Renato Velozo Ruiz

Currently based in Toronto, Canada

Welcome to my webpage! I'm a Postdoctoral Fellow in the Deparment of Mathematics at the University of Toronto. My mentors are Stefanos Aretakis and Yakov Shlapentokh-Rothman.

My research focuses on asymptotic stability results for collisionless many-particle systems. I am particularly interested in the study of particle systems in general relativity.

During 2022-2023, I was a Postdoctoral Fellow in the Laboratoire Jacques-Louis Lions at the Sorbonne Université mentored by Jacques Smulevici. I obtained my PhD in 2023 from the University of Cambridge under the supervision of Mihalis Dafermos and Clément Mouhot.

You can reach me via email at renato.velozo.ruiz@utoronto.ca.

"The universe can best be pictured as consisting of pure thought, the thought of what for want of a better word we must describe as a mathematical thinker."

Sir James Hopwood Jeans. “The Mysterious Universe”

Resume

Employment

Download Resume

Sep 2023 - Till now

Postdoctoral Fellow
at University of Toronto

Advisors: Stefanos Aretakis and Yakov Shlapentokh-Rothman

Oct 2022 - Aug 2023

Postdoctoral Fellow
at Sorbonne Université

Advisors: Jacques Smulevici

My Education

Oct 2018 - Jan 2023

PhD in Mathematics
at Univeristy of Cambridge

Advisors: Mihalis Dafermos and Clément Mouhot

Mar 2017 - Jul 2018

MS in Mathematics
at Pontificia Universidad Católica de Chile

Advisors: Jairo Bochi

Mar 2013 - Dec 2016

Bachelor in Mathematics
at Pontificia Universidad Católica de Chile

Academic Visits

Nov 2019 - Department of Mathematics at Princeton University, invited by Mihalis Dafermos.
Aug 2019 - Department of Mathematics at Pontificia Universidad Católica de Chile, invited by Duvan Henao.

Academic Honors

2018 to 2022 - Scholarship for PhD in Mathematics, CONICYT-Chile Cambridge Scholarship, Cambridge Trust, University of Cambridge.
2018 to 2022 - Scholarship for PhD in Mathematics, National Commission for Scientific and Technological Research, Chile.
2020 - Smith-Knight & Rayleigh-Knight Prize, Faculty of Mathematics, University of Cambridge.
2018 - Scholarship for Master of Science, National Commission for Scientific and Technological Research, Chile.
2017 - Scholarship for Master of Science, Department of Mathematics, PUC
2016 - Teaching Assistant of First Category, Department of Mathematics, PUC.
2013 to 2016 - Scholarship Rolando Chuaqui (awarded to the two best students of the year), Department of Mathematics, PUC.

Research

Research Interests

My research project lies at the interface between kinetic theory and general relativity. I am interested in the behavior of self-gravitating collisionless many-particle systems in the geometric framework of general relativity, which are modelled as the solutions to the Einstein-Vlasov system. Currently, my research focuses on the black hole stability problem in the context of relativistic kinetic theory. More precisely, my work concerns stability results for massless and massive collisionless many-particle systems on black hole exteriors.

I am also interested in stability results for self-gravitating collisionless many-particle systems in the framework of Newtonian gravity, which are modelled as the solutions to the Vlasov-Poisson system.

Scientific Work

Stability of Schwarzschild for the spherically symmetric Einstein–massless Vlasov system; in preparation. ⬇ Slides
Decay properties of massive Vlasov fields on Schwarzschild spacetime; in preparation.
Late-time asymptotics of small data solutions for the Vlasov-Poisson system, joint with L. Bigorgne; arXiv:2404.05812, preprint.
A note on integrated local energy decay estimates for spherically symmetric black hole spacetimes, joint with G. Holzegel, and G. Mavrogiannis; arXiv:2403.02533, preprint.
Decay properties of Vlasov fields on non-trapping asymptotically hyperbolic manifolds, joint with A. Velozo Ruiz; arXiv:2312.13496, preprint.
Modified scattering of small data solutions to the Vlasov-Poisson system with a trapping potential, joint with L. Bigorgne, and A. Velozo Ruiz; arXiv:2310.17424, preprint.
Small data solutions for the Vlasov–Poisson system with a repulsive potential, joint with A. Velozo Ruiz; Commun. Math. Phys. 405, 80 (2024).
Gevrey regularity for the Vlasov-Poisson system; Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 4, 1145-1165.
Characterization of uniform hyperbolicity for fibre-bunched cocycles; Dyn. Syst. 35 (2020), no. 1, 124-139.
Rate distorsion theory, metric mean dimension and measure theoretic entropy, joint with A. Velozo Ruiz; arXiv:1707.05762, preprint.

