Speaker:  A. Kazakov (MSU, YarSU, KFU, MIPT)

Title:  Electrical networks from the Calderon problem to the phylogenetic networks

Date and time:  13.11.2024, 17:00 (GMT +03:00)

Abstract: An electrical network is essentially a graph with positive edge weights denoting conductivities. The graph nodes are divided into two sets: inner nodes and boundary nodes. By applying voltages to the boundary nodes, we obtain a unique harmonic extension on all vertices

voltages, which might be found out by the Ohm's and Kirchhoff's laws. Investigating various properties of these harmonic extensions has given rise to many combinatorial objects, such as electrical response matrices, effective resistances, spanning trees, and groves. These objects have appeared in various mathematical theories, from Potts models (see zero Potts models [6]) to Abelian sandpile models [4].

The focus of my talk will be on the theory of planar circular electrical networks, which is closely related to the geometry of Lagrangian and Isotropic Grassmannian [2], [3], [7]. We will present two explicit constructions [2], [3] for the embedding of electrical networks to the non-negative part of the Isotropic Grassmannian using their response matrices and effective resistance matrices, respectively. Using the first construction, we will demonstrate a sketch of a new cluster solution to the discrete version of the Calderón problem, which is also known as the inverse problem of electrical impedance tomography [1]. Using the second one, we will provide a characterization of the resistance distance, which is widely used in chemistry and phylogenetic network theory [5].

The author was supported by the Russian Science Foundation grant 20-71-10110 (P).

[1] Borcea, L., Druskin, V., Vasquez, F. G.,``Electrical impedance tomography with resistor networks. Inverse Problems'', Vol.24, No.3, (2008).

[2] Bychkov, B., Gorbounov, V., Guterman, L., Kazakov, A.,``Symplectic geometry of electrical networks'', Journal of Geometry and Physics, Vol.207, (2025).

[3] Bychkov B.,  Gorbounov V.,  Kazakov A.,  Talalaev D., ``Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups,'' Moscow Mathematical Journal, Vol.23, No.2, (2023).

[4] Dhar, D., ``The abelian sandpile and related models'', Physica A: Statistical Mechanics and its applications, Vol.263, No. 1-4., (1999).

[5] Forcey, S., ``Circular planar electrical networks, split systems, and phylogenetic networks'', SIAM Journal on Applied Algebra and Geometry, Vol.7, No. 1, (2023).

[6] Fortuin, C. M., Kasteleyn, P. W.,``On the random-cluster model: I. Introduction and relation to other models'', Physica, Vol. 57, No. 4., (1979).

[7] Lam T., ``Totally nonnegative Grassmannian and Grassmann polytopes,'' arXiv preprint arXiv:1506.00603, (2015).

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com