For the detailed description of the research it is appropriate to divide the project into 6 work packages:
V. Highdimensional phenomena in mathematical physics. Develop approaches to the analysis of eigenvalues of random matrices and zeros of random analytic functions. Describe in explicit form the limiting behavior of basic models of spin glasses and neural networks (with emphasis on phase transitions). Develop a general approach to the study of smoothness properties of global attractors for infinite dimensional dissipative dynamical systems.
VI. Isoperimetric principles in geometry and probability. Develop a functional or mass transportation approach to the finite dimensional LevyGromov comparison theorem. Apply mass transportation methods to the HardyLittlewoodSobolev inequalities. Find probabilistic representation formulas in the context of the BrunnMinkowski and Ehrhard inequalities. Examine the use of functional inequalities in random matrix theory. Extend recent results on entropy growth to random variables satisfying classical probabilistic hypotheses.
The work packages are detailed in the tables below together with milestones that can be used to assess the progress of the project.
Work Package I: Asymptotic Geometric Analysis 

Leader: Partner 1, 2, 3, 5, 6, 7, 8, 10, 11, 12 

· Partner 1,12 

Milestones: 
MI.1: Solution of the duality problem for some specified spaces 
Addresses: Research Objective 1 



· Partner 5,6,7 

Milestones: 
MI.2: New examples of phase transitions and thresholds in Asymptotic Convexity 
Addresses: Research Objective 1 



· Partner 12 

Milestones: 
MI.3: Estimates of the number of MinkowskiBlaschke symmetrizations needed to approximate the Euclidean ball with a given precision 
Addresses: Research Objective 1 
Work Package II: Isometric Convex Geometry 

Leader: Partner 1, 2, 3, 4, 5, 7, 8, 9, 11, 12 

· Partner 3 

Milestones: 
MII.1: Extension of results on random approximation currently known under smoothness assumptions to general convex bodies 
Addresses: Research Objective 2 



· Partner 8 

Milestones: 
MII.2: A proof of the BrunnMinkowski inequality for Hessian capacities and for certain variational functionals 
Addresses: Research Objective 2 



· Partner 11 

Milestones: 
MII.3: Large deviation refinements of limit theorems for random configurations 
Addresses: Research Objective 2 



Milestones: MII.4: A stability version of the RogersShephard inequality 

Addresses: Research Objective 2 



Work Package III: Asymptotic Combinatorics 

Leader: Partner 2, 4, 12, 13 

· Partner 12 

Milestones: 
MIII.1: More precise determination of the asymptotic behaviour of graph eigenvalues and related parameters in random graphs 
Addresses: Research Objective 3 



· Partner 4 

Milestones: 
MIII.2: Description of new properties ensuring pseudorandomness, their comparison and equivalence 
Addresses: Research Objective 3 



· Partner 12 

Milestones: 
MIII.3: Description of new explicit constructions of pseudorandom graphs 
Addresses: Research Objective 3 
Work Package IV: Randomized Computation and Complexity 

Leader: Partner 2, 4, 11, 12 

· Partner 4 

Milestones: 
MIV.1: Rigorous connection between the mixing time of a continuous random walk (Brownian motion) and a discrete step random walk in a convex body 
Addresses: Research Objective 4 



· Partner 2 

Milestones: 
MIV.2: Improved algorithms for sampling from a convex body 
Addresses: Research Objective 4 



· Partner 4 

Milestones: 
MIV.3: Lower bounds on the complexity of sampling, integration, or volume computation 
Addresses: Research Objective 4 
Work Package V: Highdimensional Phenomena in Mathematical Physics 

Leader: Partner 1, 2, 4, 6, 8, 10, 12, 13 

· Partner 13 

Milestones: 
MV.1: Derivation of a functional equation for the limiting reproducing kernel of unitaryinvariant matrix ensembles in the local regime 
Addresses: Research Objective 5 



· Partner 10 

Milestones: 
MV.2: Concentration type estimates and the transportation tightness for counting measures and other random configurations 
Addresses: Research Objective 5 



· Partner 1, 13 

Milestones: 
MV.3: Estimates of the free energy and correlators for the SherringtonKirkpatrick model and for the new “integrate and fire” model of neural networks 
Addresses: Research Objective 5 



Milestones: MV.4: Regularity properties of the global attractors for plate and wave equations with nonlinear viscous and/or thermal dampling 

Work Package VI: Isoperimetric Principles in Geometry and Probability 

Leader: Partner 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 

· Partner 1,10 

Milestones: 
MVI.1: Simplified and unified proofs of HardyLittlewoodSobolev inequalities 
Addresses: Research Objective 6 



· Partner 10 

Milestones: 
MVI.2: New functional methods for obtaining eigenvalue distributions of random matrices 
Addresses: Research Objective 6 



· Partner 2 

Milestones: 
MVI.3: Quantitative estimates for entropy growth of random variables satisfying moment conditions 
Addresses: Research Objective 6 