Conference Talks & Seminars

Mar 2024 - Analysis Seminar, Columbia University.
Mar 2024 - Mathematical Physics Seminar, Johns Hopkins University.
Nov 2023 - Princeton Gravity Initiative's Seminar Series, Princeton University.
Nov 2023 - Analysis of Applied PDEs Graduate Student Seminar, University of Toronto.
Sep 2023 - Analysis and Applied Math Seminar, University of Toronto.
Jul 2023 - Seminari di Metodi Analitici e Probabilistici, Università di Pisa.
Jun 2023 - MathInGP Postdoctoral Research Symposium, Sorbonne Université.
Apr 2023 - PDE Seminar, DIM, Universidad de Chile.
Apr 2023 - Dynamical Systems Seminar of Santiago, Chile.
Mar 2023 - MAFRAN days, King's College, Cambridge.
Dec 2022 - Mathematical General Relativity Seminar, Sorbonne Université.
Dec 2022 - International Conference on Geometric Analysis and Hyperbolic Equations, Guangxi Center for Mathematical Research, Guangxi University.
Nov 2022 - Workshop ‘Vlasov equations: geometric and semi-classical aspects’, Institut de recherche mathématique de Rennes.
Jun 2022 - London PDE Seminar, UK.
May 2022 - Topics in General Relativity, University of Münster.
Apr 2022 - Oxbridge PDE Conference, Oxford.
Apr 2022 - Early Career Math Colloquium, University of Arizona.
Feb 2022 - PDE Seminar, Brown University.
Jan 2022 - Analysis and Geometry Seminar, PUC.
Jun 2022 - Annual Meeting of the Chilean Mathematical Society.
Dec 2021 - MAFRAN days, Université Paris-Dauphine.
Dec 2021 - Mathematical General Relativity Seminar, Sorbonne Université.
Jul 2021 - Online Mathematical GR and Hyperbolic PDE Seminar.
Jun 2021 - Junior Analysis Seminar, Imperial College London.
May 2021 - CMI Student Seminar Series, University of Cambridge.
Jan 2021 - Seminar of Kinetic Theory, University of Cambridge.
Dec 2020 - CDT Student Conference, Cambridge.
Nov 2020 - Seminar of Partial Differential Equations, DIM, Universidad de Chile.
Sep 2019 - Seminar of Analysis and Geometry, Pontificia Universidad Católica de Chile.
Sep 2019 - Graduate Student Seminar, Pontificia Universidad Católica de Chile.
Jul 2018 - Seminar of Dynamical Systems, Pontificia Universidad Católica de Chile.
Jul 2018 - Seminar of Dynamical Systems, Pontificia Universidad Católica de Valparaíso.

Teaching & Outreach

As Lecturer

Spring 2023 - MAT136 Calculus II, University of Toronto.
Fall 2023 - MAT244 Ordinary Differential Equations, University of Toronto.

As Teaching Assistant

Mich 2021 - Part III Analysis and PDEs, University of Cambridge.
Lent 2021 - Part IB Electromagnetism, University of Cambridge.
Mich 2020 - Part II Linear Analysis, University of Cambridge.
Mich 2020 - Part II Integrable Systems, University of Cambridge.
Lent 2020 - Part II Differential Geometry, University of Cambridge.
Spring 2017 - Field theory and Galois theory, PUC.
Spring 2017 - Topology, PUC.
Winter 2017 - Ordinary Differential Equations, PUC.
Spring 2016 - Field theory and Galois theory, PUC.
Spring 2015 - Introduction to Algebra, PUC.
Fall 2015 - Multivariable Calculus, PUC.
Spring 2014 - Calculus I, PUC.
Fall 2014 - Introduction to Geometry, PUC.

Outreach

2017 to 2018 - Instructor in program 'Introduction to Mathematical Reasoning' (TRM) at PUC destined to talented high school students.
2017 - Instructor in 'Mathematical Reasoning Program' (PRM). Lectures to high school teachers organized by the Deparment of Mathematics, PUC.
2016 to 2017 - Instructor of the Chilean team for international math olympiads (IMO and Iberoamerican).

Other Related Activities

May 2021 - A beginners guide to galactic dynamics (with the eyes of a mathematician). Talk for the MCR of Churchill College, University of Cambridge.
Sept 2017 - da Vinci Dome. Stand at Math Festival, San Antonio, Chile.
Aug 2017 - Chaos and Dynamical Systems. Talk at Mathematics Day, High School Padre Pedro Arrupe, Santiago.
Aug 2017 - Mathematical Research Workshop for high school students, organized by PAR Explora of CONICYT RM Norte, PUC.
Aug 2016 - Are there infinities larger than others? Talk at TRM, PUC.
Mar 2016 - Workshop on Problem Solving Strategies. Talk at II Math Camp (CaMat), Santiago, Chile.

"The universe can best be pictured as consisting of pure thought, the thought of what for want of a better word we must describe as a mathematical thinker."

Sir James Hopwood Jeans. “The Mysterious Universe”

Resume

Employment

Postdoctoral Fellow
at University of Toronto

Postdoctoral Fellow
at Sorbonne Université

My Education

PhD in Mathematics
at Univeristy of Cambridge

MS in Mathematics
at Pontificia Universidad Católica de Chile

Bachelor in Mathematics
at Pontificia Universidad Católica de Chile

Academic Visits

Academic Honors

Research

Research Interests

Scientific Work

Conference Talks & Seminars

Teaching & Outreach

As Lecturer

As Teaching Assistant

Outreach

Other Related Activities

Contact

ResearchGate

Email

Address

"The universe can best be pictured as consisting of pure thought, the thought of what for want of a better word we must describe as a mathematical thinker."

Sir James Hopwood Jeans. “The Mysterious Universe”

Resume

Employment

Postdoctoral Fellow at University of Toronto

Postdoctoral Fellow at Sorbonne Université

My Education

PhD in Mathematics at Univeristy of Cambridge

MS in Mathematics at Pontificia Universidad Católica de Chile

Bachelor in Mathematics at Pontificia Universidad Católica de Chile

Academic Visits

Academic Honors

Research

Research Interests

Scientific Work

Conference Talks & Seminars

Teaching & Outreach

As Lecturer

As Teaching Assistant

Outreach

Other Related Activities

Contact

ResearchGate

Email

Address

Postdoctoral Fellow
at University of Toronto

Postdoctoral Fellow
at Sorbonne Université

PhD in Mathematics
at Univeristy of Cambridge

MS in Mathematics
at Pontificia Universidad Católica de Chile

Bachelor in Mathematics
at Pontificia Universidad Católica de Chile