Mircea Petrache    





    
Address:  
Forschungsinstitut fuer Mathematik,
Raemistrasse 101, 8092, Zuerich, Switzerland

Office: G 36.1

Email: mircea.petrache (you know what) protonmail.com
Phone: +49 (0)1577 06 36 926
(This here is my work webpage. In case you prefer to see my old blog instead, you can find it here.)

About
In September 2017-December 2017 I am visiting the FIM in Zürich.

In 2015-17 I was at the MIS Max Planck Institute in Leipzig
and Max Planck Institute in Bonn, under a scholarship of the European Post-Doc Institute (mentors: B. Kirchheim, S. Müller)

In 2013-2015 I had a postdoctoral position at Laboratoire Jacques-Louis Lions in collaboration with Sylvia Serfaty and under a scholarship of the Fondation de Sciences Mathèmatiques de Paris.

In 2013 I obtained my Ph.D at ETH Zürich, where I completed my thesis entitled   "Weak bundle structures and a variational approach to Yang-Mills Lagrangians in supercritical dimensions" (advisor: Tristan Rivière). In 2008 I graduated at Scuola Normale Superiore, with a thesis entitled "Differential Inclusions and the Euler equation" (advisor: Luigi Ambrosio)

[for more informations see my CV]


Research

I am interested in Calculus of Variations, PDEs, Geometric Measure Theory, with special emphasis on interactions/applications to other fields, such as Probability, Optimization, Mathematical Physics.

Point-like (or more complicated) topological singularities arising in Nonlinear Sobolev Spaces and Gauge Theory can be interpreted as particles or vortices, and I study the asymptotic behavior of minimizing configurations as the number of vortices increases. The sharp asymptotics appearing in these studies are relevant in Random Matrix theory, in Approximation theory and Statistical Physics.

At the moment I am especially interested in uniformization/crystallization phenomena, where for large numbers of points one can prove/quantify that the configurations come close to forming lattice-like structures.

 Uniformization phenomena appear also in a wider class of geometric minimization problems.

Related to the previous point is also the study of asymptotics of a large number of quantum particles, which is relevant to the computations of shapes of large molecules via Density Functional Theory. Here a multimarginal Optimal Transportation problem with an exotic cost appears, and I'm interested in the asymptotics as the number of marginals grows to infinity.

Conferences and events

Publications and Preprints

Large particle systems with long-range interactions
(..see this YouTube video for a fun simulation from "Cody's lab" channel, that anybody can reproduce at home: Why do those magnets produce that pattern?)

Crystals
Supercritical Yang-Mills theory and Nonlinear Sobolev spaces
  • arxiv.org/abs/1306.2010 The resolution of the Yang-Mills Plateau problem in super-critical dimension (with T. Rivière) Advances in Mathematics, Volume 316, 20 August 2017, Pages 469-540, DOI
  • arxiv.org/abs/1508.07813 Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds (with J. Van Schaftingen) Int. Math. Res. Not., 2017 (12), Pages 3647-3683, DOI
  • arxiv.org/abs/1306.6763 A singular radial connection over B^5 minimizing the Yang-Mills energy Calc. Var. and PDE, 54 (2015) no.1, 631-642. DOI 
  • arxiv.org/abs/1302.5659 Global gauges and global extensions in optimal spaces (with T. Rivière)  Analysis and PDE, 7 (2014), No. 8, 1851–1899.  DOI
  • arxiv.org/abs/1204.0209 Interior partial regularity for minimal L^p-vectorfields with integer fluxes Ann. S.N.S. Pisa Cl. Sci., Vol. XIV (2015), issue 4, 1119-1156. DOI
  • arxiv.org/abs/1204.0175 Notes on a slice distance for sigular L^p-bundles Journ. Funct. Anal., 267 (2014), no. 2,   405–427DOI
  • arxiv.org/abs/1007.0681 An integrability result for L^p-vectorfields in the plane Adv. Calc. Var., 6 (2013), no. 3, 299–319. DOI
  • arxiv.org/abs/1007.0668 Weak closure of singular abelian L^p-bundles in 3-dimensions (with T. Rivière)  Geom. Funct. Anal. 21 (2011), 1419-1442. DOI
Optimal transport and currents

Mean-field equations for higher-order laplacians

Theses

  • PhD thesis: Weak bundle structures and a variational approach to Yang-Mills Lagrangians in supercritical dimensions  Download
  • Master's degree thesis: Differential inclusions and the Euler equation Download


Some puzzling natural phenomena: Saturn "hexagonal" vortices, Ants vortex ( video 1, video 2, a paper)