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.
Status | Finished |
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Effective start/end date | 6/1/06 → 5/31/12 |
Funding
- National Science Foundation: $710,690.00