Here you can find the websites of the most important places in Berlin where you can apply: 

Research area 1 (differential geometry, mathematical physics):

IMPRS "Geometric Analysis, Gravitation and String Theory" 

DFG Collaborative Research Center "Space - Time - Matter" (SFB 647)

Max Planck Institute for Gravitational Physics (Albert Einstein Institute, AEI)

Research area 2 (algebraic and arithmetic geometry, number theory):

RTG "Moduli and Automorphic Forms: Arithmetic and Geometric Aspects" 

Research area 3 (stochastics, statistics and finance) :

RTG "Stochastic Analysis with Applications in Biology, Finance and Physics" 

Geosim

Max Planck Institute for Molecular Genetics (MPIMG)

Weierstraß Institute for Applied Analysis and Stochastics (WIAS) 

Research area 4 (discrete mathematics and optimization):

RTG "Methods for Discrete Structures" 

Zuse Institute Berlin (ZIB)

Research area 5 (geometry, topology and visualization):

DFG Collaborative Research Center "Transregio" (SFB/TRR 109)

Research area 6 (numerical mathematics and scientific computing):

IMPRS "Computational Biology and Scientific Computing" 

DFG Collaborative Research Center "Scaling Cascades in Complex Systems" (SFB 1114)

Geosim

Nanopoly

Max Planck Institute for Molecular Genetics (MPIMG)

Zuse Institute Berlin (ZIB)

Research area 7 (applied analysis and differential equations):

DFG Collaborative Research Center "Transregio" (SFB/TRR 109)

DFG Collaborative Research Center "Control of self-organizing nonlinear systems" (SFB 910)

Weierstraß Institute for Applied Analysis and Stochastics (WIAS) 

General mathematical institutions in Berlin:

ECMath

Research Center MATHEON "Mathematics for Key Technologies"