Assistant Professor

Bio

Jon Fortney teaches mathematics at Zayed University.He joined Zayed University in 2012.  He is a member of the American Mathematical Society and the Mathematical Association of America.

EMPLOYMENT HISTORY:

Prior to returning to graduate school to obtain his PhD Jon Fortney worked as a property and casualty actuarial analist.

Office

Dubai Academic City, C- L2 -010

Phone:

+971 4 4021695

Email:

Teaching Areas

Mathematics – Differential Geometry – Geometric Mechanics
Additionally, undergraduate degrees in Actuarial Science andChemical Engineering

Research and Professional Activities

To dateJon Fortney’s work is primarily in geometric mechanics, a subfield of differential geometry:

I am interested in the interface between differential geometry and dynamical systems, a field called geometric mechanics. Three broad geometrically motivated formalisms are frequently used to model a wide range of physical systems: the Lagrangian, the Hamiltonian, and the gradient formalisms. Motivated by differing Hamiltonian and gradient formulations of circuit dynamics, my research focused on the geometric relationship between these two formalisms in a general context. My research so far has concluded that given an implicitly defined Hamiltonian system on a manifold (defined with respect to a Dirac structure) that if the Hamiltonian function satisfies certain conditions, then the implicitly defined Hamiltonian system induces either a pseudo-gradient system on a Lagrangian submanifold of the cotangent bundle or it foliates this Lagrangian submanifold into pseudo-Riemannian leaves which give rise to a pseudo-gradient systems on each leaf. Up to now my research has concentrated on the geometric relationship between the Hamiltonian and the gradient formalism; the Lagrangian formalism was not addressed. There are a number of questions that I am interested in addressing: Is a more unified mathematical approach to these formalisms possible? Do Dirac structures play any role in the Legendre transform between the Lagrangian and Hamiltonian formalisms? Could one model an interconnected system where different subsystems are modeled by different formalisms?

Jon Fortney recently complted an undergraduate textbook titled A Visual Introduction to Differential Forms and Calculus on Manifolds, currently under review for publication:

I am interested in the geometric point-of-view and interpretation of mathematical objects. Beyond a doubt, differential forms – multi-linear, skew-symmetric, covariant tensor fields – play a fundamental role in differential geometry, geometric analysis, and de Rham cohomology. As such they are well-known and well-understood and are treated extensively in a very wide range of both introductory and advanced mathematics textbooks. But despite all this; the standard treatment of differential forms in the literature is often cursory, being a topic to be passed through quickly before getting to the primary subject matter, and is thus almost purely presented from a computational point-of-view. That is, the computational rules necessary to perform the required calculations are presented and little else is. This makes sense given the purpose of most textbooks, to cover the foundational material quickly and only to the degree necessary to tackle more advanced subject matter, but it leaves students without any sort of a geometric picture or visual image as to what differential forms actually are. I am currently interested in finding more effective ways to present differential forms that will provide students a better geometric understanding of the objects. While not all students may feel the need to have a firm geometric picture when working with differential forms, many do.

MEMBERSHIP OF SCIENTIFIC AND PROFESSIONAL SOCIETIES: 

• American Mathematical Society

• Mathematical Assosiation of America

SELECTED PUBLICATIONS:

• J.P. Fortney, “Basis Independence of Implicitly Defined Hamiltonian Circuit Dynamics,” Mathematics Across Contemporary Science, Springer Proceedings in Mathematics and Statistics, T. Abualrub et al (eds.) DOI 10.1007/978-3-319-46310-0_5

• J.P. Fortney, “Dirac Structures in Pseudo-Gradient Systems with an Emphasis on Electrical Networks,” IEEE Transactions of Circuits and Systems – I: Regular Papers, Vol. 57, No. 8, Aug. 2010.