Michael Francis

photo

E-mail: mjf5726 at psu dot edu
Office: 008 McAllister
CV (updated August 2018)

Seminars

In Fall 2018, I am the organizer of the Student Geometric Functional Analysis Seminar at Penn State University.

Some old seminar pages:

Teaching

In Fall 2018, I am teaching Sections 1 and 2 of Math 026: Trigonometry at Penn State. Please visit the following links for more information.

Department Coursepage
MyMathLab
Canvas

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 Postgraduate Doctoral Scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC) and by a Teaching Assistantship from Penn State University. I am a Canadian citizen and have a masters and bachelors degree from the University of Victoria in British Columbia, Canada. Some of my non-mathematical interests include spending time outdoors and bashing out a tune on the piano/guitar.

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 manage to 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

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 to demonstrate written competency in mathematics, a prerequisite at Penn State University for the scheduling of an oral comprehensive exam. In it, we 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].

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.