Superport Networks: Generalizing Kirchhoff's Matrix-Tree Theorem

Explore a 49-minute lecture on superport networks presented by Pavlo Pylyavskyy from the University of Minnesota, Twin Cities, at IPAM's Statistical Mechanics and Discrete Geometry Workshop. Delve into the study of multiport networks, which differ from traditional electrical networks in their boundary conditions. Learn how boundary vertices are paired, with incoming currents summing to zero in each pair. Discover the generalization of Kirchhoff's matrix-tree theorem for this setup, including the complexities of different forests contributing with varying signs. Examine the use of Kenyon and Wilson's formula for response matrix minors, determinantal identities, and combinatorial bijections in proving these theorems. Gain insights into superport networks, a concept that encompasses both ordinary and multiport networks. Recorded on March 26, 2024, this talk presents joint work with Svetlana Shirokovskikh and Mikhail Skopenkov, offering a deep dive into advanced concepts in electrical engineering and network theory.