Hi! I'm Fernando Vieira Costa Júnior and I'm a Mathematician from Brazil :-)
Formed with focus on Analysis, I have a particular sympathy for Algebra, Number Theory and Logic. Teaching for me is a passion ! In this page I talk a bit about myself and about my job. My students will find here support material for my courses.
My main research interest is that of Linear Dynamics and Operator Theory. I am also interested in the implementation of mathematical algorithms on Mathematica and Python.
At the moment I am a FNRS postdoctoral researcher at the Mathematics Department of the University of Mons.
This is the small town I was born in. But I've lived in there for no more than a couple of days ! At that time my family had a peculiar tradition : every childbirth was made by my À l'époque ma famille avait une tradition : tous les accouchements étaient faits par mon uncle and obstetrician Dr. Wilson Vieira Costa, his clinic is located in this small town, 45km away from my parents house. In this picture one can see the Goiti's Christ, a kind of "mini Christ the Redeemer" statue.
June 2018 - May 2021
The subject of my thesis is the theory of Linear Dynamical Systems, specially on the concept of hypercyclicity and related topics. What mainly interested us were the algebras of hypercyclic vectors on a Fréchet sequence algebra. The adaptations of Baire criteria for vectors to the algebraic context have given us lots of material to work on, what lead to a significant increase in our understanding of this topic.
I had the opportunity to have Frédéric Bayart as my thesis advisor, which took place in the city of Clermont-Ferrand, at the Blaise Pascal Mathematics Laboratory of the Clermont Auvergne University.
Download here my thesis manuscript (in English/French)
July 2016 - February 2018
During my master I had the lucky chance of working with Daniel Pellegrino on the Bohnenblust-Hille inequality (among others). I had to learn to code in Mathematica in order to implement an algorithm which finds the extreme points of the unitary ball on the space of multilinear forms acting on the Euclidean space. My Master took place at the Mathematics Department of the University of Paraíba.
Download here my Masters dissertation (in Portuguese)
January 2012 - December 2015
The initial idea of my project was to study Lebesgue integral in advance, but while researching the story behind this integral and the general theory of integration, I found a lot of notions of integration, which improved more and more our capacity of integrating. I ended up by exploring this vast univers of integration types and their chronological interrelations. In the final project we include many historical informations while developing the theory, goins from Newton and Leibniz, passing through Cauchy and Riemann, Lebesgue and Riesz, and arriving at Henstock, Kurzweil, MacShane and others. This project was supervised by Ornan Philipe, who is, teaching-wise, one of the best professors I have ever met. My studies took place at the Federal University of Alagoas, Campus Arapiraca.
Download here my Licence monography (in Portuguese)
January 2009 - December 2011
My High School took place at my parents town Arapiraca in the state of Alagoas, in the Quintella Cavalcanti School. I had a very good mathematics professor, the very kind Roberto Da Silva.
February - June 2023 • L1 S2 in Maths, Info et Physics • TD
February - June 2023 • L1 S2 in Maths, Info et Physics • TD
February - June 2023 • L1 S2 in Physics/Chemistry • TD
September 2022 - January 2023 • L1 S1 in Mathematics • TD
February - June 2022 • L1 S2 Informatics • TD
February - June 2022 • L1 S2 Physics/Chemistry • TD
September - December 2021 • L3 S5 in Math • TD
September - December 2021 • L1 S1 in Math • TD
September - December 2021 and 2022 • L3 S5 in Humanities • CM/TD
Octobre - December 2021 • - • TD
February - July 2020 and 2021 • L1 S2 in Science
September - December 2019 and 2020 • L1 S1 in Science
2012 - 2016 • Licence in Math
In construction...
Preprint
We obtain a multi-dimensional generalization of the Costakis-Sambarino criterion for common hypercyclic vectors with optimal consequences on a large class of fractals. Applications include families of products of backward shifts parameterized by any Hölder continuous curve in R^d, for all d ≥ 1.
Journal of Mathematical Analysis and Application (2022)
In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a questionraised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on ℓ_1(N) with the convolution product for the family of backward shifts(B_{w(λ)})_{λ>0} induced by the weights w_n(λ) = 1 + λ/n.
