Research Interests:
My Ph.D. is in smooth manifolds under Craig Guilbault. Smooth manifolds are the most
general mathematical objects on which one may do calculus, and
therefore differential equations and control engineering.
Configuration spaces (c-spaces) of mechanical systems with constraints
are the archetypal examples of smooth manifolds. C-spaces are the
underlying sets of "generalized coordinates" - "generalized
coordinates" are really just a coordinate patch on a smooth manifold
to a topologist - and, with Lagrangians on them, are used in system
dynamics in mechanical and control engineering. I intend to apply my
manifold topology to optimal geometric control theory (optimal
control engineering on manifolds) real-world problems, hopefully
eventually in the autonomous robots industries. I have a blog related
to my learning experiences in these new areas of STEM, This Week's Finds in Second-Order Ordinary
Differential Equations on Manifolds and Geometric Control Theory.
Publications:
"\(\mathcal{EZ}\)-Structures for Group Extensions of Fundamental Groups
of Manifold K(G,1)'s with \(\mathcal{EZ}\)-Structures", on hold
"Some Examples of Pseudo-Collars on High-Dimensional Manifolds with
Non-Superperfect Kernel Groups", on hold
"Some Examples of Pseudo-Collars on High-Dimensional Manifolds with
Non-Split-Exact Group Extensions", on hold
"A Necessary and Sufficient Condition for a Self-Diffeomorphism of a
Smooth Manifold to be the Time-1 Map of the Flow of a Differential
Equation"
https://arxiv.org/abs/2110.12806,
unpublished
"More Examples of Pseudo-Collar Structures on High-Dimensional
Manifolds",
Topology and its
Applications, Vol. 270, February 2020
https://arxiv.org/abs/1804.02597 https://www.sciencedirect.com/science/article/abs/pii/S0166864118305716
"A Geometric Reverse to the Plus Construction and Some Examples of
Pseudo-Collars on High-Dimensional Manifold",
Michigan
Mathematical Journal, Vol. 67 No. 3, August 2018
https://arxiv.org/abs/1508.03670 https://projecteuclid.org/journals/michigan-mathematical-journal/volume-67/issue-3/A-Geometric-Reverse-to-the-Plus-Construction-and-Some-Examples/10.1307/mmj/1522980163.short
"Some Results on Pseudo-Collar Structures on High-Dimensional
Manifolds", doctoral dissertation, May 2015
http://arxiv.org/abs/1502.04338 https://dc.uwm.edu/etd/916/
