"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)
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.
Since January 2018 I am Assistant Professor at PUC Chile.
ICIAM conference, Valencia, Spain, July 15-19, 2019, -- talks in mini-symposia "Discrepancy and Minimal Energy" and "Molecular simulation: quantum mechanical models" (see a photo of me with L. Betermin at https://iciam2019.org/images/site/FotosCongreso/CommonAreas/Session(4).jpg)
BIRS workshop January 27th - February 1st, 2019 on Optimal Transport Methods in Density Functional Theory.
Workshop that I co-organized, on November 30th, 2018, London: Symmetries, asymptotics and multi-scale approaches
AIM workshop Discrete geometry and automorphic forms, San Jose, California, September 24-28, 2018
Workshop on Geometric Measure Theory in Verona, June 11-15, 2018
Workshop on Nonlocal interactions: Dislocations and beyond, Bath, 11-14 June 2018
ICERM trimester on Optimal and Random Point Configurations, Brown University, Providence, February 26th - March 2nd, 2018
Workshop that I co-organized, on February 15-17, 2017, London: New trends in Mathematical Physics at the interface of Analysis and Probability