Gerardo Barrera Vargas

About me

I am a postdoctoral researcher (2020-2023) in the Department of Mathematical and Statistical Sciences at University of Helsinki, Helsinki, Finland working under the supervision of professor Jani Lukkarinen.

I work in the group of Mathematical Physics:

https://wiki.helsinki.fi/display/mathphys/Seminars+and+Workshops

I got my Phd studies in August 2015 at the Instituto de Matematica Pura e Aplicada, IMPA, Brazil, advised by professor Milton David Jara Valenzuela.

Here is a link to my CV.

Research and activities

My research interests lie in the interface between stochastic processes, dynamical systems and mathematical physics.

In particular, I am interested in the study of the so-called abrupt convergence to the invariant "cut-off phenomenon" for random perturbations of dynamical systems driven by different kinds of noises (Brownian motion, Lévy processes, etc). Roughly speaking, I studied the abrupt convergence in the small temperature regime to the equilibrium state for stochastic differential equations.

The cut-off phenomenon was extensively studied in the eighties to describe the phenomenon of abrupt convergence that appears in models of card shuffling, Ehrenfest's urns and random transpositions. In general, it is a challenging problem to decide that an specific family of stochastic models exhibit or does not exhibit a cut-off phenomenon. It requires a complete understanding of the dynamics of the specific random process.

In addition, I have also focused my research topics on random polynomials and their relation with the random structured matrices as circulant and Toeplitz matrices. Those structure matrices are very useful in applications.

Recently, I have been working in the stability of trinomials with general complex coefficients and the so-called counting problem for the roots of harmonic trinomials.

Fellowships

2009-2011: Master Scholarship from CONACyT, Mexico. I studied my Master degree in the Center for Research in Mathematics, CIMAT, Mexico under the supervision of professor José Alfredo López Mimbela. I focused on the Renyi entropy for dynamical systems. Here is a link to my Master Thesis.

2011-2015: Ph.D. Scholarship from CNPq, Brazil. I studied my Phd in the Instituto de Matematica Pura e Aplicada, IMPA, Brazil under the supervision of professor Milton Jara Valenzuela. I focused on small random perturbations of dynamical systems. Here is a link to my PhD Thesis.

2015-2016: Postdoctoral Fellowship from CIMAT, Mexico. I worked in small random perturbations of differential equations and the asymptotics of singular values for structured random matrices via random polynomials.

2016-2017: Postdoctoral Fellowship from CONACyT, Mexico. I worked on small random perturbations of differential equations driven by Lévy noise.

2017-2019: Postdoctoral Fellowship from PIMS, Canada. I worked on the so-called shadow phenomenon for perturbations of Hamiltonian systems.

2020-2023: Postdoctoral Fellowship at University of Helsinki, Finland. I am working with professor Jani Lukkarinen in the department of Mathematics and Mathematical Physics.

2024: Docent title in Mathematical Physics, University of Helsinki, Finland.

2022-2026: National Researcher Level I, CONACyT-Mexico.

Teaching

As a graduate student, I was teaching assistant of several courses at IMPA. I hold office hours and review sessions for those classes.

Quantitative Finance Theory for the Professional Master in Finance at IMPA (3-times).
Probability and Stochastic Processes for the Professional Master in Finance at IMPA (1-time).
Summer Course: Markov Chains and Mixing Times for the Academic Master and Doctorate in Mathematics at IMPA (2-times).

In the professional master I taught classes for people who worked in Petrobras, in the Industry and Insurance Companies. In the Academic Master and Doctorate I taught classes for Master and Phd students. In both systems I needed to prepare review sessions as the specific program needed and I provided office hours.

In the University of Alberta, Department of Mathematical and Statistical Science, I taught Math 334: Introduction to Differential Equations in Winter 2018 and Spring 2019.

From September 2019 to December 2019, I was teaching the undergraduate course Introduction to Mathematics at the Polytechnic University of Yucatan, UPY.

From September 2021 to October 2021, I was teaching the undergraduate course BSMA2001: Linear Algebra II at the University of Helsinki.

From September 2022 to October 2022, I will teach the undergraduate course BSMA2001: Linear Algebra II at the University of Helsinki.

From January 2024 to March 2024, I will teach the undergraduate course Algebraic Structures I at the University of Helsinki.

Recently, I received the docent title in Mathematical Physics from the University of Helsinki, Finland.

