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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
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