Topics in Model Theory

马克计划继续他在模型理论方面的研究，专注于与其他数学领域的联系。他目前的研究方向之一是试图理解复数中的可定义集。他还对微分场的模型理论非常感兴趣——这是一个迷人的领域，需要将稳定性理论、微分代数和代数几何的思想复杂地融合在一起。在另一个方向上，马克将继续研究模型论和描述集论之间的联系。特别是，他将尝试更多地理解可数模型上的同构关系是Borel的理论。在模型理论中，人们通过研究方程组的解集来研究数学结构，而更复杂的解集可以从这些基本集中构建出来。在某些情况下，比如具有代数运算的实数或复数，构造的集合在几何上是简单的——例如，只有有限多个连通的块。在实数中，如果你加上指数函数，可定义集在几何上仍然是简单的，但在复数中，你可以构造无限的离散集，比如整数。在这种情况下，不知道构造的集合是否可以任意复杂。可能，所有的集合都是由简单的片段产生的。马克将调查这个问题。所研究的集合在动力系统和控制理论的许多应用中自然出现，理解它们的约束将是有用的。

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.