I am member of editorial commission for Textuniversitários Editora Livraria da Física São Paulo.

Publications

Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems.
Journal of Statistical Physics 2016, Volume 163, Number 1, 113-138.
Joint work with Milton Jara, IMPA, Brazil.
Abrupt convergence for a family of Ornstein-Uhlenbeck processes.
Brazilian Journal of Probability and Statistics 2018, Volume 32, Number 1, 188-199.
Thermalisation for small random perturbations of dynamical systems.
Annals of Applied Probability 2020, Volume 30, Number 3, 1164-1208.
Joint work with Milton Jara, IMPA, Brazil.
Cut-off phenomenon for the Ornstein-Uhlenbeck processes driven by Lévy processes.
Electronic Journal of Probability 2020, Volume 25, Number 15, 1-33.
Joint work with Juan Carlos Pardo, CIMAT, Mexico.
A switch convergence for a small perturbation of a linear recurrence equation.
Brazilian Journal of Probability and Statistics 2021, Volume 35, Number 2, 224-241.
Joint work with Shuo Liu, University of Alberta, Canada.
Cutoff phenomenon for the maximum of a sampling of Ornstein-Uhlenbeck processes.
Statistics and Probability Letters 2021, Volume 168, 108954.
Zero-free neighborhood around the unit circle for Kac polynomials.
Periodica Mathematica Hungarica 2021, 84, 159-176.
Joint work with Paulo Manrique, National Polytechnic Institute, Mexico.
Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance.
Journal of Statistical Physics 2021, Volume 184, Number 27, 1-54.
Joint work with Juan Carlos Pardo, CIMAT, Mexico and Michael Hoegele, University of Los Andes, Colombia.
The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise.
Electronic Journal of Probability 2021, Volume 26, Number 119, 2021, 1-76.
Joint work with Juan Carlos Pardo, CIMAT, Mexico and Michael Hoegele, University of Los Andes, Colombia.
The cutoff phenomenon in Wasserstein distance for nonlinear stable Langevin systems with small Lévy noise.
Journal of Dynamics and Differential Equations 2022, 1-28.
Joint work with Juan Carlos Pardo, CIMAT, Mexico and Michael Hoegele, University of Los Andes, Colombia.
On the number of roots for harmonic trinomials.
Journal of Mathematical Analysis and Applications 2022, Volume 514, Number 2, 126313.
Joint work with Waldemar Barrera and Juan Pablo Navarrete, Autonomous University of Yucatan, Mexico. Corrigendum.
The cutoff phenomenon for the stochastic heat and wave equation subject to small Lévy noise.
Stochastics and Partial Differential Equations: Analysis and Computations 2022.
Joint work with Juan Carlos Pardo, CIMAT, Mexico and Michael Hoegele, University of Los Andes, Colombia.
Salem-Zygmund inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices.
Boletín de la Sociedad Matemática Mexicana 2022, Volume 28, Number 45.
Joint work with Paulo Manrique, National Polytechnic Institute, Mexico.
Limit behavior of the invariant measure for Langevin dynamics.
Probability and Mathematical Statistics 2022.
The asymptotic distribution of the condition number for random circulant matrices.
Joint work with Paulo Manrique, National Polytechnic Institute, Mexico.
Extremes, 2022.
Quantitative control of Wasserstein distance between Brownian motion and the Goldstein-Kac telegraph process.
Joint work with Jani Lukkarinen, University of Helsinki, Finland.
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2023, Volume 59, Number 2, 933-982.
Egerváry's theorems for harmonic trinomials.
Joint work with Waldemar Barrera and Juan Pablo Navarrete, Autonomous University of Yucatan, Mexico.
Acta Mathematica Hungarica, 2024.
Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability.
Joint work with Michael Hoegele, University of Los Andes, Colombia.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2023, Volume 33, Number 11, 113124.
The stability region for Schur stable trinomials with general complex coefficients.
Joint work with Waldemar Barrera and Juan Pablo Navarrete, Autonomous University of Yucatan, Mexico.
Journal of Dynamics and Differential Equations 2023.

PREPRINTS

Here is a list of projects that are submitted for publication.

