Spectral Graph Theory over Ordered Fields
The goal of this course is to present different aspects of spectral graph theory and random walks on graphs. Together with classical weighted graphs with positive real edge-weights we will discuss graphs, whose edge-weights belong to an arbitrary real-closed ordered field. The non-Archimedean fields contain infinitely small elements, which leads to the phenomenon of existence of strictly monotone bounded non-converging sequences. The latest leads to new effects on graphs, in particular, on infinite ones.
We will discuss such aspects as capacity of graphs, type of a vertex in the corresponding random walk (recurrence/transience) and spectral properties. The concepts will be illustrated with numerous examples.
We will discuss such aspects as capacity of graphs, type of a vertex in the corresponding random walk (recurrence/transience) and spectral properties. The concepts will be illustrated with numerous examples.
Lecturer
Anna Muranova
Date
7th April ~ 21st May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 10:40 - 12:15 | A3-4-101 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Preliminary knowledge in analysis, probability theory, linear algebra and abstract algebra are required.
Syllabus
(1) Finite real weighted graphs.
Dirichlet forms. Laplace operator. Spectral properties. Dirichlet problem. Electrical networks. Energy. Minimization of energy. Related Markov chain and random walk.
(2) Infinite real weighted graphs.
Dirichlet problem. Capacity. Markov chain: recurrence and transience. Equivalent definitions of capacity. Relation between capacity and type of state.
(3) Non-Archimedean ordered fields.
Non-Archimedean property. Infinitesimals. Order topology. Convergence. Examples. The Levi-Civita field.
(4) Concept of graphs over non-Archimedean ordered fields.
Laplace operator. Spectral properties. Dirichlet problem. Energy. Electrical networks. Capacity and type of graph.
(5) Graph over non-Archimedean ordered fields and Markov chain.
Projection of non-Archimedean weights on reals. Directed graphs. Markov chains: essential and irreducible classes. Projection of capacity vs. type of state.
Dirichlet forms. Laplace operator. Spectral properties. Dirichlet problem. Electrical networks. Energy. Minimization of energy. Related Markov chain and random walk.
(2) Infinite real weighted graphs.
Dirichlet problem. Capacity. Markov chain: recurrence and transience. Equivalent definitions of capacity. Relation between capacity and type of state.
(3) Non-Archimedean ordered fields.
Non-Archimedean property. Infinitesimals. Order topology. Convergence. Examples. The Levi-Civita field.
(4) Concept of graphs over non-Archimedean ordered fields.
Laplace operator. Spectral properties. Dirichlet problem. Energy. Electrical networks. Capacity and type of graph.
(5) Graph over non-Archimedean ordered fields and Markov chain.
Projection of non-Archimedean weights on reals. Directed graphs. Markov chains: essential and irreducible classes. Projection of capacity vs. type of state.
Reference
[1] F.Fischer, M. Keller, A. Muranova, and N. Nicolussi. Capacity of infinite graphs over non-Archimedean ordered fields. Journal of Mathematical Analysis and Applications. Volume 543, Issue 2, Part 2, 15 March 2025.
[2] M. Keller, D. Lenz, and R. K. Wojciechowski. Graphs and discrete Dirichlet spaces, volume 358 of Grundlehren der mathematischen Wissenschaften. Springer, 2021.
[3] M. Keller, A. Muranova. Recurrence and transience for non-Archimedean and directed graphs. arXiv: 2406.17344.
[4] S. Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Revised Third Edition. Springer, 2002.
[5] A. Muranova. On the notion of effective impedance. Operator and Matrices,Volume 14, Number 3(2020), p. 723-741
[6] W. Woess.Denumerable Markov Chains. EMS, 2009
[2] M. Keller, D. Lenz, and R. K. Wojciechowski. Graphs and discrete Dirichlet spaces, volume 358 of Grundlehren der mathematischen Wissenschaften. Springer, 2021.
[3] M. Keller, A. Muranova. Recurrence and transience for non-Archimedean and directed graphs. arXiv: 2406.17344.
[4] S. Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Revised Third Edition. Springer, 2002.
[5] A. Muranova. On the notion of effective impedance. Operator and Matrices,Volume 14, Number 3(2020), p. 723-741
[6] W. Woess.Denumerable Markov Chains. EMS, 2009
Video Public
Yes
Notes Public
Yes
Language
English