SFB 647:  Space - Time - Matter: Analytic and Geometric Structures

• Subject Area: Mathematics ; Physics
• Term: from 2005 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5486209

In 1918, Hermann Weyl published his famous book Raum - Zeit - Materie in which he intended to present a comprehensive treatment of Einstein s theory of general relativity. The discovery of quantum mechanics, in 1925, produced an enormous scientific challenge: to overcome the incompatibility between the quantum world and Einstein s view of the universe. This goal has not been reached yet, but with the advent of the string theory it seems that the structure of matter and the structure of the universe are about to be expressed in a common language and, possibly, in the framework of a single theory. The process of developing quantum field theory and, in particular, the new geometric concept of matter ( geometrisation of matter ) has also brought about a new relationship between mathematics and physics that seems to have paved the way for the emergence of a new field, with a common language and a common collection of examples and methods, and with the ability to solve difficult old problems and to formulate entirely new, far reaching theories.
The areas of expertise planning to cooperate here, geometry, analysis, and string theory, are naturally associated with space, time, and matter. Nevertheless, the title of the project should not be read as a promise to produce a final answer, but to launch a serious effort to further clarify the intricate interplay between the relevant elements of both triads, and to reveal new concepts of common value.
The projects fall into two groups, the first one addressing the geometry of matter , and the second one being devoted to the evolution of geometric structures . Under the first heading, roughly half of the projects will be devoted to the detection, construction, and classification of special geometric structures, which have become of interest in string theory and cosmology. The second group of projects will examine special solutions, deformations, singularity formation, and stability properties of the relevant partial differential equations, with a view of shedding more light on the structures discussed in the first group.
Since the joint competence spectrum covers a remarkably large and coherent area, only rarely to be found in a single place, it is to be expected that cooperation along these lines will unleash a considerable potential for innovation.

• DFG Programme: Collaborative Research Centres

Completed projects

• A01 - Strings, D-Branes, and Manifolds of Special Holonomy (Project Heads Farkas, Gavril ;  Schmitt, Alexander )
• A02 - Special Geometries and Fermionic Field Equations (Project Heads Agricola, Ilka ;  Friedrich, Thomas )
• A03 - Singularities in Manifolds with Special Holonomy (Project Heads Altmann, Klaus ;  Jahnke, Priska )
• A04 - Quasilinear Wave Equations, Membranes and Supermembranes (Project Heads Ecker, Klaus ;  Huisken, Gerhard ;  Nicolai, Hermann )
• A06 - Conformal Symmetry, Supergravity, and teh AdS/CFT Correspondence (Project Heads Baum, Helga ;  Erdmenger, Johanna ;  Plefka, Jan )
• A07 - Holonomy Theory of Indefinite Metrics, Conformally Invariant Differential Operators and Q-Curvature (Project Head Baum, Helga )
• A08 - Tests and Applications of the AdS/CFT Correspondence (Project Heads Dorn, Harald ;  Plefka, Jan ;  Theisen, Stefan )
• A09 - Enumerative Geometry of Moduli Spaces (Project Head Farkas, Gavril )
• A11 - Algebraic Varieties and Principal Bundles: Semistable Objects and their Moduli Spaces (Project Heads Farkas, Gavril ;  Schmitt, Alexander )
• A12 - Topological Rigidity and Dynamics (Project Head Reich, Holger )
• B01 - Almost One-Dimensional Systems, Spectral Analysis, and Evolution Equations (Project Head Brüning, Jochen )
• B03 - Singularity Structure, Longtime Behaviour and Dynamics of Solutions of Nonlinear Evolution Equations (Project Heads Ecker, Klaus ;  Fiedler, Bernold ;  Schnürer, Oliver )
• B04 - Geometry and Physics of Spacelike Hypersurfaces in Lorentzian Manifolds (Project Heads Bär, Christian ;  Ecker, Klaus ;  Huisken, Gerhard )
• B05 - Evolution of Geometrical Structures in Classical and Quantum Cosmology (Project Heads Plefka, Jan ;  Rendall, Alan ;  Teschner, Jörg )
• B06 - Analytic and Spectral Properties of Geometric Operators (Project Heads Bär, Christian ;  Schüth, Dorothee )
• B07 - Dynamics of Cosmological Models: The Tumbling Universe (Project Heads Fiedler, Bernold ;  Liebscher, Stefan ;  Rendall, Alan )
• B08 - Integrable Structures in the Gauge/String Theory Correspondence (Project Heads Beisert, Niklas ;  Plefka, Jan ;  Staudacher, Matthias )
• C01 - Algebraic Topology: Rigidity and Dynamics (Project Head Reich, Holger )
• C02 - Differential Geometry: Geometric and Spectral Invariants of Riemannian, Lorentzian and Conformal Manifolds (Project Heads Baum, Helga ;  Schüth, Dorothee )
• C03 - Algebraic Geometry: Deformations, Moduli and Vector Bundles (Project Heads Altmann, Klaus ;  Farkas, Gavril ;  Schmitt, Alexander )
• C04 - Structure of Quantum Field Theory: Hopf Algebras versus Integrability (Project Heads Kreimer, Dirk ;  Staudacher, Matthias )
• C05 - AdS/CFT Correspondence: Integrable Structures and New Observables (Project Heads Forini, Valentina ;  Plefka, Jan ;  Staudacher, Matthias )
• C06 - Scattering Amplitudes: Symmetries and Interrelations in Maximal Supergravity and Yang-Mills Theory (Project Heads Nicolai, Hermann ;  Plefka, Jan )
• C07 - Differential Operators of Mathematical Physics: Spectral Theory and Dynamics (Project Heads Brüning, Jochen ;  Bär, Christian ;  Rendall, Alan )
• C08 - Mathematical Physics: Dynamics and Nonlinear Evolution Equations in General Relativity (Project Heads Ecker, Klaus ;  Fiedler, Bernold ;  Huisken, Gerhard ;  Rendall, Alan )
• C09 - Geometric Variational Problems: Direct Methods, Associated Flows and the Influence of Ambient Geometry and Physics (Project Heads Bär, Christian ;  Ecker, Klaus ;  Huisken, Gerhard ;  Metzger, Jan ;  Schulze, Felix )
• MGK - Integrated Research Training Group "Space - Time - Matter" (Project Head Brüning, Jochen )
• Z - Central Tasks of the Collaborative Research Centre (Project Heads Brüning, Jochen ;  Staudacher, Matthias )

• Applicant Institution: Humboldt-Universität zu Berlin
• Participating University: Freie Universität Berlin; Universität Potsdam
• Participating Institution: Max-Planck-Institut für Gravitationsphysik; (Albert-Einstein-Institut)
• Spokespersons: Professor Dr. Jochen Brüning, until 12/2014; Professor Dr. Matthias Staudacher, since 12/2014