This monograph describes some of the most interesting results obtained by the mathematicians and physicists collaborating in the CRC 647 "Space – Time – Matter", in the years 2005 - 2016. The work presented concerns the mathematical and physical foundations of string and quantum field theory as well as cosmology. Important topics are the spaces and metrics modelling the geometry of matter, and the evolution of these geometries. The partial differential equations governing such structures and their singularities, special solutions and stability properties are discussed in detail.

Contents
Introduction

Algebraic K-theory, assembly maps, controlled algebra, and trace methods

Lorentzian manifolds with special holonomy – Constructions and global properties

Contributions to the spectral geometry of locally homogeneous spaces

On conformally covariant differential operators and spectral theory of the holographic Laplacian

Moduli and deformations

Vector bundles in algebraic geometry and mathematical physics

Dyson–Schwinger equations: Fix-point equations for quantum fields

Hidden structure in the form factors ofN = 4 SYM

On regulating the AdS superstring

Constraints on CFT observables from the bootstrap program

Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities

Yangian symmetry in maximally supersymmetric Yang-Mills theory

Wave and Dirac equations on manifolds

Geometric analysis on singular spaces

Singularities and long-time behavior in nonlinear evolution equations and general relativity