## Nicolas Juillet
IRIMAS - Département de Mathématiques |

Research |
Pictures and
animations |
Talks, surveys and slides |
Page
d'enseignement (Teaching) |
Links |

Announcement :

Bienvenue sur ma page web! / Welcome to my web page !

I'm a "Professeur des Universités" at Université de Haute-Alsace (UHA) in Mulhouse at IRIMAS. CV 01/22 (French)

On the period 2009--2021 I've been a "Maître de conférence" of the "Université de Strasbourg" at the "Institut de Recherche Mathématique Avancée" (IRMA).

Research

- May 2021: In my second paper with Charles Boubel we specify what is the Markov-quantile process (that we introduced in the first paper) in the case of absolutely continuous curves in the Wasserstein space. Doing this we obtain a canonical process that is a minimizer solution of the Lagrangian continuity equation and is Markovian. .pdf.

- With Martin Brückerhoff we prove that the value function and the set of solutions of the martingale transport problems are not continuous from dimension 2. .pdf, DOI. Published in Electronic Communications in Probability, 2022, Vol. 27, paper no. 24, 1-10.

- Oct. 2020: In a side project with Moreno Andreatta and Corentin Guichaoua we revisit with metric measure spaces and probabilities the famous "Hexachordal Theorem", a surprising identity coming from the theory of music. .pdf

- Collaboration with Martin Brückerhoff and Martin Huesmann. A peacock is a family of real measures indexed in a way the measures are increasing in convex order. Kellerer's theorem tells us that this is exactly a condition, so that there exists a martingale with these measures as marginals. Using ideas from martingale and stochastic optimal transport we show how to define in systematic way such a martingale. No special assumption is made on the initial measures. .pdf , DOI.Published in Electronic Journal of Probability Vol 27 (2022), paper no. 127, 62p.

- In this short paper written with Jean Bérard, we compare in convex order two sums of positive random variables with randomly many terms M and N. It is known that if M and N are in convex order, so are the corresponding sums. Our (new) proof is effective since it is based on a coupling. We also provide an algorithm .pdf. Published in Electronic Communications in Probability 2020, Vol. 25, paper no. 45, 1-9.

- The Monge problem on the real line can still be the subject of a paper! We select a uniquely determined solution approximating the problem by the problem for concave power costs .pdf. To appear in SIAM Journal on Mathematical Analysis.

- This is a proof that Brunn-Minkowski inequalities (and CD(K,N) conditions are not satisfies on subRiemannian structures) generalizing my thesis result to a more general geometric setting .pdf. To appear in Revista Matematica Ibroamericana.

- In this paper written with Nathael Gozlan we investigate a generalized transport problem .pdf. Published in Proceedings of the London Mathematical Society 120 (2020), no. 3, 434-463.

- In the following paper (joint work with Charles Boubel) we define the Markov-quantile process attached to a curve of probability measures. This has connections with the peacock problem and the transport equation from an optimal transport perspective .pdf. To appear in Journal de l'Ecole Polytechnique 9 (2022).

- In this short paper written with Jean Bérard, we examine the reconciliation of probability measures defined each on a sigma-algebra of the sape space Omega .pdf. Published in European Journal of Applied Mathematics 30 (2019), no. 6, 1300-1310 (special issue ``Applied Optimal Transport'').

- In this joint work with Michel Bonnefont we try to couple two Brownian motions of the Heisenberg group in a way that the Wasserstein distance induced by the coupling does not explode as time goes to infinity, .pdf. Published in Annales de l'Institut Henri Poincaré. Probabilités et Statistiques 2020, Vol 56, No 1, 543-565.

- As new martingale optimal transports we introduce the shadow couplings, generalizing the left-curtain coupling. A connexion is made with the Skorokhod embedding problem. This is a joint work with Mathias Beiglböck, .pdf. To appear in Trans. Amer. Math. Soc.

- In this joint work with Mario Sigalotti we look at generalizations of the Whitney extension theorem for Carnot groups. We introduce the notion of pliable Carnot group .pdf. Published in Analysis & PDE, No. 7, 1637-1661 (2017).

- This following paper with Matthias Erbar is in relation with an optimal transport approach of the Ricci flow that makes sense for metric spaces. We look at Euclidean cones and the Heisenberg group .pdf. Published in Journal de Mathématiques Pures et Appliquées (9) 110 (2018), 123--154.

- A paper on examples/counterexamples for peacocks and martingales indexed on the plane .pdf. Published in Séminaire de Probabilités XLVIII, 13--32.

- An optimal transport approach of the peacock problem is proposed .pdf. It used to be the last part of the next item of this list. Published in Electronic Journal of Probability Vol 23 (2018), paper 8.

- The left-curtain coupling is continuous with respect to its marginals: "Stability of the shadow projection and the left-curtain coupling".pdf. Published in Annales de l'Institut Henri Poincaré. Probabilités et Statistiques 2016, Vol 52, No 4, 1823-1843. The last part of the corresponding preprint on ArXiv has become the paper one item above.

