Mathematics Research

Interests and Background

In general I have a very broad interest in geometric control theory, PDE, applied mathematics, geometric analysis, and contact geometry. However, my attention is currently focused on the use of Exterior Differential Systems (EDS), Cartan's method of Equivalence, and moving frames to study problems in geometric control theory and PDE. My thesis is in the area of geometric control theory, specifically relating to the theory of cascade feedback linearization of control systems with symmetry, which in turn has important implications for the study of dynamic feedback linearizable control systems. My thesis draws on subjects such as EDS, Goursat bundles, Calculus of Variations, and Lie algebras. It's a very enjoyable overlap of subjects and I hope to continue finding more tools from other areas in geometry to answer questions in control theory. My advisor is the excellent Professor Jeanne Clelland with Peter Vassiliou as my wonderful co-advisor.

Before my current work, my first two years of graduate school were spent studying fluid equations from the perspective of infinite-dimensional Riemannian geometry. For example, the Euler equations for incompressible fluid flow on a finite dimensional manifold are precisely the Euler-Arnold equations on the space of volume preserving diffeomorphisms (endowed with an appropriate metric, that is). I still find this area interesting!

In a related area, my advisor and authored this paper in the Archive for Rational Mechanics. It gives further explicit results on the ``rarity" of Beltrami fields with non-constant proportionality factor. These types of solutions are of interest to the areas of hydrodynamics and plasma physics.

I have also done some work directly related to the geometry of Plasma physics with Dr. Joshua W. Burby while at Los Almaos National Laboratory via the NSF mathematical sciences graduate internship program (NSF-MSGI). Ultimately, this resulted in a first order asymptotic approximation to the Vlasov-Maxwell equations using formal slow manifold methods.

Here is my C.V.

Published and Accepted Work

  1. Burby, J. W., and T. J. Klotz. ``Slow manifold reduction for plasma science." Communications in Nonlinear Science and Numerical Simulation (2020): 105289.
  2. Clelland, Jeanne N., and Taylor Klotz. "Beltrami fields with nonconstant proportionality factor." Archive for Rational Mechanics and Analysis 236.2 (2020): 767-800.
  3. G. Dean, T. Klotz, B. Prinari and F. Vitale, ``Dark-Dark and Dark-Bright Soliton Interactions in the two-component Defocusing Nonlinear Schrodinger Equation'', Applic. Anal., Vol.92, pp. 379-397 (2013)

In Progress and Submitted

  1. T. Klotz, ``Geometry of Cascade Feedback Linearizable Control Systems," thesis - arXiv preprint arXiv:2102.08521, 2021
  2. J. Clelland, T. Klotz, P. Vassiliou, ``Dynamic Feedback Linearization of Control Systems with Symmetry" Submitted - arXiv preprint arXiv:2103.05078, 2021
  3. T. Klotz, ``Dynamic and Cascade Static Feedback Linearization for the PVTOL System," in preparation