On the Role of Functorial Constructions in Computational Electromagnetics and Associated Pedagogical Insights
Forty years ago, computational electromagnetics struggled to make the transition from 2D problems to 3D problems, and many false starts based on 2D intuition hampered progress. It was during this time that the author stopped thinking about dimensionality inductively, and started to count 1, 2, n, 3, 4, … . Embracing an n-dimensional perspective enabled to one to identify differential forms and the deRham complex, Hodge theory on manifolds with boundary, simplicial complexes, Whitney forms and algebraic topology as key ingredients in the transformation of how we think about the finite element analysis of electromagnetic fields.
In addition to the category theory perspective that comes with the use of algebraic topology, there is an appreciation of the middle dimension(s) of manifolds that comes from differential topology, and the realization that there is a mod(4) dependence on dimension when it comes to generating “geometric understanding”. Thus, it is hopeless to think of dimensions inductively if one is not interested in going beyond dimension 4!
In this talk, I’ll consider the functorial constructions which make finite element algorithms simple and effective, and how the analysis of “near force-free magnetic fields” has become a key testing ground for showcasing a wide variety of topological ideas in 3D. Returning to the mod(4) dependence on dimension, the talk will end by considering an analogous 7D problem.
Speaker’s Bio
Prof. Kotiuga received his B.Eng., M. Eng., and Ph.D. from McGill University in 1981, 1982, and 1985 respectively. After a post-doc at MIT, he joined Boston University in 1987. Over the years he has held visiting appointments at MIT (Cambridge MA), ETH (Zurich), U. Pau (France), TUT (Finland), Univ of Trento (Italy), and other shorter appointments.
Prof. Kotiuga’s research focuses on topological aspects of 3-dimensional problems in computational electromagnetics, the use of Whitney forms and simplicial data structures in the context of the finite element method. His earlier work on cuts for magnetic scalar potentials and helicity functionals has, in recent years, led to a topological characterization of near force-free magnetic fields. His early topological work in the context of vertical Bloch line memories now informs topological considerations in nanoscale MRAM devices. More recently, informed by psychoacoustics, he is revisiting issues of transient modeling in electroacoustics.
Two books tied to his research are:
- Gross, P.W., Kotiuga, P.R., Electromagnetic Theory and Computation: A Topological Approach. Cambridge U. Press, 2004; reissued in 2011, and there’s an e-book version.
- Kotiuga, P. R., (editor), A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings & Lecture Notes, Vol.: 50, Amer. Math. Soc., 2010.