Introduction to Khovanov Homology by Paul Wedrich
Knot homologies are the subjects of a young, fast-paced research field at the intersection of low-dimensional topology, symplectic geometry, representation theory and mathematical physics. This course will give an introduction to knot homologies in the spirit of Khovanov's categorification of the Jones polynomial and discuss their relationship to higher representation theory and applications. The schedule will be roughly as follows:
Part 1: The Jones polynomial and Khovanov homology.
We start with the 'cube of resolutions' state-sum model for the Jones polynomial, see how Khovanov homology categorifies it and then discuss what it means for a knot homology to be functorial under knot cobordisms.
Part 2: Knot polynomials via quantum group representation theory.
We meet the Reshetikhin-Turaev/Chern-Simons knot invariants of type A, an infinite family of knot polynomials defined using the representation theory of quantum groups, which generalise the Jones polynomial. We then discuss how to compute them using the quantum skew Howe duality of Cautis-Kamnitzer-Morrison.
Part 3: Categorified quantum groups.
With a view towards categorifying the type A knot invariants, we meet the diagrammatic categorified quantum groups of Khovanov-Lauda.
Part 4: Generalisations of Khovanov homology.
We see how categorified skew Howe duality leads to a combinatorial definition of type A knot homologies and finish by discussing the relationships to geometric approaches to knot homologies, applications and open problems.
Dror Bar-Natan: Khovanov's homology for tangles and cobordisms. Geom. Topol., 9:1443--1499, 2005. Sections 1 -- 7.
Aaron D. Lauda: An introduction to diagrammatic algebra and categorified quantum sl(2). Bull. Inst. Math. Acad. Sin., 7:165-270, 2012. Sections 1 and 3.
Sabin Cautis, Joel Kamnitzer, and Scott Morrison: Webs and quantum skew Howe duality. Math. Ann., 360(1-2):351--390, 2014.
Aaron D. Lauda, Hoel Queffelec, and David E. V. Rose: Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m). Algebr. Geom. Topol., 15: 2517-2608, 2015.
Multilayer Networks: Structure and Dynamics by Ginestra Bianconi
Multilayer Networks INTRODUCTION [PDF 32,134KB]
Multilayer Networks STRUCTURE 2 [PDF 21,017KB]
Multilayer Networks MODELS 3 [PDF 7,209KB]
Multilayer Networks PERCOLATION 4 [PDF 9,419KB]
Multilayer Networks DIFFUSION 5 [PDF 14,117KB]
Multilayer networks describe interacting complex systems formed by different interacting networks. these networks are ubiquitous and include social networks, financial markets, multimodal transportation systems, infrastructures, the brain and the cell. The multilayer structure of these networks strongly affects the properties of dynamical and stochastic processes defined on them, which can display unexpected characteristics. For example, interdependencies between different networks of a multilayer structure can cause cascades of failure events that can dramatically increase the fragility of these systems; while the interplay between the multiplexity and diffusion has major consequences for applications in transportation, and in general navigation of multilayer networks.
In these lectures this new emerging topic of network theory will be introduced focusing on a statistical mechanics perspective. We will cover generative models for multilayer networks, generalized percolation transition describing the emergence of the mutually connected giant component, and diffusion properties of multilayer networks. Always amble room will be given to the discussion of the interdisciplinary applications of these results.
Boccaletti, S., Bianconi, G., Criado, R., del Genio, C.I., Gómez-Gardeñes, J., Romance, M., Sendiña-Nadal, I., Wang, Z. and Zanin, M., 2014. The structure and dynamics of multilayer networks. Physics Reports, 544(1), pp.1-122.
Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y. and Porter, M.A., 2014. Multilayer Networks. Journal of Complex Networks, 2(3), pp.203-271.