# Joel Villatoro - Mathematician

I am an American mathematician currently working as a Lecturer at Indiana University Bloomington. In the past I have worked as an NSF Fellow at Washington University in St. Louis and 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: (first name)david(last name) (at) gmail.com

### 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.

## Papers and Preprints

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 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.