Curriculum Vitae

Experience

• PhD Research Project 

2022 - today

Title: Singular differential homology

Description: The main aim of this project consists in the construction of a natural differential extension of singular homology, dualizing the analogous extension in the framework of cohomology. In particular, the groups we are going to define are supposed to fit in suitable exact sequences, analogous to the cohomological ones. Moreover, we expect that the corresponding version of Poincaré duality holds. Depending on the time left, it would be natural to investigate how to generalize the previous construction to any (co)homology theory.

Advisor: Fabio Ferrari Ruffino

Financial support: FAPESP Scholarship (2022/00676-3)

• Research Visit

March 1, 2024 - February 28, 2025

Title: Twisted Borel K-theory and generalizations

Description: The present project is the natural continuation of the one realized by the former PhD student Alffer Gustavo Hernandez Posada, who computed some interesting twisted K-theory groups up to extension, i.e. he showed that they are the central term of a certain short exact sequence. This is a meaningful result, but it has to be completed by computing the groups themselves, not only up to extension. Moreover, these computations have been realized in the topological framework, and we aim to extend them to the differential setting.

Advisor: Fabio Ferrari Ruffino

Host: José María Cantarero López

Financial support: FAPESP Scholarship (2023/03876-6)

• MSc Research Project 

2020 - 2022

Title: Ordinary and twisted K-theory

Description: The candidate intends to deepen his knowledge on K-theory, considering its various significant versions and their main applications. Moreover, he is going to analyse in detail the construction of the most relevant models of twisted K-theory. In particular, he will concentrate on the model through twisted vector bundles, trying to show directly that it represents a twisted cohomology theory.

Advisor: Fabio Ferrari Ruffino

Financial support: FAPESP Scholarship (2019/22159-8)

• Undergraduate Research Project 

2019 - 2020

Title: An introduction to $\infty$-categories

Description: The candidate, during his previous bachelor projects, learnt the basic concepts of Algebraic Topology (fundamental group and covering spaces, singular homology and cohomology, first notions about generalized cohomology theories). In this project we are going to study the language of $\infty$-categories, which is essential to understand the classification of topological field theories achieved by Jacob Lurie. At a first stage the candidate will have to deepen his knowledge about basic tools, in particular about higher homotopy groups and simplicial (co)homology within the categorical framework. Then, he will be introduced to higher category theory and he will complete the project dealing with the language of $\infty$-categories.

Advisor: Fabio Ferrari Ruffino

Financial support: FAPESP Scholarship (2018/24174-1)

• Undergraduate Research Project 

2018 - 2019

Title: Topics on Algebraic Topology

Description: During the first six months the student will learn about singular homology and cohomology theories, in order to complete his knowledge about basic algebraic topology. Then we will start the study of topological K-theory and we will show explicitly that it is a generalized cohomological theory that obeys all the axioms of Eilenberg-Steenrod, except for the dimension one.

Advisor: Fabio Ferrari Ruffino

Financial support: FAPESP Scholarship (2017/21378-2)

• Undergraduate Research Project 

2017 - 2018

Title: Unidimensional dynamical systems: lexicon, elementary concepts and some important theorems

Description: The project intends to give to the student an elementary background on discrete unidimensional dynamical systems, preparing him to research on this subject. For this purpose it proposes the study of the first chapter of  ''An introduction to chaotic dynamical systems'' by R. Devaney, complemented by more advanced texts wrote to researchers in this area and providing scientific maturity to the student. The candidate will study fundamental concepts of dynamical systems including Symbolic Dynamics, Devaney's Definition of Chaos, the Sarkovskii's Theorem, Structural Stability, dynamics of maps in $S^1$ and fragments of Kneading Theory. In the end the student will have been seen a complete description of the illustrative behavior of the family of real maps $F_\mu(x)= \mu x(1-x)$, $\mu \in \mathbb{R}$, $\mu \geq 1$, including the phenomenon of period-doubling, the Poincaré's Classification Theorem for homeomorphisms in $S^1$.

Advisor: Liane Bordignon

Financial support: FAPESP Scholarship (2016/21492-7)

• Undergraduate Research Project 

2016 - 2017

Title: Fundamental group and covering spaces

Description: The project intends to study the fundamental group of general topological spaces and some of its classical applications, especially the Brouwer Fixed Point Theorem and the Bosuk-Ulam Theorem, together with a topological proof of the Fundamental Theorem of Algebra, as well as its relations with covering spaces. 

Advisor: Daniel Vendrúscolo

Financial support: FAPESP Scholarship (2016/00470-5)