About me
I am currently a Postdoc in the Complex Geometry and Holomorphic Foliations group of IMPA, working with Hossein Movasati.
My main area of research is Algebraic Geometry, with emphasis in Hodge Theory and algebraic cycles. I am interested in developing techniques to compute periods of Hodge cycles explicitly, and use this information to study algebraic varieties. I am particularly interested in the Hodge conjecture, and its variational version. My work have concentrated in the case of hypersurfaces, where we can associate to every non-trivial primitive Hodge cycle an Artinian Gorenstein algebra. Lately this relation has specially dragged my attention.
Here you can find my CV.
Articles
- Small codimension components of the Hodge locus containing the Fermat variety.
Pre-print. - Periods of Complete Intersection Algebraic Cycles.
Pre-print. - Integral Hodge conjecture for Fermat varieties
Joint with E. Aljovin and H. Movasati.
Journal of Symbolic Computation, Vol. 95 (2019), pp. 177--184. - Periods of linear algebraic cycles
Joint with H. Movasati.
Pure and Applied Mathematics Quarterly, Vol. 14, No. 3-4 (2018), pp. 563--577.
Thesis
- Periods of Algebraic Cycles.
19.03.19. IMPA. Slides.
Recorded Talks
- Periods of algebraic cycles and Hodge locus.
Iberoamerican Webminar of Young Researchers in Singularity Theory and related topics.
24.02.21. Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid. Online Seminar. Slides. - Constructing algebraic cycles on hypersurfaces, an explicit approach to Hodge conjecture.
Brazilian Algebraic Geometry Seminar.
05.08.20. Universidade Estadual de Campinas. Online Seminar. Slides. - How to construct Hodge cycles on hypersurfaces? (Spanish).
Seminario de Geometría Algebraica.
16.06.20. Universidad Católica de Chile. Online Seminar. Slides. - Small codimension components of the Hodge locus and periods of the Fermat variety.
Geometry, Arithmetics and Differential Equations of Periods Seminar.
12.06.20. IMPA. Online Seminar. Slides. - On Variational Hodge Conjecture for Hypersurfaces (Spanish).
Math Department Colloquium.
10.07.19. Universidad de Chile. Santiago, Chile. - Artinian Gorenstein Algebras and Components of the Hodge Locus.
Hodge Theory Day.
21.03.19. IMPA. Rio de Janeiro, Brazil. - Periods of Complete Intersection Algebraic Cycles.
Hodge and Noether Lefschetz Loci Seminar.
28.11.18. CMSA, Harvard University. Cambridge, US. - Periods of Linear Cycles in Fermat Varieties.
Worhshop: Picard-Fuchs Equations and Hypergeometric Motives.
26.03.18. HCM, Bonn University. Bonn, Germany.
Teaching
Instructor
- Introduction to Modular Forms and Elliptic Curves.
Joint with H. Movasati. Teacher assistant J. Duque.
Detailed content of the course. My lecture notes. Exercises.
Summer Term 2021. IMPA. - Introduction to Mixed Hodge Structures.
My lecture notes.
Spring Term 2020. IMPA. - Computational Commutative Algebra.
My lecture notes. Exercises.
Spring Term 2020. IMPA. - Hodge Theory.
Recorded lectures at IMPA YouTube channel.
Detailed content of the course. Exercises.
Spring Term 2019. IMPA. - An Introduction to Algebraic de Rham Cohomology and Infinitesimal Variations of Hodge Structures.
Detailed content of the mini-course.
Spring Term 2018. IMPA. - Introduction to Number Theory.
Joint with H. Movasati and M. Torres.
Detailed content of the course. My lecture notes.
Summer Term 2017. IMPA.
Research Schools TA
- Stable Birrational Invariants and the Stable Lüroth Problem.
Prof. Claire Voisin.
02.12.2019 - 06.12.2019. Universidad de Talca.
Fourth Latin American School of Algebraic Geometry and Applications.
Graduate TA
- Topics in Hodge Theory
Prof. Hossein Movasati.
Spring Term 2017. IMPA. - Introduction to Complex Geometry.
Prof. Hossein Movasati.
Fall Term 2017. IMPA. Exercises. - Schemes and Cohomology.
Prof. Karl-Otto Stöhr.
Fall Term 2017. IMPA. Exercises. - Introduction to Number Theory.
Prof. Hossein Movasati.
Summer Term 2017. IMPA. Exercises. - Classical Algebraic Geometry.
Prof. Karl-Otto Stöhr.
Spring Term 2016. IMPA. Exercises. - Classical Algebraic Geometry.
Prof. Eduardo Esteves.
Spring Term 2015. IMPA. - Abstract Algebra.
Prof. Carolina Araujo.
Fall Term 2015. IMPA.
Undergraduate TA
- Linear Algebra.
Prof. Alejandro Maass.
Spring Term 2012. Universidad de Chile. Exercises. - Multivariable Calculus.
Prof. Juan Dávila.
Fall Term 2012. Universidad de Chile. Exercises. - Graph Theory.
Prof. Martín Matamala.
Fall Term 2012. Universidad de Chile. - Introduction to Calculus.
Prof. José Soto.
Spring Term 2011. Universidad de Chile. Exercises. - Introduction to Algebra.
Prof. Alejandro Maass.
Spring Term 2011. Universidad de Chile. Exercises.
Notes
Book
- A Course in Hodge Theory: Periods of Algebraic Cycles.
Book in progress, joint with Hossein Movasati (second volume of Hossein's book A Course in Hodge Theory: With Emphasis on Multiple Integrals).