Quelques prépublications, notes de cours, et exposés.

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 ; was once supposed to be published 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 generalizalization 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. See GAFA 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. See the Annales de la faculté des sciences Toulouse 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. See Memoir of the AMS 1012 (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, et même un bout de représentations conforme. Relu de manière variable et toujours un peu en désordre.

11. 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.

12. 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. See Advances in Analysis, Princeton 2014.

13. 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.

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

15. 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.

16. 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.

17. 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.

18. Regularity of almost minimizers with free boundary. This is the beginning of an attempt with T. Toro to re-prove some of the celebrated results of Alt, Caffarelli, and Friedman, on the regularity of free boundaries, in the context of almost minimizers. See https://arxiv.org/abs/1306.2704 or Calc. Var. And PDE 2015.

19. 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 https://arxiv.org/abs/1408.6645 or Math. Annalen.

20. 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 https://arxiv.org/abs/1411.2512 or Math. Z. 2017.

21. Sliding10.pdf A quite long paper about the regularity at the boundary of quasiminimal and almost minimal sets subject to sliding boundary conditions. Astérisque 2019.

22. 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 https://arxiv.org/abs/1408.7093 or Publicacion Matematiques 2016.

23. A free boundary problem for the localization of eigenfunctions. 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 with many phases. See https://arxiv.org/abs/1406.6596 or Astérisque 392 (2017).

24. 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. See https://arxiv.org/abs/1505.02684 or PRL 2016.

25. Computing spectra without solving eigenvalue problems, with Doug Arnold, Marcel Filoche, David Jerison, and Svitlana Mayboroda, where we explain how to use the landscape function to get a good idea of the spectrum of an elliptic operator. See https://arxiv.org/abs/1711.04888.

26. AgmonAndWeyl.pdf Localization of eigenfunctions via an effective potential, with Doug Arnold, Marcel Filoche, David Jerison, and Svitlana Mayboroda, where we prove decay, for eigenfunctions of an elliptic operator, in the regions where the effective potential (the inverse of the landscape function) is large. See https://arxiv.org/abs/1712.02419v2.

27. Free boundary regularity for almost-minimizers, with Max Engelstein and Tatiana Toro, where we continue in the context of 18 and prove the $C^1$ regularity of the free boundary in the one-phase problem. See https://arxiv.org/abs/1702.06580.

28. Elliptic theory for sets with higher co-dimensional boundaries, with Joseph Feneuil and Svitlana Mayboroda, where we introduce an elliptic measure for domains whose boundary is Ahlfors regular of co-dimension larger than 1, based on an appropriate degenerate elliptic operator. See https://arxiv.org/abs/1702.05503.

29. Dahlberg’s theorem in higher co-dimension, with Joseph Feneuil and Svitlana Mayboroda, where we show that for small Lipschitz graphs (of higher codimensions), the elliptic measure introduced in the previous paper is $A_\infty$-absolutely continuous with respect to the Hausdorff measure. See https://arxiv.org/abs/1704.00667.

30. Square functions, non-tangential limits and harmonic measure in codimension 1, with Max Engelstein and Svitlana Mayboroda, where we find an intriguing case of higher codimension elliptic measure (as above) which is proportional to Hausdorff measure. See https://arxiv.org/abs/1808.08882.

31. MinimalPark3 Sliding almost minimal sets and the Plateau problem, lecture notes of PCMI (Park city) in july 2018. More of a survey, about 56 pages.

32. Local Sliding Regularity A local description of 2-dimensional almost minimal sets bounded by a curve. An attempt to prove results like J. Taylor, but at the boundary. 337 pages (sorry !) but the introduction is shorter and more precise than the PCMI notes above.

33. The landscape law for the integrated density of states With M. Filoche and S. Mayboroda. Estimates on the number of eigenvalues for some Schrödinger operators ; uses the number of boxes where the effective potential (the reciprocal of the Landscape function)(an error in the abstract) takes a small value.

34. Regularity for the almost-minimizers of variable coefficient Bernouilli-type functionals With M. Engelstein, M. Smit Vega Garcia, and T. Toro. A continuation of the study of free boundary problems of the Alt-Caffarelli-Friedman, this time with non isotropic and variable coefficients.

35. Elliptic theory in domains with boundaries of mixed dimensions or Astérisque 442. With J. Feneuil and S. Mayboroda. Like Item 28, but we found out that we wanted to allow boundaries that are composed of pieces of various dimensions. Now the boundary comes with a doubling measure, with compatibility properties.

36. Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets With S. Mayboroda. Like Item 29, but we finally got what we wanted in the first place : the analogue in higher co-dimensions of the A∞ absolute continuity results for UR boundaries.

37. Good elliptic operators on Cantor sets With S. Mayboroda. A counterexample : a suitable elliptic operator (not close to the Laplacian) that gives an absolutely continuous elliptic measure on the Garnett-Ivanov Cantor set.

38. Mon cours accéléré de M2 sur la théorie spectrale et un peu d’analyse. Fait en 2022 pour la première fois, donc plein d’erreurs.

38. Cantor sets with absolutely continuous harmonic measure With A. Julia and C. Jeznach. Some lower dimensional (non self-similar) Cantor sets in the plane, with a harmonic measure absolutely continuous w.r.t. the Hausdorff measure.