Joel Villatoro - Mathematician
About MeAcademic CV
I am an American mathematician currently working as a postdoctoral researcher at the Max Plank Institute for Mathematics in Bonn. I received my PhD at the University of Illinois at Urbana-Champaign under the supervision of Rui Loja Fernandes. My primary research interests are in the theory of Lie groupoids, Lie algebroids, differentiable stacks and connections with Poisson and symplectic geometry.
Contact: (lastname) (at) mpim-bonn.mpg.de
(2021-Present) Postdoctoral Fellow - Max Plank Institute for Mathematics - Bonn
(2018-2021) Postdoctoral Fellow - KU Leuven, Belgium
(2012-2018) Doctor of Philosophy in Mathematics - University of Illinois at Urbana Champaign
(2008-2012) Bachelors of Science in Mathematics - University of Oklahoma
About My Research
My research is motivated by a deep interest in the study of the relationship between symmetry and singularities. At the most basic level, a symmetry is a permutation of a mathematical object which preserves a desirable structure. For example, in Newton's laws of motion one might notice that if one were to shift the entire universe to the in one direction, any valid trajectory of an object would be sent to a new valid trajectory. This is an example of translational symmetry. Emmy Noether, in a monumental contribution to the study of physics, noticed that there was a relationship between symmetries of physical systems and conserved quantities. Going back to our example of Newtonian mechanics, it turns out that the translational symmetry of Newtonian trajectories gives rise to conservation of momentum.
In geometry, a singularity is a point in space where a mathematical model ceases to function. The most famous example of this is perhaps the singularities that appear in the study of Einsteins theory of gravitation. A less mathematically intensive example of a singular geometric object might be that of a 'teardrop'. A single sharp point on a otherwise perfect sphere. Suppose one was interested in understanding paths on such an object. Away from the sharp point, differential geometers can easy make sense of the notion of the 'direction' a path on the teardrop. If one imagines that the teardrop is sitting inside of a three dimensional space, one can define the direction of a path at a given point by constructing the unique tangent plane which touches the teardrop at exactly one point. However, at the 'sharp point' there are many planes which touch the teardrop at only one point. It turns out that by understanding the symmetries of the teardrop, one can 'unwind' the sharp point after which geometry can proceed as usual. This process is sometimes called desingularization.
In my research, I try to study singular objects not unlike the teardrop example mentioned above. The main tool I use is the theory of Lie groupoids and Lie algebroids which can be thought of as a very general kind of desingularzation. A common theme in my research is to try to understand how much of differential geometry and algebraic topology can still be performed for spaces with singularities. Other tools I occasionally make use of include sheaves, diffeology and category theory depending on the needs of the problem. For examples, I frequently turn to Poisson and symplectic geometry and my work is often motivated by trying to obtain a 'macro level' understanding of the geometric constructions and objects that appear in these fields.
Papers and Preprints
On sheaves of Lie-Rinehart algebras
Abstract: We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher) homotopy groups and groupoids for such objects which directly generalize the homotopy groups and Weinstein groupoids of Lie algebroids. We consider, the special case of sheaves of Lie-Rinehart algebras over smooth manifolds. We show that, under some reasonable assumptions, such sheaves induce a partition of the underlying manifold into leaves and that these leaves are precisely the orbits of the fundamental groupoid.
Integration of Singular foliations via paths
Co-author(s): Alfonso Garmendia
Status: Published in International Mathematics Research Notices
Abstract: We give a new construction of the holonomy and fundamental groupoids of a singular foliation. In contrast with the existing construction of Androulidakis and Skandalis, our method proceeds by taking a quotient of an infinite dimensional space of paths. This strategy is a direct extension of the classical construction for regular foliations and mirrors the integration of Lie algebroids via paths (per Crainic and Fernandes). In this way, we obtain a characterization of the holonomy and fundamental groupoids of a singular foliation that more clearly reflects the homotopic character of these invariants. As an application of our work, we prove that the constructions of the fundamental and holonomy groupoid of a foliation have functorial properties.
Stacks in Poisson Geometry (PhD Thesis)
Status: Dissertation accepted by University of Illinois
Abstract: This thesis is divided into four chapters. The first chapter discusses the relationship between stacks on a site and groupoids internal to the site. It includes a rigorous proof of the folklore result that there is an equivalence between the bicategory of internal groupoids and the bicategory of geometric stacks. The second chapter discusses standard concepts in the theory of geometric stacks, including Morita equivalence, stack symmetries, and some Morita invariants. The third chapter introduces a new site of Dirac structures and provides a rigorous answer to the question: What is the stack associated to a symplectic groupoid? The last chapter discusses a remarkable class of Poisson manifolds, called b-symplectic manifolds, giving a classification of them up to Morita equivalence and computing their Picard group.
Poisson manifolds and their associated stacks
Status: Published in Letters in Mathematical Physics
Abstract: We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic. We also discuss the non-integrable case.
Picard groups of b-symplectic manifolds
Status: Published in Journal of Symplectic Geometry.
Abstract: We compute the Picard group of a stable b-symplectic manifold M by introducing a collection of discrete invariants Gr which classify M up to Morita equivalence.
A Geometric proof of the Fundamental Theorem of Algebra (with visualizations)
Abstract: This is a brief iPython notebook hosted on google's Colab service. It exposits a geometric proof of the fundamental theorem of algebra. The proof is supplemented with visualizations to illustrate the key concepts of the proof.
On the algebraic geometry and homotopy theory of sheaves of Lie-Rinehart structures
Host: Junior Global Poisson Workshop 2020
Abstract: Lie-Rinehart algebras are an algebraic generalization of the notion of a Lie algebroid. In this talk we will first survey the basics of Lie-Rinehart algebras and the natural notions of morphisms between them. A natural spectrum functor which sends Lie-Rinehart algebras to sheaves of Lie-Rinehart algebras over locally ringed spaces will then be defined. In the manifold setting, we will see that examples of non-algebroid sheaves of Lie-Rinehart algebras arise quite naturally. In conclusion, there will be an explanation of how several classical invariants of Lie algebroids such as characteristic foliations, homotopy groups, and Crainic-Fernandes style integration theory might be generalized to this setting.
Poisson manifolds and their associated stacks
Host: BIRS Workshop 17w5023 "Geometric structures on Lie groupoids"
Abstract: Lie groupoids with Morita equivalence are a convenient model for the study of singular manifolds. This talk will explain how one can interpret a symplectic groupoid and symplectic Morita equivalence as a model for a singular Dirac manifold. To that end, we will define a site (a category with a topology) whose objects are Dirac manifolds, DMan, and explain how to associate a stack over DMan to any symplectic groupoid. Furthermore, we will relate isomorphisms of such stacks with symplectic Morita equivalences.