Logic, Group theory, Combinatorics and Ergodic theory

Project Details

Description

Groups of finite Morley rank arise in model theory as a substantial

generalization of the class of algebraic groups. It has been conjectured

that the simple groups in this category are all algebraic, and Cherlin is

working on this conjecture with a team of international collaborators.

This involves techniques used in the classification of the finite simple

groups as well as some ideas from black box group theory. In graph theory,

using a mix of model theoretic and combinatorial techniques, Cherlin and

Shelah are developing techniques to determine, for a given finite set of

forbidden graphs, whether there is a universal graph meeting the

constraints. The ultimate question here is whether the entire problem is

algorithmically decidable. Thomas works on the theory of countable Borel

equivalence relations, combining the methods of descriptive set theory with

techniques related to superrigidity. The methods of descriptive set theory

cast considerable light on classical classification problems, and

conversely powerful methods coming from group theory illuminate and advance

the general theory. Cherlin and Thomas also host a dynamic visitor program

at Rutgers, coordinated with annual visits by Shelah.

Infinite group theory provides a tool for studying and exploiting the

symmetries of a mathematical model or a physical system. Cherlin and his

collaborators are aiming at the classification of the groups associated

with well-behaved algebraic systems, while Simon Thomas approaches the

study of infinite groups from the point of view of their actions and the

analysis of one action in terms of another. A particularly strong role is

played here by ideas coming from the theory of dynamical systems. Graphs

are the mathematical abstraction of networks in general, and the problems

under consideration relate to the analysis, preferably by a general

(computable) algorithm, of classes of graphs characterized by forbidding a

fixed set of patterns.

StatusFinished
Effective start/end date6/1/065/31/12

Funding

  • National Science Foundation: $710,690.00

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