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 and/or autonomous
vehicles
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unpublished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/
