45, Rue d'Ulm
75005 Paris, France
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.
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
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
some semiclassical arguments, that are
possible to quantify.
Precompactness of radial extremizing sequence for a k-plane transform
, submitted to Journal of Functional Analysis
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.
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.
General relativity and scattering theory:
I am currently
mathematical aspects of the Hawking effect. The Hawking effect
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
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
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,
completely satisfying theorem would be to prove that all extremizing
sequences are precompact modulo affine maps, as in this
of Christ. A first step is to study the quasi-extremizers for this
inequality. Our study is based on this
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
can easily be generalized.
and there are some pictures of some of my trips.