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Spectral Structures and Topological Methods in Mathematics
EMS Series of Congress Reports

Spectral Structures and Topological Methods in Mathematics

Michael Baake (Universität Bielefeld, Germany)
Friedrich Götze (Universität Bielefeld, Germany)
Werner Hoffmann (Universität Bielefeld, Germany)

ISBN print 978-3-03719-197-2, ISBN online 978-3-03719-697-7
DOI 10.4171/197
July 2019, 433 pages, hardcover, 17 x 24 cm.
88.00 Euro

This book is a collection of survey articles about spectral structures and the application of topological methods bridging different mathematical disciplines, from pure to applied. The topics are based on work done in the Collaborative Research Centre (SFB) 701.

Notable examples are non-crossing partitions, which connect representation theory, braid groups, non-commutative probability as well as spectral distributions of random matrices. The local distributions of such spectra are universal, also representing the local distribution of zeros of $L$-functions in number theory.

An overarching method is the use of zeta functions in the asymptotic counting of sublattices, group representations etc. Further examples connecting probability, analysis, dynamical systems and geometry are generating operators of deterministic or stochastic processes, stochastic differential equations, and fractals, relating them to the local geometry of such spaces and the convergence to stable and semi-stable states.

Keywords: Universal distributions, free probability, Markov processes, Schrödinger operators, heat kernel, spatial ecology, metastability, numerical analysis, critical regularity, aperiodic order, dynamical systems, special Kähler structure, non-crossing partitions, localising subcategory, braided groups, zeta functions, subgroup growth, representation growth, Brumer–Stark conjecture, p-divisible groups