Indiana University Mathematics Journal (2022) • In collaboration with F. Bayart et Q. Menet
We investigate the existence of a common hypercyclic vector for a family (T_λ)_{λ∈Λ} of hypercyclic operators acting on the same Banach space X. We give positive and negative results involving the dimension of Λ and the regularity of each map λ ∈ Λ → T_λ^n(x), x ∈ X , n ∈ N.
Israel Journal of Mathematics (2022) • In collaboration with F. Bayartet D. Papathanasiou
Over time the concept of hypercyclicity has been explored in different manners and contexts, gaining new forms and applications. In particular when the spacehas an adjacent structure, we can always look for sets of hypercyclic vectors compatible with that framework. In this paper we deal with hypercyclic operators acting on Fréchet sequence algebras and give criteria for the existence of common and disjoint hypercyclic algebras.
Avances in Mathematics (2020) • In collaboration with F. Bayartet D. Papathanasiou
The question of whether a hypercyclic operator T acting on a Fréchet algebra Xadmits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterisations in the context of convolution operators acting on H(C) and backward shifts acting on a general Fréchet sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on c_0 admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered.
Numerical Functional Analysis and Optimization (2018)
The Bohnenblust–Hille inequality for m-linear forms was proven in 1931 as a generalization of the famous 4/3-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as m grows) is unknown, but significant for applications. A recent result, obtained by Cavalcante, Teixeira and Pellegrino, provides a kind of algorithm, composed by finitely many elementary steps, giving as the final outcome the optimal truncated Bohnenblust–Hille constants of any order. But the procedure of Cavalcante et al. has a fairly large number of calculations and computer assistance cannot be avoided. In this paper we present a computational solution to the problem, using the Wolfram Language. We also use this approach to investigate a conjecture raised by Pellegrino and Teixeira, asserting that C_m=2^{1−1/m} for all m ∈ ℕ and to reveal interesting unknown facts about the geometry of B_{ℒ(^3ℝ^3)}.
26-30 of June 2023 • Thessaloniki, Greece • Participation with talk
19-23 of June • Caserta, Italy • Participation with talk
Page of the conference • My talk
12-13 of June 2023 • Mons, Belgium • Invited talk
CIRM • Marseille - France • November 29 to December 3, 2021 • Participation with poster
Page of the conférence • My poster on hypercyclic algebras
Université Paris-Est Marne-la-Vallée • Paris, France • June 17-21, 2019 • Participation with talk
The 2019 edition of the international conference ACOTCA took place in Marne-la-Valée, France. In this occasion, I've given a talk on the existence of common hypercyclic algebras.
Page of the conference • Slides of my talk
University of the Littoral Côte d'Opale, Calais, France • October 19, 2023
Talk : Introduction to hypercyclic algebras (download)
University of Lille, Lille, France • Octobre 13, 2023
Talk : Simultaneous hypercyclicity in 2D (download)
Aix-Marseille University, Marseille, France • September 26, 2022
Talk : Common hypercyclicity in several dimensions (download)
Aix-Marseille University, Marseille, France • June 24, 2022
Talk : The origins of Linear Dynamics (download)
University of Lille, France • Novembre 26, 2021
Talk : In the search of closed hypercyclic algebras (download)
Avignon University, Avignon, France • Novembre 23, 2021
Talk : Hypercyclic vectors and algebras (download)
Online talk • June 11, 2021
Talk : Hypercyclic algebras for convolution operators (download)
Clermont-Auvergne University, Clermont-FD, France • April 3, 2019
Talk : Linear Chaos theory (download)
Federal University of Alagoas, Arapiraca, Brazil • August 24-28, 2015 • Short course
Course notes (in portuguese) - Slides
Federal University of Manaus, Manaus, Brazil • august 13-17, 2014 • Short course
Course notes (in portuguese) - Slides
Clermont-Auvergne University, Clermont-FD, France • June 2019
My academic email address is Fernando.VIEIRACOSTAJUNIOR@umons.ac.be and my personal (and permanent) email address is fv.costajunior@gmail.com. Usually you can find me at the Mathematics Department of the University of Mons.
Last update: October 25, 2023