I want a list of the sort of mathematics/mathematical tools that are applied to the study of complex and probabilistic systems in order to make quantitative and qualitative observations about their behaviors.

These mathematical tools need not be exact (in fact, I expect most will draw from statistics), but I want a list of methods that are rigorous and useful beyond a specific problem domain.

For instance, what are the mathematical tools behind:

-Statistical physics (Renormalization group, for instance)

-Large-scale differential equation solvers used in modeling

-Image processing techniques in computer vision (for instance)

-Social Network Analysis

-Compression technologies

-Visualizations of large-scale data

In other words, what are the most interesting and rigorous mathematics applied to large-scale information processing?

A list of techniques and tools:

Sampling techniques in Applied Probability and Stochastic Modeling

Principle component analysis based on eigenvalues of matrix representations of data (plus loads of other stuff related to matrix representations, like singular value decomposition, rank minimization, etc.)

Belief propagation in neural networks

The PageRank metric on networks

Random Matrix Theory (for statistical physics)

The finite element method (much more of engineering flavor, but mathematically based and enhanced in its efficiency by all sort of deep mathematics, e.g. from differential geometry)

hugespan that you ambitiously ask about? $\endgroup$ – Yemon Choi Apr 9 '10 at 6:27