Cutoff stability of multivariate geometric Brownian motion.
Joint work with Juan Carlos Pardo, CIMAT, Mexico and Michael Hoegele, University of Los Andes, Colombia.
Submitted.
Gradual convergence for Langevin dynamics on a degenerate potential.
Joint work with Milton Jara, IMPA, Brazil and Conrado da Costa, Durham University, UK.
Submitted.
Cutoff ergodicity bounds in Wasserstein distance for a viscous energy shell model with Lévy noise.
Joint work with Juan Carlos Pardo, CIMAT, Mexico, Michael Hoegele, University of Los Andes, Colombia and Ilya Pavlyukevich, Jena University, Germany.
Minor corrections, Journal of Statistical Physics.
Profile cut-off phenomenon for the ergodic Feller root process.
Joint work with Liliana Esquivel, University of Valle, Colombia.
Submitted.
The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem.
Joint work with Eyleifur Bjarkason and Sigurdur Hafstein, University of Iceland, Iceland.
Submitted.

Work in progress

Here is a list of projects that are in preparation and should become papers soon.

Limit theorems for coagulations equations: simple kernels. Joint work with Marina Amado Ferreira, University of Helsinki and Alessia Nota, Università degli Studi dell'Aquila.
Cut-off phenomenon for expanding maps on the circle.

Outreach

Notas sobre algunos teoremas de Jenö Egerváry.
Joint work with Waldemar Barrera, Miguel Angel Can Ek and Juan Pablo Navarrete, Autonomous University of Yucatan, Mexico.
Abstraction And Application, 2022, Volume 38, Number 4, 50-59.

sELECTED Invited talks and MINI-CURSE

Switch type convergence to equilibrium for coercive Langevin dynamics driven by Lévy processes in Probability Seminar at University of Durham. Durham, U.K., March, 2021, Online.
Mixing times in dynamical systems in III International Conference on Stochastic Processes, Random Phenomena and Their Applications CIPEFA, December, 2020, Online.
Profile of the convergence to equilibrium for Langevin diffusions under a potential with a unique degenerate minimum in Mathematical and Physics Seminar at University of Helsinki. Helsinki, Finland, November, 2020, Online.
Abrupt convergence to equilibrium for random perturbations of dynamical systems in Mathematical and Physics Seminar at University of Helsinki. Helsinki, Finland, January, 2020.
Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes in the XV Latin American Congress of Probability and Mathematical Statistics. Merida, Yucatan, Mexico, December, 2019.
On the abrupt convergence for Ornstein-Uhlenbeck processes driven by a Lévy noise in the Retreat for Young Researchers in Stochastics. Banff, Alberta, Canada, October, 2018.
Mini-course Thermalisation for dynamical systems in the Pontificia Universidad Católica de Chile 2018. Santiago, Chile, August, 2018.
Fokker-Planck equations arising from SDE: convergence of their densities in the International Conference on Stochastic Partial Differential Equations. Edmonton, Alberta, Canada, September-October, 2018.
Thermalisation for perturbations of dynamical systems in the Retreat for Young Researchers in Stochastics at Banff International Research Station. Banff, Alberta, October, 2017.
Cut-off phenomenon for small perturbations of hyperbolic dynamics in the thematic PIMS Workshop on Stochastic Nonlinear Dynamics. Edmonton, Alberta, August, 2017.
Cut-off phenomenon for stochastic small random perturbations of hyperbolic dynamical systems in the thematic trimester Stochastic Dynamics out of Equilibrium. Paris, France, June, 2017

SELECTED CONTRIBUTED TALKS

An example of no-cutoff phenomenon for a non-hyperbolic random dynamical system in Bernoulli-IMS One World Symposium 2020, August, 2020, Online.
Shuffling dynamical systems in CIMAT-Merida Colloquium. Merida, Yucatan, Mexico, November, 2019.
Switch type convergence for random dynamical systems in the Probability and Statistics Seminar at Autonomous University of Yucatan. Merida, Yucatan, Mexico, January 2019.
On the threshold convergence for Fokker-Planck equations arising from SDE in the PIMS Workshop on stochastic and deterministic modelling in Biology. Jasper, Alberta, Canada, September, 2018.
On the switch-type convergence for stochastic differential equations in the International Congress of Mathematical Physics 2018. Montreal, Canada, July, 2018.
On the cut-off phenomenon for hyperbolic attractors in the Young Researchers Symposium YRS 2018. Montreal, Canada, July, 2018.
Cut–off phenomenon for stochastic small perturbations of m-dimensional dynamical systems in the XII Symposium on Probability and Stochastic Processes. Merida, Yucatan, Mexico, November, 2015.