Mircea Petrache

  • "Crystallization to the square lattice for a two-body potential" with L. Betermin and Lucia De Luca https://arxiv.org/abs/1907.06105 got accepted in Archiv for Rational Mechanics and Analysis

  • New preprint "Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings" with Giacomo Del Nin https://arxiv.org/abs/2101.11977

  • Fondecyt Regular grant (4 years 2021-2025). Project title: "Rigidity, stability and uniformity for large point configurations"

  • En Enero 11-15 dicte' el mini-curso (en espanol) "Teoria de la Rigidez y Tensegridades" (playlist Youtube: disponible en enlace arriba)

  • New preprint "Sharp discrete isoperimetric inequalities in periodic graphs via discrete PDE and Semidiscrete Optimal Transport" with Matias Gomez (master student UTFSM) https://arxiv.org/abs/2012.11039

  • "Classification of uniformly distributed measures of dimension 1 in general codimension" with Paul Laurain https://arxiv.org/abs/1905.09601 - accepted in Asian J. Math.

(last updated: Jan. 29th, 2021)

Contact:

Facultad de Matemáticas, Avda. Vicuña Mackenna 4860, Macul, Santiago, 6904441, Chile

Email: mpetrache (you know what) mat.puc.cl

Cellphone: +56 9 3686 3545

Office: 142, Maths Department. Office Phone: 23544038

Mathematics is there to interact with other sciences. I'm actively searching new ways to apply it in real-world problems. I will not work on a topic unless I actually believe that the gained knowledge can later be applied outside mathematics.


I was trained in PDEs, Calculus of Variations, Geometric Analysis and Geometric Measure Theory. Which I now use to study emergent behavior and structures, especially (but not only) for large point configurations.


A song making fun of maths

A storyline including some of my interests

  • In geometry, physics, computer science "optimization principles" (which play the role of "rulles" in our models) produce new structures not imposed in the model,and therefore called emergent structures.

  • If the new structure can be topologically detected, we talk of topological singularities (e.g. "charges", "vortices", "defects", etc. , depending on the field). Mathematically, these form e.g. in nonlinear variational problems, or steady or limit states in dynamical systems.

  • If point vortices appear, they can be described as interacting point configurations, made of singularities/defects.

  • Such point configurations organize themselves in relation with the geometry/shape of the ambient space.

  • The key limiting factor in working with large systems of points is their exponential complexity (=the space of configurations grows in complexity exponentially in the number of points).

How to understand/control exponential complexity?

Quantify the group behavior of the points, through problem-specific structures, that in turn instruct low-complexity approximations. Examples:

  • Statistical mechanics measures group behavior and correlations within the theory of crystallization and in the study of other phases of matter.

  • In Material science in macroscopic/continuum limits the properties of large numbers of atoms are summarized by (fewer) continuum variables.

  • In probability theory, ad-hoc large deviation principles including concentration of measure ideas.

  • In computer science and analysis, multi-scale analysis and metric dimension reduction are studied. Interesting insights come from compressed sensing.

  • Neural networks are assumed to "learn" the structure of datasets: it is interesting to find quantitative principles that explain how they do it.

CV

Since January 2018 I am Assistant Professor at PUC Chile.

September 2017-December 2017: visited FIM in Zürich and Vanderbilt University.

2015-17: MIS Max Planck Institute in Leipzig and Max Planck Institute in Bonn. (Funding: European Post-Doc Institute. Mentors: B. Kirchheim, S. Müller)

2013-2015: Postdoc at Laboratoire Jacques-Louis Lions (Funding: Fondation de Sciences Mathèmatiques de Paris. Mentor: Sylvia Serfaty)

2013: Ph.D at ETH Zürich, (Thesis: "Weak bundle structures and a variational approach to Yang-Mills Lagrangians in supercritical dimensions". Advisor: Tristan Rivière).

2008: MSc Scuola Normale Superiore, (Thesis: "Differential Inclusions and the Euler equation". Advisor: Luigi Ambrosio)


cv.pdf

Past events: