Below is a description of the main topics I’ve worked on. You can find a simple list of my publications on my CV page.

Topological rigidity of polynomial foliations

My earliest work concerns topological rigidity of holomorphic foliations on the complex projective plane. For my undergraduate thesis at UNAM (directed by Laura Ortiz-Bobadilla) I considered holomorphic foliations on $\mathbb{P}^2$ which are generated by a generic quadratic vector field on $\mathbb{C}^2$ and showed that the previously known rigidity results could be slightly improved. During my first year at Cornell I continued to explore these ideas under the supervision of Yulij S. Ilyashenko. We proved that the paradigm of topological rigidity can be formalized for generic quadratic foliations on $\mathbb{C}^2$: two such foliations are topologically equivalent if and only if they are affine equivalent (note that we do not make any assumptions on the proximity of the foliations nor on the conjugating homeomorphism).

[1] “Strong topological invariance of the monodromy group at infinity for quadratic vector fields”. J. Singul. 9, 193-202 (2014). doi: 10.5427/jsing.2014.9n.

[2] “The utmost rigidity property for quadratic foliations on $\mathbb{P}^2$ with an invariant line”. Bol. Soc. Mat. Mex. 23, no.2, 759-813 (2017). doi: 10.1007/s40590-016-0127-5.

Non-algebraizability of germs of holomorphic foliations

In the summer of 2014 I spent a few weeks at the Institut de Recherche Mathématique de Rennes. Together with Frank Loray we constructed the first known explicit example of the germ of an analytic foliation on $(\mathbb{C}^2,0)$ which is not analytically conjugate to the germ of an algebraic foliation. The existence of such foliations was established by Yohann Genzmer and Loïc Teyssier in 2010, but no concrete examples were known before our work.

[3] “An example of a non-algebraizable singularity of a holomorphic foliation”. Enseign. Math. 62, 7-14 (2016). doi: 10.4171/LEM/62-1/2-3.

Spectra of singularities for polynomial vector fields on $\mathbb{C}^2$

Another topic I’m interested in is the following: Consider a polynomial vector field on $\mathbb{C}^2$ with non-degenerate singularities, and having an invariant line at infinity. At each singular point, the linearization matrix of the vector field has two non-zero eigenvalues (the spectrum of such matrix). The extended spectra of singularities of a vector field is defined to be the collection of all such eigenvalues, together with the Camacho-Sad indices of the singular points at infinity (of the extended foliation on $\mathbb{P}^2$). Note that these numbers are preserved under affine equivalence of vector fields (i.e. they are analytic invariants of the vector field).

In the paper [4] below is a first step in the study of the spectra of quadratic vector fields. In particular, we show that two generic quadratic vector fields are affine equivalent if and only if their extended spectra coincide, and that the generic fiber of the finite spectra map consists of exactly two points.

In [6], a collaboration with Yury Kudryashov, we describe all the algebraic relations among the extended spectra of quadratic vector fields. What’s more, we show that the “hidden relations” do not come from an index theorem (as is the case for the previously known relations coming from the Euler-Jacobi formula, the Camacho-Sad theorem and the Baum-Bott theorem).

[4] “Twin vector fields and independence of spectra for quadratic vector fields”. J. Dynam. Control Syst. 23, 623-633 (2017). doi: 10.1007/s10883-016-9344-5.

[a] “The Woods Hole trace formula and indices for vector fields and foliations on $\mathbb{C}^2$”. Preprint: arXiv: 1608.05321.

[5] (With Y. Kudryashov) “Spectra of quadratic vector fields on $\mathbb{C}^2$: The missing relation”. To appear in: Mosc. Math. J. Preprint: arXiv: 1705.06340.

Multipliers at fixed points of self-maps on projective space

The above ideas are very related to a similar situation about self-maps of $\mathbb{P}^2$. Indeed, a generic endomorphism has finitely many non-degenerate fixed points, and the derivative of the endomorphism at such fixed points carries two eigenvalues which are not equal to 1. Define the spectra of fixed points to be the collection of such eigenvalues. In fact, the question on the spectra of singularities of vector fields can be understood as a very particular case of self-maps. This is discussed in [5], and also how all the index theorems for vector fields and endomorphisms that we know (holomorphic Lefschetz, Baum-Bott, Camacho-Sad, and so on) can be reduced to a particular case of the so-called Woods Hole trace formula (aka the Atiyah-Bott fixed point theorem).

In collaboration with Adolfo Gillot we have studied the case of quadratic self-maps on the projectiv plane. We have described how to recover all the algebraic relations among the multipliers at fixed-points and proved that, in the generic case, these maps are completely determined by their multipliers (up to linear changes of coordinates). As an application, we have used the previous constructions to describe the Kowalewski exponents for a particular class of quadratic homogeneous vector fields on $\mathbb{C}^3$ having exclusively single-valued solutions.

[6] (With A. Guillot) “On the multipliers of fixed points of quadratic self-maps of the projective plane with an invariant line”. Comput. Methods Funct. Theory. (2019). doi: 10.1007/s40315-019-00293-w.

A map between moduli spaces of connections

My most recent research project at IRMAR is about rank 2 logarithmic connections on an elliptic curve with two punctures. We looked at how certain connections on the Riemann sphere transform when they are pulled-back to the elliptic curve via a ramified double cover. This procedure defines a map between the corresponding moduli spaces of connections. This is a collaboration with Frank Loray. Below you can find a preprint.

[7] (With F. Loray) “A map between moduli spaces of connections”. SIGMA Symmetry Integrability Geom. Methods Appl. 16, n.125, 42p (2020).
doi: 10.3842/SIGMA.2020.125.

 

Note: The code to perform all the computations for several of these projects is hosted on my GitHub page.