How model theory looks at Lie groups and Lie Algebra

  Model theory typically looks at classical mathematical structures in novel ways. The guiding principle is to understand what relations are definable, and there are usually related questions of effectivity. In the case of Lie theory, there are two current lines of research, both of which I will describe, but with more emphasis on the first. The most advanced work concerns exponentials and logarithms, in both real and complex situations. To understand the definable relations, and to show various natural problems are decidable, one uses a mixture of analytic geometry with number-theoretic conjectures related to Schanuel's Conjecture. More recent work, not yet closely connected to the preceding, concerns the limit behaviour (model-theoretically), of finite -dimensional modules over semisimple Lie algebras, and here again, for decidability, one seems obliged to consider number-theoretic decision problems, around Siegel's Theorem.