University of Western Ontario

Department of Mathematics

Middlesex College, Room 133

E-mail: mfranc65 at uwo dot ca

CV (updated November 2023)

I am a postdoctoral fellow at the University of Western Ontario with faculty mentors Tatyana Barron and Masoud Khalkhali. I obtained my PhD from the Pennsylvania State University in 2021 under the supervision of Nigel Higson. I also hold masters and bachelors degrees from the University of Victoria. My broad research areas are operator theory and differential geometry, especially foliations and smooth groupoids. I am a Canadian citizen.

Most days of the week, an operator algebra is a collection of continuous linear transformations of a normed vector space, closed with respect to compositions, linear combinations, and some choice of topology. The fun starts when one sees how many constructions from far flung areas of mathematics land in this arena. Prominently, one has the whole industry of noncommutative geometry, established by Alain Connes. Here, we seek out ways to understand geometric situations, especially those resistant to classical methods, by understanding associated operator algebras. In this way, we are able to bring to bear the powerful theorems of functional analysis, as well as powerful invariants, especially operator K-theory and its friends.

These days, one feels more and more that most of the constructions which give us interesting operator algebras pass--not always in a obvious way--through the world of (topological, smooth, etc) groupoids. Accordingly, the business of attaching groupoids to geometrical situations is very much deserving of attention from people interested in applying operator theory to geometry, or vice versa!

Here are some old pages of seminars for which I was the organizer or co-organizer:

This Fall/Winter I am teaching Math 0110A - Introductory Calculus. All course content is hosted on OWL.

On singular foliations tangent to a given hypersurface (arXiv). This article introduces and analyzes a specific family of singular foliations it calls transverse order k-foliations. There is only one singular leaf, and an integer parameter k encodes the ``order of tangency'' of the leafwise vector fields to L. It turns out a loop in L determines a well-defined holonomy map at the level of (k−1)-jets. It is shown the resulting holonomy map is a compete invariant and that all possible values of this invariant occur. Concrete descriptions of associated groupoids and algebras are given as well.

The Newlander-Nirenberg theorem for complex b-manifolds, with Tatyana Barron (arXiv). Melrose introduced b-manifolds and the b-tangent bundle as tools for studying differential operators that suffer a first-order degeneracy along a given hypersurface. Many classical geometries admit "b-analogues" in which the b-tangent bundle fills the role of the usual tangent bundle (so one has symplectic b-geometry, Riemannian b-geometry, etc). Complex b-geometry was introduced by Mendoza. This article establishes the "b-Newlander-Nirenberg theorem": every complex b-manifold is locally isomorphic to some standard model. This means it makes sense to define complex b-manifolds using "b-holomorphic charts".

On automorphisms of complex b^k-manifolds, with Tatyana Barron (arXiv). The b-calculus of Melrose is a tool for studying structures on a smooth manifold with a first order degeneracy at a given hypersurface. In this framework, Mendoza defined complex b-manifolds. In the spirit of work of Scott, we extend Mendoza's definition to the case of higher-order degeneracies, introducing the notion of a complex b^k-manifold for k a positive integer. We then investigate the local and global automorphisms of complex b^k-manifolds. We also propose b^k-analogues for some classical spaces of holomorphic functions.

H-Unitality of smooth groupoid algebras (arXiv). In this article, I show that the convolution algebra of smooth, compactly supported functions on a Lie groupoid is homologically unital in the sense of Wodzicki. I also prove homological unitality of certain ideals associated to invariant closed subsets of the unit space, leading to an excision property in Hochschild and cyclic homology around invariant, closed subsets. This extends my previous work establishing the Dixmier-Malliavin theorem in this setting.

Elementary number theory lecture notes. In Spring 2023, I taught "Math 2156: Mathematical Structures II" at University of Western Ontario. This course covered a mixture of Elementary Number Theory and Graph Theory. These are my lecture notes from the number theory component.

A Dixmier-Malliavin theorem for Lie groupoids, J. Lie Theory 32, no. 3, 879–898, (2022). Dixmier and Malliavin proved that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. In this article, I establish that the same holds in a Lie groupoid and derive, as a corollary, some results on the multiplication structure of certain ideals. Elsewhere, I have applied this work to study the smooth convolution algebra of certain singular foliations, in the sense of Androulidakis and Skandalis.

Groupoids and algebras of certain singular soliations with finitely many leaves. My PhD Thesis, completed in Summer 2021, under the supervision of Professor Nigel Higson.

The smooth algebra of a one-dimensional singular foliation (arXiv). Androulidakis and Skandalis showed how to associate a holonomy groupoid, smooth convolution algebra and C*-algebra to any singular foliation. In this article, I study the groupoids and algebras associated to the singular foliations of a one-dimensional manifold given by vector fields that vanish to order k at a point. I show that, whereas the C*-algebras of these foliations are divided into two isomorphism classes according to the parity of k, the smooth algebras are pairwise nonisomorphic.

Subgraph-avoiding minimum decycling sets and k-conversion sets in graphs, with Kieka Mynhardt and Jane Wodlinger, Australas. J. Combin. 74, 288–304 (2019). A minimum decycling set in a (finite) graph G is a set of vertices which breaks every cycle and has as few vertices as possible subject to this constraint. This article proves that, in a graph of maximum degree r bigger than 3 that is not a complete graph, one can always find a minimum decycling set which, moreover, does not contain any (r-2)-regular subgraph. This result has several corollaries, including Brooks' theorem.

Introduction to C*-algebra homology theories. This is an expository essay on the axiomatic approach to C*-algebra homology theories, especially Cuntz's proof via an infinite swindle that any stable, homotopy-invariant, half-exact functor from C*-algebras to abelian groups satisfies Bott periodicity.

Darboux's theorem and Euler-like vector fields. This essay was written as a prerequisite for the scheduling of an oral comprehensive exam at Penn State University. In it, I explain how to prove the Darboux theorem of symplectic geometry using the linearizability of so-called Euler-like vector fields. This method was taken from "Splitting theorems for Poisson and related structures" by Bursztyn-Lima-Meinrenken.

Linear Galois theory. These are some notes on the fundamental correspondence theorem of Galois theory. Their main purpose is to emphasize the not especially difficult, but conceptually rather attractive point that, by enlarging each Galois group to the ring of transformations of the extension field that are linear with respect to the subfield, one gets a version of the correspondence theorem which is valid for all finite extensions, instead of just the Galois ones.

A first look at geometric group theory. This report, and the talk which accompanied it, were prepared for the Graduate Student Seminar course at Pennsylvania State University, administered by Professor Sergei Tabachnikov.

Traces, one-parameter flows and K-theory. My masters thesis, completed in Summer 2014, under joint supervision of Professors Heath Emerson and Marcelo Laca.

Two topological uniqueness theorems for spaces of real numbers (arXiv). This report, and the talk which accompanied it, were prepared for the Graduate Student Seminar course at University of Victoria, administered by Professor Kieka Mynhardt.