Model theory of expansions of valued fields

我的研究领域是模型理论代数。这是数理逻辑的一个子领域，其中模型论技术应用于数学的其他部分，包括数论、代数几何、表示论和实解析几何。过去二十年来，这一领域取得了一些非常令人兴奋的进展，包括 Pila 将 o 极小性应用于分析簇计数、Hrushovski 对特征 p 中的 Manin-Mumford 猜想的证明以及 Cluckers、Hales 和 Loeser 在动机整合方面的工作，他们使用该工作将 Ngo 对特征 p 中朗兰兹纲领的基本引理的证明转移到特征 0 中的陈述。上述工作要么使用价值域理论，要么在价值域理论中进行类比。我自己的工作是继续发展有价值域的模型理论的基础。 *****我的研究计划的总体目标是扩大模型论的可能应用范围，以解决数学其他领域中涉及有价值域上的解析函数的更多问题。这需要深入了解这些结构的模型理论特性。在接下来的五年中，我将通过解决分为两大类的项目来实现这一计划。第一类是对有限分析函数的有价值域中的虚数（即可定义的等价关系的商）的研究。基于我与合作者的早期工作，我们现在相当了解代数值域论中的虚数。合作仍在继续，以解决具有有限分析功能的有价值领域的更丰富的结构。第二类项目是对具有估值和凸排序的领域的研究。通过与无序情况类比，我计划发展代数结构和解析结构的留数场支配理论。这将对创建有序贝尔科维奇空间理论产生影响。 **

My area of research is model-theoretic algebra. This is a subfield of mathematical logic, in which techniques of model theory are applied to other parts of mathematics including number theory, algebraic geometry, representation theory and real analytic geometry. The last twenty years have seen some very exciting developments in this area, including Pila's applications of o-minimality to counting points on analytic varieties, Hrushovski's proof of the Manin-Mumford conjecture in characteristic p and the work of Cluckers, Hales and Loeser on motivic integration which they used to transfer Ngo's proof of the Fundamental Lemma of the Langlands program in characteristic p to also have the statement in characteristic 0. All of the above work either uses the theory of valued fields or has analogies in the theory of valued fields. My own work is in continuing to develop the foundations of the model theory of valued fields.******The overall goal of my research program is to increase the range of possible applications of model theory to address more problems in other areas of mathematics which involve analytic functions on a valued field. This requires a deep understanding of the model-theoretic properties of these structures. In the next five years, I will pursue this program by addressing projects which fall into two broad categories. The first category is the study of imaginaries (that is, quotients by definable equivalence relations) in valued fields with restricted analytic functions. Based on earlier work of myself with collaborators, we now understand fairly well the imaginaries in the algebraic valued field theory. The collaboration continues in order to tackle the much richer structure of the valued field with restricted analytic functions. The second category of projects is the study of fields with both a valuation and a convex ordering. By analogy with the non-ordered case, I plan to develop the theory of residue field domination for both the algebraic and the analytic structures. This will have consequences for creating a theory of ordered Berkovich space. **