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Ma liste de prépublications.

  1. MumShah.dvi Open questions on the Mumford-Shah functional, a 12-pages survey with questions, showed up in 2005 in « Perspectives in Analysis », the proceedings of the conference in honour of L. Carleson.

  2. Notes-Parkcity.dvi A booklet of notes on uniform rectifiability; maybe this will be published one day by the AMS, as a Park City Lecture notes

  3. Purdue.dvi Quasiminimal sets for Hausdorff measures, published in the proceeding of the analysis symposium in Purdue, a description of how we may try to use Almgren restricted sets and a concentration lemma of Dal Maso, Morel, Solimini to prove existence results.

  4. A generalization of Reifenberg's theorem in $R^3$, with Thierry De Pauw and Tatiana Toro, where Reifenberg's topological disk theorem about how to parameterize subsets of Euclidean space that are uniformly close to planes at all scales and locations is generalized to sets of dimension 2 in 3-space (mainly) that are close to minimal cones at all scales and locations. Now see GAFA in 2008.

  5. Holder regularity of two dimensional almost-minimal sets in $R^n$ (A slightly simpler proof of the biHolder part of J. Taylor's regularity theorem for two-dimensional almost-minimal sets in 3-space, now generalized to $n$-space, plus a characterization of minimal sets in 3-space, but only with the Mumford-Shah notion of competitors. Now see the Annales de la faculté des sciences Toulouse, in 2009.

  6. $C{^1+\alpha$-regularity for two-dimensional almost-minimal sets in $R^n$. Epiperimetry results and a slight extension of J. Taylor's regularity theorem; now see the J. Geometric Analysis, in 2010.

  7. Evian-Proc-08.pdf Lecture notes for a series of lectures in Evian, June 2008. This is a description of the two papers above (J. Taylor's theorem) and potential applications.

  8. MS-IHP-08.pdf Transparents pour une série d'exposés à l'IHP (2007) sur la fonctionnelle de Mumford-Shah.

  9. Reifenberg parameterizations for sets with holes, with T. Toro. More precise estimates on the parameterizations obtained by Reifenberg's construction. We also allow flat sets with holes. Now see Memoir of the AMS 1012 (in 2012).

  10. CoursM2.pdf Résumés de cours de M2 faits au cours des dernières années; techniques d'analyse, un peu de rectifiabilité, un peu d'intégrales singulières, sujets connexes. Partiellement relu, toujours en désordre.

  11. Notesagreg2015.pdf Des résumés de cours d'agrégation faits à Orsay. Aussi SGDG.

  12. Montreal011.pdf Lecture notes on the proof above of J. Taylor's regularity theorem and connected topics, for the proceedings of a summer school in Montreal (2011), published by the CRM in 2013.

  13. SteinLecture.pdf A survey on Plateau's problem, and why I think it should be solved again. This is the first draft of a paper for the proceedings of the E. Stein conference in 2011. Now see Advances in Analysis, Princeton 2014.

  14. PaloAlto12.pdf Transparencies for 2 lectures on paremeterizations, and mostly Reifenberg's topological disk theorem, in Palo Alto (January 2012). Note : this was never proofread seriously.

  15. PlateauMons012.pdf Les transparents pour un exposé de vulgarisation sur le problème de Plateau (à Mons, Mars 2012).

  16. Rectifiability of self-contracted curves in the Euclidean space and applications, A joint paper with A Daniilis, E. Durand-Cartagena, and A. Lemenant. See the Journal of Geometric Analysis, 2015.

  17. Approximation of a Reifenberg-flat set by a smooth surface. A short paper with a sufficient condition for Approximation of a closed set by a smooth surface (at scale 1), in the Reifenberg sense. See the Bulletin of the Belg. Math. Soc Simon Stevin 2014.

  18. A non-probabilistic proof of the Assouad embedding theorem with bounds on the dimension. A short paper with Marie Snipes, where we give a constructive proof of part of a theorem of Naor and Neiman, which itself improves the Assouad Embedding Theorem. The point is to get bounds on the dimension that do not depend on the desired Holder exponent < 1. See Analysis and Geometry in Metric spaces, 2012.

  19. Regularity of almost minimizers with free boundary. This is the beginning of an attempt with T. Toro to reprove some of the celebrated results of Alt, Caffarelli, and Friedman, on the regularity of free boundaries. See ArXiv or soon Calc. Var. And PDE.

  20. Wasserstein Distances and Rectifiability of doubling measures : Part I, with Jonas Azzam and T. Toro. First of a series of 2 where we try to relate the regularity of (the support of a) measure with its Wasserstein distances to flat measures. See ArXiv or soon Math. Annalen.

  21. Wasserstein2.pdf Wasserstein Distances and Rectifiability of doubling measures : Part II, with Jonas Azzam and T. Toro. Second of the series. Here we compare the measure to images of it by dilations+rotations. See ArXiv:1411.251 again.

  22. Sliding10.pdf A very long paper about the regularity at the boundary of quasiminimal and almost minimal sets subject to sliding boundary conditions.

  23. MonoSlide-2015.pdf A monotonicity formula for minimal sets with a sliding boundary condition, where the point is that the ball is not necessarily centered on the boundary set, but on the other hand the nondecreasing quantity is rarely constant. See ArXiv:1408.7093 or soon Publicacion Matematiques.

  24. DFJM.pdf A long paper with Marcel Filoche, David Jerison, and Svitlana Mayboroda where, motivated by the localization of eigenfunctions for some operators, where we study a variant of the Alt, Caffarelli, and Friedman free boundary problem, but wih many phases.

  25. EffectivePotential.pdf The effective confining potential of quantum states in disordered media, with Doug Arnold, Marcel Filoche, David Jerison, and Svitlana Mayboroda, where we say that the inverse of the landscape function of Filoche-Mayboroda acts as an effective potential for the localization of eigenfunctions for some operators. The red part is corrections compared to the ArXiv text.