- My first article with Mathias Beiglböck, ''On a problem of optimal transport under marginal martingale constraints'' .pdf. Published in Annals of Probability 2016, Vol 44, No. 1, 42-106.

- Here is my paper about hypoelliptic diffusion seen from the point of view of optimal transport, still for Heisenberg groups. Published in Calculus of Variations and PDEs, 2014, Volume 50, Issue 3-4, pp 693-721. It is quite different from the original preprint of 2009 (.pdf)

- My small article on the structure of the Euclidean Wasserstein space, entitled "On displacement interpolation of measures involved in Brenier's Theorem" has appeared in Proc. of AMS, 139(10) (2011). (.pdf)

- "A counterexample to the geometric traveling salesman problem in the Heisenberg group" has appeared in the Revista Matematica Ibroamericana, 26(3) (2010). It presents a rectifiable curve that does not satisfy a criterion that was supposed to be true for all rectifiable curves of the Heisenberg group. (see .pdf)

- In a paper with Alessio Figalli we answer an open question of L. Ambrosio and S. Rigot (JFA 2004) about the optimal transport in the Heisenberg group : during an optimal transport starting from an absolutely continuous measure, the interpolated intermediate measures are absolutely continuous too. This paper is published in Journal of Functional Analysis, 255 (2008), no 1. see (.pdf)

- There are certain new synthetic Ricci curvature bounds whose definition is adapted to the metric measure spaces. If you want to know what happens when you consider them for the Heisenberg group, you can have a look at Int. Math. Res. Not. IMRN 2009, no. 13, 2347-2373. or (.pdf). NB: this paper is called "Geometric inequalities and generalized Ricci bounds in the Heisenberg group". It is sometimes still referred under its preliminary name: "Ricci curvature bounds and geometric inequalities in the Heisenberg group".

In December 2018, I've been defending my habilitation thesis "Transport optimal en théorie des processus et en géométrie" (other link).

In December 2008, I've been defending my doctoral thesis in mathematics at the Institut Fourier (Université Grenoble 1) and the Institut für angewandte Mathematik (Universität Bonn). My advisors were Hervé Pajot and Karl-Theodor Sturm. My thesis is called "Optimal transport and geometric analysis in Heisenberg groups" (other link).

Pictures and animations (Up)

I have written some animations with Geogebra: In this animation, you can examine the shape of the triangles of the Heisenberg group. The second animation shows how the third coordinate of an horizontal curve evolves with respect to the swept algebraic area. Another one: animation. On the left, the .gif file illustrate the "sunset coupling" of between two uniform measures defined in this paper .pdf. In this animation one can experiment several shadow couplings between a discrete measure (three atoms) and a continuous measure. Here are the sunset coupling, the left-curtain coupling and the middle-curtain coupling between two uniform measures. |

Talks, surveys and slides (Up)

A list of talks with sometimes a survey in proceedings. I've added some slides: these are not "self-contained" diaporamas but rather the companions of a talk with chalk at blackboard.

