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-flows" 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 topological singularities (such as as "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
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