Research

Research interests

I am mostly interested in quantum topology, more precisely on quantum invariants of knots and 3-manifolds, and its relationship with finite type invariants. I am also interested in categorification in the context of low-dimensional topology. Some keywords are: ribbon Hopf algebra, universal tangle invariant, Kontsevich invariant.

At a non-research level, I am still interested in (un)stable homotopy theory.

Preprints and publications

1. Strictification and non-strictification of monoidal categories, arXiv:2303.16740

2. On Bar-Natan - van der Veen's perturbed Gaussians. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 46 (2024). arvix

3.  A combinatorial PROP for bialgebras, arXiv:2106.13107. To appear in European Journal of Combinatorics.

4. An Introduction to Knot Homology Theories, TEMat monográficos, 2 (2021): Proceedings of the 3rd BYMAT Conference, pp. 11-14. ISSN: 2660-6003.

5. Ascenso y Descenso Finito, Infinito y Torcido [Finite, Infinite and Twisted Ascent and Descent] (jw/ JMF Castillo), La Gaceta de la RSME, Vol. 21, Nº 3, 2018, pages 497-516. Expository paper

Work in progress:

• A Hopf-algebraic construction of the 2-loop polynomial of genus one knots.

• On the naturality of the universal tangle invariant.

• Classification of lax-TQFTs in dimensions 1 and 2.

Theses

• Universal quantum knot invariants (PhD thesis, supervised by Roland van der Veen). This thesis  focuses on two knot invariants: the universal invariant subject to a ribbon Hopf algebra —that dominates the Reshetikhin-Turaev invariants coming from the representation theory of the Hopf algebra— and the Kontsevich invariant —which is universal among finte type invariants—. More precisely, Gaussian calculus techniques developed by Bar-Natan and van der Veen are exploited to study a family of knot polynomial invariants that determines the universal invariant subject to a specific algebra, and to investigate how the most elementary polynomial of this collection is closely related to the so-called 2-loop polynomial of the knot, an invariant that encodes a certain part of its Kontsevich invariant.

• K-theory with Reality (Master's thesis, supervised by Lennart Meier).  This was mainly a project on (Equivariant) Stable Homotopy Theory,  where we specially studied the spectra KO and K of real and complex K-theory, and the Z/2-spectrum KR of K-theory with Reality, arising from considering vector bundles with an anti-linear involution in the category of Z/2-spaces. 

• Theory of Relativity (Bachelor's thesis, supervised by Juan A. Navarro). It is a beautiful walk from Classical Mechanics to General Relativity, written from the point of view of pseudo-Riemannian geometry. The goal was to describe in a mathematical way the structures of Classical Mechanics (Galilean Spacetime, Newtonian Gravitation) and the Theory of Relativity (Minkowski Spacetime, Newton-Cartan Gravitation). An overview in English can be found here).