Complexity and Networks

Uitgebreide vaknaam	Complexity and Networks
Leerdoelen	At the end of the course, the student is able to: 1. find bases in root systems and act by their automorphism groups; express the classification of semisimple Lie algebras in terms of Dynkin diagrams; find Weyl bases in these algebras; 2. bound controllable space by using graph partitions, verify whether a given set of nodes is a zero forcing set and apply the theory to specific classes of graphs in order to select leaders rendering the system controllable; 3. bound controllable space by using graph partitions, verify whether a given set of nodes is a zero forcing set and apply the theory to specific classes of graphs in order to select leaders rendering the system controllable; 4. characterise the transition to synchronization in the Kuramoto model using the coupling strength and order parameter; compute the critical coupling and order parameter for the Cauchy distribution; model (numerically) the emergence of synchronization; perform the reduction of the Kuramoto-Sakaguchi model of identical oscillators and show its integrability for special parameter values
Omschrijving	This is an interdisciplinary course that presents different aspects of the theme Complexity and Networks from both the pure and applied mathematics points of view. The course consists of four parts. Part I of the course, by A.V.Kiselev, is about root systems, i.e. finite groups of reflections w.r.t. vectors in Euclidean spaces. Root systems give the main classification of objects' Complexity in Mathematics: from Platonic solids and Lie algebras in fundamental interactions, paving here a way to Mendeleev's periodic table of elements, to singularities in bifurcation or catastrophe theory, to multidimensional Fourier transform, and cryptography (using encryption schemes based on quasicrystals).. In the second part, by K. Camlibel, the notion of system controllability is discussed with emphasis on systems defined on graphs, diffusively coupled leader/follower systems, and concepts such as controllable space, leader selection, and targeted controllability. In the third part, by D. Valesin, the topic is the Erdös-Renyi random graph model; the main result that is stated and proved is the phase transition, with respect to the connectivity probability parameter, of the size of the largest connected component of this graph. In the fourth part, by N. Martynchuk, we discuss the phenomenon of synchronization in the context of the Kuramoto model. We address the original model with the mean-field coupling and discuss some of its generalisations, such as the Kuramoto-Sakaguchi model and Kuramoto models on weighted networks. The appearance of integrability in the Kuramoto-Sakaguchi model and a few physical examples will also be discussed.
Uren per week
Onderwijsvorm	Hoorcollege (LC), Werkcollege (T)
Toetsvorm	Opdracht (AST) (The assessment consists of 4 homework assignments, one for each part of the course. The final grade is the average of the grades of the 4 homework grades, provided that all grades are above 4. If one of the homework grades is 4 or less then the final grade is the minimum of the homework grades.)
Vaksoort	master
Coördinator	prof. dr. D. Rodrigues Valesin
Docent(en)	prof. dr. M.K. Camlibel , A.V. Kiselev ,prof. dr. D. Rodrigues Valesin
Verplichte literatuur	Titel Auteur ISBN Prijs (suggestion, not mandatory) Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag. J.E. Humphreys 0-387-90053-5 (suggestion, not mandatory) Synchronization in Complex Networks, Physics Reports, vol. 469, pp. 93-153, 2008. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou (suggestion, not mandatory) "Random graphs and complex networks." Available on http://www.win.tue.nl/rhofstad/NotesRGCN.pdf (2009). Van Der Hofstad, Remco (suggestion, not mandatory) Random graph dynamics. Vol. 200, no. 7. Cambridge: Cambridge university press, 2007. Durrett, Richard (suggestion, not mandatory) Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press. J.E. Humphreys 0-521-37510-X (suggestion, not mandatory) The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77:137, 2005 Juan A. Acebrón, L. L. Bonilla, Conrad J. Pérez Vicente, Félix Ritort, and Renato Spigler (suggestion, not mandatory) Constants of motion for superconducting Josephson arrays, Physica D: Nonlinear Phenomena, vol 74, issues 3-4, pp. 197-253, 1994 S. Watanabe and S.H. Strogatz
Entreevoorwaarden	The course assumes prior knowledge in: - Real and complex analysis - Probability theory - Linear algebra (vector spaces, invariant subspaces, quotient spaces, eigenvalues, eigenvectors) - Group Theory (permutation group, conjugacy) - Fourier series - Dynamical systems (equilibria, linear stability) - Systems theory (state and controllability) - Programming skills
Opmerkingen	This course was registered last year with course code WMMA16000
Opgenomen in	Opleiding Jaar Periode Type MSc Applied Mathematics: Computational Mathematics  ( MSc Applied Mathematics: Computational Mathematics ) - semester I a verplicht MSc Applied Mathematics: Systems and Control  (MSc Applied Mathematics: Systems and Control) - semester I a verplicht MSc Courses for Exchange Students: AI - Computing Science - Mathematics - semester I a MSc Mathematics and Physics (double degree)  (Mathematics and Complex Dynamical Systems (40 ects)) - semester I a verplicht MSc Mathematics: Mathematics and Complex Dynamical Systems - semester I a verplicht MSc Mathematics: Science, Business and Policy  (MSc Mathematics: Science, Business and Policy) - semester I a verplicht MSc Mathematics: Statistics and Big Data  (MSc Mathematics: Statistics and Big Data) - semester I a verplicht