2019/2020
Научно-исследовательский семинар "Электрические многообразия"
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Преподаватели:
Горбунов Василий Геннадьевич
Язык:
английский
Кредиты:
3
Контактные часы:
36
Course Syllabus
Abstract
We will discuss the new mathematical features of the classical theory of electrical networks developed by Ohm and Kirkhoff more than 100 years ago. These include the new type of the cluster algebras which were originally introduced for studying the variety of totally positivity matrices, new types of discrete integrable systems, the representations of the Temperley–Lieb algebra, the important algebra for knot theory and the theory of quantum integrable systems.
Learning Objectives
- To be introduced to the theory modern theory of networks and an important structure of cluster algebras networks possess sometimes
Expected Learning Outcomes
- To gain an understanding of this part of modern mathematics through studying several key examples of such a structure
- To understand and be able to identify the main features of the networks: the graph defining the network, the boundary measurement matrix, the cluster transformations preserving the boundary measurement matrix, the inverse problem for networks
Course Contents
- The modern approach to the theory of totally positive matrices developed by G. Lusztig
- TheworkofA.Berenstein,S.Fomin,A.Zelevinnskyontheclusteralgebrarelated to the general linear group
- Introduction to the theory of electrical networks: response matrix, inverse problem, approach via symplectic geometry
- Electrical varieties as the new approach to the theory of electrical networks
Bibliography
Recommended Core Bibliography
- Gorbounov, V., & Talalaev, D. (2019). Electrical varieties as vertex integrable statistical models. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1905.03522
Recommended Additional Bibliography
- Lam, T., & Pylyavskyy, P. (2011). Electrical networks and Lie theory. https://doi.org/10.2140/ant.2015.9.1401