Topics in Model Theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Marker plans to continue his research in model theory, focusing on the connections with other areas of mathematics. One direction of his current research focuses on trying to understand definable sets in the complex numbers with exponentiation. He also remains very interested in the model theory of differential fields--a fascinating area requiring a sophisticated mixture of ideas from stability theory, differential algebra and algebraic geometry. In a different direction Marker will continue his work on connections between model theory and descriptive set theory. In particular, he will try to understand more about theories where the isomorphism relation on countable models is Borel. In model theory one studies mathematical structures by looking at solution sets to systems of equations and more complicated sets you can build from these basic sets. In some situations, like the real or complex numbers with algebraic operations, the sets constructed are geometrically simple-for example there are only finitely many connected pieces. In the real numbers if you add the exponential function, the definable sets are still geometrically simple, but in the complex numbers you can construct infinite discrete sets like the integers. In this case it is unknown if the sets constructed can be arbitrarily complicated. Possibly, all the sets arise from simple pieces. Marker will investigate this problem. The sets studied arise naturally in many applications in dynamical systems and control theory and it would be useful to understand their constraints.