Michael Francis

Research

My research is mainly focused on operator algebras and their applications to 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!

Publications

  1. Michael Francis, H-unitality of smooth groupoid algebras, J. Noncommut. Geom. (2025). [journal] [arXiv]
  2. Michael Francis, On singular foliations tangent to a given hypersurface, J. Noncommut. Geom. (2024). [journal] [arXiv]
  3. Tatyana Barron and Michael Francis, On automorphisms of complex bk-manifolds, Geometric Methods in Physics XL, 2024, pp. 199–207. [journal] [arXiv]
  4. Michael Francis, A Dixmier-Malliavin theorem for Lie groupoids, J. Lie Theory 32, no. 3, 879–898 (2022). [journal] [arXiv]
  5. Michael Francis, Christina M. Mynhardt and Jane L. Wodlinger, Subgraph-avoiding minimum decycling sets and k-conversion sets in graphs, Australas. J. Combin. 74, 2019, pp. 288–304. [journal]

Preprints

  1. Tatyana Barron and Michael Francis, The Newlander-Nirenberg theorem for complex b-manifolds. [arXiv]
  2. Michael Francis, The smooth algebra of a one-dimensional singular foliation. [arXiv]
  3. Michael Francis, Two topological uniqueness theorems for spaces of real numbers. [arXiv]

Theses

  1. Michael Francis Groupoids and Algebras of Certain Singular Foliations with Finitely Many Leaves, PhD Thesis (2021). [Penn State library]
  2. Michael Francis Traces, one-parameter flows and K-theory, MSc Thesis (2014). [UVic library]

Other writings

  1. Michael Francis, Elementary Number Theory Lecture Notes (2023). [PDF]
  2. Michael Francis, Darboux's theorem and Euler-like vector fields (2018). [PDF]
  3. Michael Francis, Introduction to C*-algebra homology theories (2018). [PDF]
  4. Michael Francis, Linear Galois Theory (2018). [PDF]
  5. Michael Francis, Meeting notes from von Neumann algebra learning seminar (2017). [HTML]
  6. Michael Francis, A first look at geometric groups theory (2017). [PDF]