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BIMSA-HSE Joint Seminar on Data Analytics and Topology
Inverse problems related to electrical networks and the geometry of non-negative Grassmannians
Inverse problems related to electrical networks and the geometry of non-negative Grassmannians
Organizers
Speaker
Anton Kazakov
Time
Monday, January 20, 2025 8:00 PM - 9:00 PM
Venue
A6-101
Online
Zoom 468 248 1222
(BIMSA)
Abstract
An electrical network is just a graph equipped with positive edge weights denoting conductivities, which nodes are divided on two sets: a set of inner nodes and a set of boundary nodes. Applying voltages $\mathbf{U}\colon V_B \to\mathbb{R}$ to its boundary nodes, we obtain the unique harmonic extension on all vertices voltages $U\colon V \to \mathbb{R}$, which might be found out by the Ohm's and Kirchhoff's laws. Studying different properties of these harmonic extensions has given rise to many combinatorial objects: electrical response matrices, effective resistances and partition functions of spanning groves. All of them have appeared in many theories from the statistical physics (see, for instance, $q \to 0$ Potts models [6] and its relation to Abelian sandpile models [4]) to some areas of chemistry [8].
In the focus of my talk will be the theory of the planar circular electrical networks, which closely relates to the geometry of non-negative Grassmannians [2], [3], [9]. We will present the explicit construction [2], [3] of the embedding of electrical networks to the non-negative part of Grassmannian $\mathrm{Gr}(n − 1, 2n)$ by their effective resistance matrices. Using it, we will provide the sketch of the cluster solution of the network topology reconstruction problem [7], which has the application in phylogenetic network theory [5].
The author was supported by the Russian Science Foundation grant 20-71-10110 (P).
REFERENCES
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. Gorbounov V., Kazakov A. “Electrical networks and data analysis in phylogenetics”, arXiv preprint arXiv:2501.01383, (2025).
8. Klein D. J., Randi´c M., “Resistance distance”, J. Math. Chem., Vol. 12, (1993).
9. Lam T., “Totally nonnegative Grassmannian and Grassmann polytopes,” arXiv preprint arXiv:1506.00603, (2015).
In the focus of my talk will be the theory of the planar circular electrical networks, which closely relates to the geometry of non-negative Grassmannians [2], [3], [9]. We will present the explicit construction [2], [3] of the embedding of electrical networks to the non-negative part of Grassmannian $\mathrm{Gr}(n − 1, 2n)$ by their effective resistance matrices. Using it, we will provide the sketch of the cluster solution of the network topology reconstruction problem [7], which has the application in phylogenetic network theory [5].
The author was supported by the Russian Science Foundation grant 20-71-10110 (P).
REFERENCES
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. Gorbounov V., Kazakov A. “Electrical networks and data analysis in phylogenetics”, arXiv preprint arXiv:2501.01383, (2025).
8. Klein D. J., Randi´c M., “Resistance distance”, J. Math. Chem., Vol. 12, (1993).
9. Lam T., “Totally nonnegative Grassmannian and Grassmann polytopes,” arXiv preprint arXiv:1506.00603, (2015).