# Mircea Petrache

## Research summer school Point Configurations: Deformations and Rigidity , London is Postponed to summer 2021

- New preprint
**"Crystallization for Coulomb and Riesz Interactions as a Consequence of the Cohn-Kumar Conjecture"**with S. Serfaty https://arxiv.org/abs/1908.09714 **The first PUC-Bath workshop on "Nonlinear PDE's and Applications"**takes place at PUC on 10-13 September 2019- New preprint
**"Crystallization to the square lattice for a two-body potential"**with L. Betermin and Lucia De Luca https://arxiv.org/abs/1907.06105 - New preprint "
**Classification of uniformly distributed measures of dimension 1 in general codimension**" with Paul Laurain https://arxiv.org/abs/1905.09601 - "
**Cyclically monotone non-optimal N-marginal transport plans and Smirnov-type decompositions for N-flow**s" https://arxiv.org/abs/1903.09817 **"Unconstrained polarization (Chebyshev) problems: basic properties and Riesz kernel asymptotics"**paper with Ed Saff and Doug Hardin appeared on https://arxiv.org/abs/1902.08497

(last updated: Nov 30th, 2019)

*Office: *142, edificio rolando Chaqui

*Email: *mpetrache [usual symbol] mat.puc.cl

### 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 storyline including some of my interests

- In geometry, physics, computer science, and other fields, "optimization principles" (or more generally "the rules of the system") sometimes produce new
**emergent structures**. - New structure takes the form of
such as as**topological singularities**(*"charges", "vortices", "defects", etc.*, depending on the field*).*Mathematically, these form e.g. in**nonlinear variational problems**, or as**steady or limit states of dynamical systems**. - The above belong to the topic of
**interacting point configurations,**made of singularities/defects or otherwise. In general such configurations organize themselves in relation with**the geometry/shape of the ambient space.** **Exponential complexity,**is the landmark property of the point configurations I work with, and is a key limiting factor in working with large systems of points.- A way to understand/control such high complexity, is to find new clever ways to
**quantify**the**group behavior in large systems**, and find new**problem-specific**structures that furnish**low-complexity approximations**

Within statistical physics, I am interested in **crystallization **and **phase transition** phenomena. Material science has a long tradition of studying **rigidity** of materials and various macroscopic properties (both quantum and classical) as **emergent behaviour** of systems of defects. In probability theory, one builds ad-hoc **large deviation principles** and quantifies **long-range correlations**. In computer science, there are useful techniques known as **metric dimension reduction,** while other useful methods come from **compressed sensing.**

Look at my papers for the results I got so far.

And here for a selection of topics I'd like to work on, for example with student.

Slides from a recent talk

A song making fun of maths

## 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, 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**