Michael Francis

photo

Pennsylvania State University
Department of Mathematics
McAllister Building, Room 008
E-mail: mjf5726 at psu dot edu

CV (updated November 2020)

Research statement

About Me

I am a math PhD student at Penn State University. My doctoral advisor is Nigel Higson. My broad research area is operator algebras and geometry. I am supported by a Teaching Assistantship from Penn State University and was supported by a Postgraduate Doctoral Scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC) from 2015-2018. I am a Canadian citizen and have a masters and bachelors degree from the University of Victoria in British Columbia, Canada.

Operator Algebras and Geometry

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!

Writings

The smooth algebra of a one-dimensional singular foliation (2020): Androulidakis and Skandalis showed how to associate a holonomy groupoid, smooth convolution algebra and C*-algebra to any singular foliation (https://arxiv.org/abs/math/0612370v4). 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.

A Dixmier-Malliavin theorem for Lie groupoids (2020): 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.

Subgraph-avoiding minimum decycling sets and k-conversion sets in graphs, Francis, M. D.; Mynhardt, C. M.; Wodlinger, J. L., Australas. J. Combin. 74 (2019), 288–304.: 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 is taken from arXiv:1605.05386 [math.DG].

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: 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.

Seminars

Pages of old seminars for which I was the organizer:

Teaching

In Fall 2020, I am teaching Section 002 of Math 232 and Section 009 of Math 230. Students may see the Canvas page for more information.