|Date: Friday, February 21, 2014
Location: 3866 East Hall (4:10 PM to 5:00 PM)
Title: Positivity Phenomena in Circular Planar Electrical Networks
Abstract: Each planar electrical network embedded in a disk has a corresponding response matrix, which maps voltages at boundary vertices to induced currents. Curtis, Ingerman, and Morrow characterize these response matrices in terms of the nonnegativity of some minors called circular minors. In this talk, after introducing the notions relevant to circular planar electrical networks, I will investigate more closely some combinatorial properties relating to response matrices and circular minors. First, extending the work of Postnikov, I will introduce electrical positroids, which are the sets of circular minors which can simultaneously be positive in a response matrix. I will give a self-contained axiomatic description of these electrical positroids. Second, in investigating tests for the positivity of circular minors, I will discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam and Pylyavskyy. Building off of initial work by Kenyon and Wilson, I will investigate the clusters of this algebra using an analogue of weak separation, as was originally introduced by Leclerc and Zelevinsky.
This talk is based on joint work with Carl Lian and Brandon Tran.
Speaker: Josh Alman