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
Author(s): Michael Baake, Friedrich Götze, Werner Hoffmann (editors)
Series: EMS Series of Congress Reports
Publisher: European Mathematical Society
Year: 2019
Language: English
Pages: 435
Preface......Page 6
Introduction......Page 8
Contents......Page 12
Introduction......Page 16
Symmetric random matrices......Page 20
Non-symmetric random matrices......Page 23
Local spectral distributions......Page 27
Connections between probability theory and number theory......Page 31
Analogies between classical and free probability......Page 35
References......Page 38
Fokker–Planck–Kolmogorov equations......Page 44
Three selected results on SPDEs......Page 55
References......Page 67
Analysis on manifolds......Page 70
Analysis on metric measure spaces......Page 78
Homology theory on graphs......Page 83
References......Page 86
Introduction......Page 90
Complex systems......Page 92
Markov evolutions......Page 96
Birth-and-death evolutions......Page 99
Vlasov-type scalings......Page 103
Kinetic equation......Page 107
References......Page 118
Large deviations......Page 122
Kramers' Law......Page 128
Parabolic SPDEs......Page 132
Unstable periodic orbits......Page 136
Mixed-mode oscillations......Page 140
References......Page 141
Equivariant evolution equations......Page 144
The freezing method......Page 148
Applications......Page 152
Relative Equilibria......Page 159
Nonlinear eigenvalue problems......Page 167
References......Page 171
Introduction......Page 174
Nonlinear Schrödinger equations on compact manifolds......Page 180
Nonlinear systems on Euclidean space......Page 188
References......Page 194
Introduction......Page 198
Variational solutions to the Dirichlet problem......Page 200
Ellipticity and coercivity of nonlocal operators......Page 204
(Weak) Harnack inequalities, and Hölder regularity......Page 206
References......Page 209
Introduction......Page 212
Weak model sets......Page 213
A decorated quasiperiodic tiling with mixed spectrum......Page 218
Random inflations......Page 225
Enumeration of lattices......Page 228
References......Page 232
Introduction......Page 236
Special Kähler geometry in local coordinates......Page 238
Some global aspects of special Kähler geometry on P 1......Page 245
References......Page 247
The poset of non-crossing partitions......Page 250
Non-crossing partitions in free probability......Page 255
Braid groups......Page 262
Non-crossing partitions in Coxeter groups......Page 274
The Hurwitz action......Page 280
Non-crossing partitions arising in representation theory......Page 282
Generalised Cartan lattices......Page 284
Braid group actions on exceptional sequences......Page 285
References......Page 287
Introduction......Page 290
Preliminaries......Page 291
Types of localisation......Page 294
Cohomological localisations for the projective line......Page 300
Exotic localisations......Page 307
References......Page 311
Introduction: From group theory to topology......Page 314
Brown's criterion......Page 316
Combinatorial Morse theory......Page 318
Matching complexes for graphs and surfaces......Page 320
Higher generation in symmetric groups and braid groups......Page 323
The braided Thompson group Vbr......Page 325
A cube complex for Vbr......Page 327
The Morse function and its descending links......Page 329
Connectivity of descending links......Page 331
References......Page 335
Introduction......Page 338
Zeta integrals......Page 339
The Trace formula......Page 349
Unipotent terms in the trace formula......Page 353
References......Page 357
Zeta functions associated to groups and rings......Page 360
Submodule zeta functions......Page 362
Representation zeta functions for unipotent group schemes......Page 370
References......Page 376
Conjectures of Brumer, Gross and Stark......Page 380
Preliminaries......Page 381
The abelian case......Page 383
The general case......Page 385
Relations to further conjectures and results......Page 394
References......Page 401
Introduction......Page 404
Frames......Page 405
Displays......Page 408
Classification of p-divisible groups......Page 412
The nilpotency condition......Page 413
Isogenies......Page 417
References......Page 422
List of contributors......Page 424
Index......Page 426
