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.
Marker has continued his research in model theory, a branch of mathematical logic. In particular, he has concentrated on finding applications of model theoretic ideas to problems and structures arising in other areas of mathematics, such as algebra, algebraic geometry and analytic geometry. For example, there are many things we do not understand about the field of complex numbers with exponentiation, such as whether there are any automorphism other than the trivial one and complex conjugation. Zilber gave a model theoretic construction of an exponential field that is easy to understand from a model theoretic perspective and it is an open question whether this field is the complex exponential field. One direction of Marker's research is to take properties of Zilber's field to see if they hold in the complex numbers and take properties of the complex numbers to see if they hold in Zilber's field. The hope is that these investigations will give us game changing new insights into the classical structure of complex exponetiation and the geometry of solutions to systems of exponential equations. Marker has been very active as a mentor to graduate students and postdocs. During the period of the grant he was mentor to two postdocs. Two graduate students received their Ph.D. under his directiona and he is currently supervising six additional Ph.D. studnets.
