Research Statement Model Theory and Algebraic Groups

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. So, in practice, questions about algebraic groups, and spaces they act upon, often reduce to questions about simple groups which are well understood thanks to the classification machinery. 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 C, as opposed to a dimension over the real numbers R. 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. This conjecture occurred as a special case of Zilber’s thematic trichotomy principle, which posits an essential connection between pure model theory and algebraic structures. The role of the Cherlin-Zilber conjecture within the Zilber theme can be made more precise by observing two facts. First, the simple groups of finite Morley rank are exactly the uncountably categorical simple groups, meaning simple groups whose uncountable 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 [33, 2.25]. Furthermore, Zilber’s theme and the experience of the classification program for finite simple groups suggests that any sporadic counterexamples to the conjecture should have a natural role outside of logic. Advances towards the Cherlin-Zilber Algebraicity Conjecture have generally involved the transfer of powerful ideas from the classification of finite simple groups, an approach known as the Borovik program. The imported methods are often those of “local analysis” where local examination of the group reveals global structure. We also exploit geometric machinery such as Tits’ theory of buildings or the Curtis-Tits-Phan theorem when groups are finally “identified” as being algebraic. The success of Borovik’s program makes the Cherlin-Zilber algebraicity conjecture the most approachable problem within the Zilber theme which does not require ambient geometric hypotheses. Alexandre Borovik and Simon Thomas have separate work showing that the classification of finite simple groups is “stronger” than the classification of simple algebraic groups.