first to describe the multisoliton manifold for the 1-d cubic NLS flow, and then to consider their stability. This continues recent work, joint with Herbert Koch, on energy estimates for this and other related integrable evolutions.

]]>Title: Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem with and without surface tension.

Abstract: We consider the two-dimensional water wave problem in an infinitely long canal of

finite depth both with and without surface tension. In order to describe the evolution

of the envelopes of small oscillating wave packet-like solutions to this problem the

Nonlinear Schrödinger equation can be derived as a formal approximation equation.

The rigorous justification of the Nonlinear Schrödinger approximation for the water

wave problem was an open problem for a long time. In recent years, the validity

of this approximation has been proven by several authors only for the case without

surface tension.

In this talk, we present the first rigorous justification of the Nonlinear Schrödinger approximation for the two-dimensional water wave problem which is valid for the

cases with and without surface tension by proving error estimates over a physically

relevant timespan in the arc length formulation of the water wave problem. Our

error estimates are uniform with respect to the strength of the surface tension, as the

height of the wave packet and the surface tension go to zero.

Title: Local energy estimates on black hole backgrounds.

Abstract: Local energy estimates are a robust way to measure decay of solutions to linear wave equations. I will discuss several such results on black hole backgrounds, such as Schwarzschild, Kerr, and suitable perturbations converging at various rates, and briefly discuss applications to nonlinear problems. The most challenging geometric feature one needs to deal with is the presence of trapped null geodesics, whose presence yield unavoidable losses in the estimates. This is joint work with Lindblad, Marzuola, Metcalfe, and Tataru.

]]>Title: Almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems in two space dimensions.

Abstract: We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We systematically investigate all the possible quadratic null form type quasilinear strong coupling nonlinearities, and provide a new, robust approach for the proof.

]]>Title: Resonances over a potential well in an island.

Abstract: Recent work with M. Zerzeri. Let V : R^n → R be a sufficiently analytic potential which tends to 0 at infinity. Assume that for an E > 0 we have V^{-1}(]- ∞ ,E[)=U(E) ⊔ S(E), where U(E) ∩ S(E) = ∅ , with U(E) connected and bounded (the well) and S(E) connected (the sea). The distribution of the resonances of -h^2 Δ + V near E has been thoroughly studied since more than 30 years. If we increase E a natural scenario is that the decomposition persists until the closures of U(E) and S(E) intersect at a critical energy E = E_0. Under some natural assumptions we show that near E_0 most of the resonances are close to the real axis and obey a Weyl law. In one dimension there are more detailed results (Fujiie-Ramond ’98).

]]>Title: Box condition versus Chang–Fefferman condition for weighted multi-parameter paraproducts.

Abstract: Paraproducts are building blocks of many singular integral operators and the main instrument in proving “Leibniz rule” for fractional derivatives (Kato–Ponce). Also multi-parameter paraproducts appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgues spaces with respect to a measure in the polydisc. The latter problem (without loss of information) can be often reduced to boundedness of weighted dyadic multi-parameter paraproducts. We find the necessary and sufficient condition for this boundedness in n-parameter case, when n is 1, 2, or 3. The answer is quite unexpected and seemingly goes against the well known difference between box and Chang–Fefferman condition that was given by Carleson quilts example of 1974.

]]>Title: Dynamical zeta functions at zero on surfaces with boundary

Abstract: The Ruelle zeta function counts closed geodesics on a Riemannian manifold of negative curvature. Its zeroes are related to Pollicott-Ruelle resonances which have been heavily studied in the setting of Anosov dynamical systems. In 2016, Dyatlov-Zworski proved an unexpected result relating the structure of the zeta function near the origin to the topology of the manifold. This extended a formula previously only known to hold in the constant curvature setting.

This talk will consider the situation where the manifold has boundary. A similar story can be told and the ultimate result extends the constant curvature setting (understood in 2001) to the variable curvature setting.

The microlocal tools required to consider this problem had been well developed in earlier papers (principally Dyatlov-Guillarmou 2016) and it remained to manipulate correctly relative cohomology (in this case à la Bott-Tu) in order to understand the space of 1-form Pollicott-Ruelle resonances.

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