1.2 Workplan: detailed objectives and milestones

For the detailed description of the research it is appropriate to divide the project into 6 work packages:

I. Asymptotic geometric analysis. Solve the « slicing problem » concerning the volumes of sections of convex bodies. Solve the Duality of Entropy Problem as generally as possible. Investigate further the phase transitions and thresholds in Asymptotic Convexity. Understand the central limit properties of isotropic convex bodies. Determine the number of steps needed to achieve almost isometric symmetrization.

II. Isometric convex geometry. Compare random versus optimal approximation of bodies by polytopes. Analyse the asymptotic behaviour of random geometric configurations using methods from ergodic theory, Poisson approximation and the limit theory of (mixing) dependent random variables. Develop a unified approach to inequalities of Brunn-Minkowski type for variational functionals. Pursue stability estimates for classical inequalities

III. Asymptotic combinatorics. Study the asymptotic behaviour of random graphs and other combinatorial structures: in particular, spectrum-related parameters. Use pseudo-random graphs to capture deterministically the properties of random graphs (especially for use in algorithms). Improve estimates for Turan and Ramsey numbers.

IV. Randomized computation and complexity. Find improved volume algorithms and eliminate the sensitive dependence on the starting point of the random walk. Analyse the dependence on the step size and find transfer principles between walks with different step sizes. Find lower bounds on the complexity of volume computation. Find applications of randomized methods to the computation of geometric quantities other than the volume.

V. High-dimensional 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 Levy-Gromov comparison theorem. Apply mass transportation methods to the Hardy-Littlewood-Sobolev inequalities. Find probabilistic representation formulas in the context of the Brunn-Minkowski 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

 

Task I.1: Duality of entropy (Month: 0-12)

·         Partner 1,12

Milestones:

MI.1: Solution of the duality problem for some specified spaces

Addresses: Research Objective 1

 

Task I.2: Phase transitions and thresholds (Month: 0-36)

·         Partner 5,6,7

Milestones:

MI.2: New examples of phase transitions and thresholds in Asymptotic Convexity

Addresses: Research Objective 1

 

Task I.3: Symmetrization (Month: 0-24)

·         Partner 12

Milestones:

MI.3: Estimates of the number of Minkowski-Blaschke 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

 

Task II.1: Approximation of convex bodies by polytopes (Month: 0-12)

·         Partner 3

Milestones:

MII.1: Extension of results on random approximation currently known under smoothness assumptions to general convex bodies

Addresses: Research Objective 2

 

Task II.2: Inequalities of Brunn-Minkowski type for variational functionals (Month: 0-24)

·         Partner 8

Milestones:

MII.2: A proof of the Brunn-Minkowski inequality for Hessian capacities and for certain variational functionals

Addresses: Research Objective 2

 

Task II.3: Limit theorems for random configurations (Month: 0-48)

·         Partner 11

Milestones:

MII.3: Large deviation refinements of limit theorems for random configurations

Addresses: Research Objective 2

 

Task II.4: Stability versions of classical inequalities (Month: 0-36)

·         Partner 4

Milestones:          MII.4: A stability version of the Rogers-Shephard inequality

Addresses: Research Objective 2

 

Task II.5: Valuations (Month: 0-36)

·         Partner 9

Milestones:          MII.5: Characterization of affine invariant valuations

Addresses: Research Objective 2

 

 

Work Package III: Asymptotic Combinatorics

Leader: Partner 2, 4, 12, 13

 

Task III.1: Asymptotic behaviour of random graphs (Month: 0-24)

·         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

 

Task III.2: Pseudo-randomness (Month: 0-36)

·         Partner 4

Milestones:

MIII.2: Description of new properties ensuring pseudo-randomness, their comparison and equivalence

Addresses: Research Objective 3

 

Task III.3: Constructions of pseudo-random graphs (Month: 0-48)

·         Partner 12

Milestones:

MIII.3: Description of new explicit constructions of pseudo-random graphs

Addresses: Research Objective 3

 

 

 

 

 

Work Package IV: Randomized Computation and Complexity

Leader: Partner 2, 4, 11, 12

 

Task IV.1: Mixing times of random walks in convex bodies (Month: 0-36)

·         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

 

Task IV.2: Sampling from a convex body (Month: 0-12)

·         Partner 2

Milestones:

MIV.2: Improved algorithms for sampling from a convex body

Addresses: Research Objective 4

 

Task IV.3: Complexity of volume computation (Month: 0-24)

·         Partner 4

Milestones:

MIV.3: Lower bounds on the complexity of sampling, integration, or volume computation

Addresses: Research Objective 4

 

 

 

 

Work Package V: High-dimensional Phenomena in Mathematical Physics

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

 

Task V.1: Random matrices (Month: 0-24)

·         Partner 13

Milestones:

MV.1: Derivation of a functional equation for the limiting reproducing kernel of unitary-invariant matrix ensembles in the local regime

Addresses: Research Objective 5

 

Task V.2: Concentration estimates (Month: 0-36)

·         Partner 10

Milestones:

MV.2: Concentration type estimates and the transportation tightness for counting measures and other random configurations

Addresses: Research Objective 5

 

Task V.3:  Limiting behaviour of models of spin glasses and neural networks  (Month: 0-12)

·         Partner 1, 13

Milestones:

MV.3: Estimates of the free energy and correlators for the Sherrington-Kirkpatrick model and for the new “integrate and fire” model of neural networks

Addresses: Research Objective 5

 

Task V.4: Infinite dimensional dissipative dynamical systems (Month: 0-48)

·         Partner 13

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

Addresses: Research Objective 5

 

 

 

 

 

 

 

Work Package VI: Isoperimetric Principles in Geometry and Probability

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

 

Task VI.1: Mass transportation and Hardy-Littlewood-Sobolev inequalities (Month: 0-24)

·         Partner 1,10

Milestones:

MVI.1: Simplified and unified proofs of Hardy-Littlewood-Sobolev inequalities

Addresses: Research Objective 6

 

Task VI.2: Eigenvalue distribution of random matrices (Month: 0-36)

·         Partner 10

Milestones:

MVI.2: New functional methods for obtaining eigenvalue distributions of random matrices

Addresses: Research Objective 6

 

Task VI.3: Entropy growth of random variables (Month: 0-12)

·         Partner 2

Milestones:

MVI.3: Quantitative estimates for entropy growth of random variables satisfying moment conditions

Addresses: Research Objective 6