Joel Villatoro - Mathematician

For Non-Experts

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.  

• Convolution algebras of double groupoids and strict 2-groups
  Co-author(s): Angel Roman
  Status: Preprint
  Abstract: Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally, strict 2-groups. Groupoid structures give rise to convolution operations on the space of arrows. Therefore, a double groupoid comes equipped with two product operations on the space of functions. In this article we investigate in what sense these two convolution operations are compatible. We use the representation theory of compact Lie groups to get insight into a certain class of 2-groups.
  
  

• On the integrability of Lie algebroids by diffeological spaces
  Co-author(s): -
  Status: Preprint
  Abstract: Lie's third theorem does not hold for Lie groupoids and Lie algebroids. In this article, we show that Lie's third theorem is valid within a specific class of diffeological groupoids that we call `singular Lie groupoids.' To achieve this, we introduce a subcategory of diffeological spaces which we call `quasi-etale.' Singular Lie groupoids are precisely the groupoid objects within this category, where the unit space is a manifold.

Our approach involves the construction of a functor that maps singular Lie groupoids to Lie algebroids, extending the classical functor from Lie groupoids to Lie algebroids. We prove that the Ševera-Weinstein groupoid of an algebroid is an example of a singular Lie groupoid, thereby establishing Lie's third theorem in this context.

• On sheaves of Lie-Rinehart algebras
  Co-author(s): -
  Status: Preprint
  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)
  Co-author(s): -
  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
  Co-author(s): -
  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
  Co-author(s): -
  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)
  Coauthor(s): -
  Status: Finished
  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.