# 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/2101.11977Fondecyt 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:

**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)

## Past events:

**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, 2018Workshop on

**Geometric Measure Theory in Verona**, June 11-15, 2018Workshop on

**Nonlocal interactions: Dislocations and beyond**, Bath, 11-14 June 2018ICERM 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**