Topics in Model Theory

马克计划继续他在模型理论的研究，重点是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具有幂运算的复数中的可定义集合。他也仍然非常感兴趣的模型理论的微分领域-一个迷人的领域需要一个复杂的混合思想稳定理论，微分代数和代数几何。在不同的方向标记将继续他的工作之间的联系模型理论和描述集理论。特别是，他将试图了解更多关于理论的同构关系的可数模型是博雷尔。在模型论研究数学结构，通过研究解决方案集系统的方程和更复杂的集，你可以建立从这些基本的一套。在某些情况下，如真实的或复数的代数运算，所构造的集合在几何上是简单的-例如，只有1000个连接的部分。 在真实的数中，如果你加上指数函数，可定义的集合在几何上仍然是简单的，但在复数中，你可以构造像整数一样的无限离散集合。 在这种情况下，不知道所构造的集合是否可以任意复杂。 可能所有的集合都来自简单的片段。 Marker将研究这个问题。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.