Intended Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining the motivation behind their own research.
In this episode we meet Thomas Lam, professor of mathematics at the University of Michigan, who studies electrical networks (among other topics). Thomas gives a great introduction to one of the main problem in electrical networks as well as an application of electrical networks to medicine. At the end, we ask Thomas when he knew he wanted to become a mathematician. Did you know that there is a “math olympics” for high school students?
The main problem presented in this video is the motivation behind several papers about electrical networks, including the paper, “Electrical networks and Lie theory,” by Thomas Lam and Pavlo Pylyavskyy. Here is the abstract to their paper:
We introduce a new class of “electrical” Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis-Ingerman-Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. Rather surprisingly, we show that the type A electrical Lie group is isomorphic to the symplectic group. The nonnegative part of the electrical Lie group is a rather precise analogue of the totally nonnegative subsemigroup of the unipotent subgroup of . We establish decomposition and parametrization results for , paralleling Lusztig’s work in total nonnegativity, and work of Curtis-Ingerman-Morrow and de Verdière-Gitler-Vertigan for networks. Finally, we suggest a generalization of electrical Lie algebras to all Dynkin types.
While the video above is for a general audience, Thomas Lam’s paper is not (it’s written for other mathematicians). However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.
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©2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708