Друзья!
В этот вторник(05.05.26) на семинаре в 16 20 у нас выступит Михаил Скопенков с рассказом об электрических сетях с гомологическими граничными условиями.
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Cohomological networks


In many applications, electrical networks appear with boundary conditions other than the usual Dirichlet and Neumann ones, and their mixture. For instance, in networks on surfaces, one often prescribes voltage drops along topologically nontrivial loops on the surface. Special boundary conditions also arise in electromagnetic circuits, introduced by Milton and Seppecher.

We develop a general framework for working with such cohomological boundary conditions in the discrete setup. We adopt the classical concepts and theorems, such as the response matrix and the existence and uniqueness theorem. Our main result is a generalization of all-minors Kirchhoff's matrix-tree theorem, a combinatorial formula for the minors of the response matrix in terms of certain subgraphs, in the spirit of Kenyon and Wilson's enumeration of groves. This generalization is challenging because the subgraphs can now contribute with arbitrary integer coefficients. The proof uses tools from statistical physics, such as Smirnov's parafermionic observables and the double-dimer model.

This is joint work with P. Pylyavskyy and S. Shirokovskikh.

ссылка на семинар


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#нис_complex_networks