Groups of Finite Morley Rank

One of the great achievements of twentieth century mathematics is the classification of simple algebraic groups. These groups appear throughout mathematics and physics, as well as in chemistry and computer science. Groups often arise as symmetries of another object both inside and outside mathematics. A few specific cases include number theory, cryptography, algebraic geometry, algebraic topology, quantum mechanics,spectroscopy, image recognition, and Galois groups of differential equations.

Simple groups may be viewed as the building blocks of all groups, much as prime numbers are the building blocks of all integers. As the classification says that all simple algebraic groups arise from groups of matrices, questions about arbitrary groups, and often the objects they act upon, may be reduced to questions about linear algebra.

Groups of finite Morley rank lie on the border between model theory and algebraic groups. Morley rank is a notion of dimension arising naturally in model theory, which is a branch of mathematical logic. Morley rank behaves much like a dimension function for constructible sets over the complex numbers, as opposed to a dimension over the real numbers. Indeed, Hilbert’s Nullstellensatz shows that the Morley rank of an algebraic variety is equal to its Krull dimension.

The longstanding algebraicity conjecture in groups of finite Morley rank, due independently to Gregory Cherlin and Boris Zilber, is that all simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields, i.e. arise from matrix groups.

The role of the Cherlin-Zilber Conjecture in the model theory can be made more precise by observing two facts. First, the simple groups of finite Morley rank are exactly the simple groups whose uncountably models are determined up to isomorphism by their first-order theory. Secondly, any sufficiently complex uncountably categorical theory involves a group of finite Morley rank.

The Cherlin-Zilber Conjecture is proved in the important case of of groups of even type: they happen to be algebraic groups over algebraically closed fields of characteristic 2. but remains open in the general case, despite a remarkable progress of the last decade. The current line of attack is shifting towards strange relations between the so-called "black box groups" of computational group theory, pseudofinite groups and groups of finite Morley rank.

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