Train Einstein

Alexis Drouot
45, Rue d'Ulm
75005 Paris, France
alexis.drouot@ens.fr


Welcome on my website !

I am currently a fourth year student at the Ecole Normale Supérieure de Paris, starting my PhD at the Institut Fourier in Grenoble, with Professor Dietrich Häfner. My main work concerns general relativity, more precisely mathematical aspects of the Hawking effect. I also work on some problems in
integral geometry and classical and harmonic analysis, related to some k-plane transform inequalities. More can be found about it below. A CV is available here.


Papers

A quantitative version of the Catlin-D'Angelo-Quillen theorem (with Maciej Zworski), published in Analysis and Mathematical Physics (2013).
In this paper we give a quantitative proof of
the Catlin-D'Angelo-Quillen theorem, that asserts that a bihomogeneous complex form can be written as a sum of square of  polynomials, when multiplied by a certain power of the complex norm. The proof has the advantage to lead to quantitative upper bounds on this power. The main idea of the proof is to transform the compactness method of Quillen into some semiclassical arguments, that are possible to quantify.

Precompactness of radial extremizing sequence for a k-plane transform inequality, submitted to Journal of Functional Analysis (2012).
In this paper we extend the precompactness result stated in a previous work to the general radial case. The approach uses arguments of classical analysis: refined inequalities, concentration-compactness results, and quantitative results on weak-interaction.

Best constant for a k-plane transform inequality, submitted to Analysis and PDE (2011).
In this paper we study the endpoint case of the inequalities satisfied by the k-plane transform. We prove a restricted precompactness result, that nevertheless gives the existence of extremizers.  We also find some explicit extremizers and derive the value of the best constant, solving a conjecture of Baernstein and Loss. The approach uses arguments of classical analysis and the existence of a hidden conformal symmetry for the inequality.


Current work

General relativity and scattering theory: I am currently working on mathematical aspects of the Hawking effect. The Hawking effect is the apparent creation of particle that appears when we consider quantum fields in collapsing stars backgrounds. This (absolutely fascinating) phenomenon makes them look as Bose-Einstein or Fermi Dirac gas. My first project is to formulate and prove a rigourous theorem about the emission of bosons by Kerr-De Sitter black holes (rotating black holes in universes with positive cosmological constant). Compared to the work of Bachelot and the one of Häfner, that concern respectively the emission of bosons by Schwarzchild black holes and of fermions by Kerr black holes, the main difficulty is here the abcense of positive conserved energy for the Klein-Gordon equation. The recent work of Georgescu, Gerard and Häfner for scattering in Krein spaces is a first step to develop the required scattering theory of scalar field by Kerr-De Sitter collapsing star.

There are two further aspects of my PhD that I would like to develop. One concerns the study of the back reaction of the scalar field on the metric in an imposed spherical background, and its influence on particle creation. It would start with the work done by Christodoulou in the 80's - 90's about spherical solution of the Einstein-scalar equation and would require the development of a scattering theory for this nonlinear problem. This would lead to some work on black holes evaporation. Another direction would be to develop more realistic physical model for the study of the Hawking effect. In particular I would like to study the influence of the interior of the star on the particle creation. My complete PhD project can be found here (in French).

Classical and harmonic analysis: Related to my previous work with Christ, I am currently trying to extend my result about precompactness of extremizing sequences for the endpoint k-plane transform inequality to the non-radial case. Taryn Flock showed in a - still unpublished - paper that extremizers are unique modulo affine maps. As a consequence, a completely satisfying theorem would be to prove that all extremizing sequences are precompact modulo affine maps, as in this paper of Christ. A first step is to study the quasi-extremizers for this inequality. Our study is based on this paper of Christ. Another point is the complete resolution of the conjectures made by Baernstein and Loss. However, because of the abcense of the conformal symetry that I mentionned earlier, neither my work nor Christ's can easily be generalized. 


Pictures

Here and there are some pictures of some of my trips.