- Dijon, séminaire SPOC, June 2023
- Oberwolfach, Heat Kernels, Stochastic Processes and Functional Inequalities, November 2022
- Bonn, New Trends in Dirichlet forms and Optimal Transport, September 2022
- Grenoble, AMS-EMS-SMF Meeting - special session on "Quantitative Geometry of Transportation Metrics", July 2022
- Paris, séminaire "Analyse-proba", April 2022
- Banff, Stochastic Mass Transport, March 2022
- Paris, "2e journée MAS MODE de la la SMAI", March 2022
- Münster, Oberseminar Stochastik, December 2021
- Marseille, "Schrödinger Problem and Mean-field PDE Systems", November 2021
- Kristiansand, Young researchers between geometry and stochastic analysis, June 2021
- Mulhouse, séminaire de mathématiques, December 2020
- Lorraine, séminaire généraliste de l'équipe probabilités et statistiques, December 2020
- Lille, séminaire de probabilités et statistique, November 2020
- Grenoble, séminaire de probabilités, March 2020
- Bergen, Young researchers between geometry and stochastic analysis, February 2020
- Osaka, Probability and Geometry, September 2019
- Bielerhöhe, summer school on stochastic and geometric analysis, July 2019
- London, stochastic analysis seminar, Jun 2019
- Vienna, probability seminar, February 2019
- Lyon, séminaire EDP-Analyse, November 2018
- Toulouse, séminaire "mathématiques de la décision", October 2018
- Toulouse, séminaire de probabilités, October 2018
- Marseille, séminaire de probabilités et statistique, September 2018
- Tours, "journées de probabilités", June 2018
- Bâle, Seminar in Analysis and Probability theory, May 2018
- Durham, Yorshire and Durham Geometry Day, March 2018
- Durham, Geometry and Topology seminar, March 2018
- Durham, Study group on optimal transport, March 2018
- Paris 5, Groupe de travail en probabilités du MAP5, October 2017 (.pdf)
- Grenoble, Journée sous-Riemanniennes 2017, October 2017 (video)
- Bordeaux, Journées du GDR AFHP, October 2017
- Bonn, IAP on Metric Measure Spaces and Ricci curvature, September 2017
- Oxford, Martingale Optimal Transport (and friends), September 2017
- Luxembourg, conference GeoProb2017, July 2017
- Nancy, séminaire de probabilités et statistiques, March 2017
- Jyväskylä (Finland), geometry seminar, February 2017
- Nancy, séminaire de géométrie différentielle, December 2016
- Bonn, follow-up workshop to JTP "optimal transportation", August 2016 (video)
- Orléans, séminaire du MAPMO, March 2016
- Paris, "groupe de travail courbure, transport optimal et probabilités", February 2016
- Montpellier, séminaire Darboux, January 2016
- Zürich, "Frontiers in analysis and probability", October 2015
- Nantes, séminaire de géométrie, September 2015
- Paris, séminaire d'analyse fonctionnelle, June 2015
- Grenoble, séminaire de probabilités, Mai 2015
- Bonn, workshop "Optimal transport and stochastics", March 2015 (video)
- Bonn, conference "New trends in optimal transport", March 2015
- Bordeaux, séminaire d'analyse, February 2015
- Marseille, groupe de travail Calcul des variations et EDP, February 2015
- Paris, thematic day "Optimal transport" of the subRiemannian trimestre, October 2014
- Palaiseau, groupe de travail de mathématiques financières de l'école polytechnique, October 2014 (.pdf)
- Luxembourg, probability seminar, October 2014
- Paris, conference on "Geometric analysis on subRiemannian manifolds", October 2014
- Bordeaux, conférence ProbaGéo, June 2014
- Karlsruhe, "5th Geometry day of Heidelberg-Karlsruhe-Strasbourg", June 2014
- Moscou, "International Youth Conference in Geometry and Control", April 2014 (video)
- Eindhoven, YEP XI "Young European Probabilists", March 2014
- Rennes, séminaire "probabilités et modèles aléatoires", March 2014
- Evry, séminaire "probabilités et maths financières", January 2014
- Bonn, Arbeitsgruppe Sturm, July 2013
- Orléans, "journées de probabilités", June 2013
- Marseille CIRM, rencontre "ANR Géométrie et transport", May 2013
- Toulon, séminaire "analyse non-linéaire et modélisation", December 2012
- Toulouse, séminaire de géométrie, December 2012
- Nice, séminaire "géométrie et analyse", December 2012
- Paris, séminaire "géométrie sous-riemannienne", December 2012
- Pisa, "ERC Workshop on Optimal transport theory" (short talk), November 2012 (.pdf)
- Dijon, journée "Contrôle optimal géométrique et transport optimal", septembre 2012
- Montpellier, rencontre "ANR Géométrie et transport", June 2012
- Poitiers, ''Conference ProbaGeo'', June 2012
- München, "Perspectives in Optimal transport", December 2011 (.pdf)
- Bonn, Intern. Conf. on Stoch. Anal. and App. (invited
session), September 2011

- Wien, Seminar Finanzmathematik, May 2011

- Dijon, séminaire "Statistiques, probabilités et applications",
May 2011

- Dijon, Conference in Memory of Paul Malliavin, October 2010
- Nice, "New trends in subRiemannian geometry", April 2010
- Maynooth (Ireland), colloquium, March 2010 (.pdf)
- IHES, "Analysis and geometry on groups", September 2009
- Grenoble, conférence interdisciplinaire "Repère et espace", April 2009. Text for the proceedings, in french (.pdf).
- Marseille, séminaire d'analyse appliquée, March 2009
- Strasbourg, séminaire "GT3", January 2009. I've never used these slides! In french (.pdf)
- Nice, séminaire "Géométrie et analyse", January 2009
- Orsay, séminaire d'analyse harmonique, March 2009
- Kyoto, contributed talk in the conference "Probabilistic approach to Geometry", August 2008. This survey is published in the conference proceedings of the ASPM 57 . (.pdf)
- Luxembourg, mathematics seminar, June 2008
- CIRM (French mathematical center in Marseille), workshop "Espaces métriques mesurés, courbure de Ricci minorée et transport optimal", March 2008
- Nantes, séminaire de géométrie, January 2008
- Bologna, seminario di analisi matematica, November 2007
- Lausanne, séminaire de géométrie et théorie ergodique, September 2007
- Levico Terme (close to Trento), conference "Analysis and Geometry in metric measure spaces", June 2007
- Orsay (Paris sud), séminaire d'analyse harmonique, Mai 2007
- Toulouse, groupe de travail "optimal transport", January 2007
- Grenoble, séminaire de théorie spectrale et géométrie, October 2006. The related survey was published in the"Actes du Séminaire de Théorie Spectrale et Géometrie"of the Institut Fourier. (.ps)

Last updated September 2021.