Model theory of expansions of valued fields

我的研究领域是模型论代数。这是数理逻辑的一个子领域，其中模型论的技术被应用于数学的其他部分，包括数论、代数几何、表示理论和实解析几何。在过去的二十年里，这一领域出现了一些非常令人兴奋的发展，包括Pila将o极小性应用于解析变量的点数，Hrushovski在特征p中对Manin-Mumford猜想的证明，以及Cluckers， Hales和Loeser在动机积分方面的工作，他们利用这些工作将Ngo对特征p中Langlands程序的基本引理的证明转移到特征0中的陈述。以上所有的工作要么使用了值场理论，要么在值场理论中有类比。我自己的工作是继续发展价值领域模型理论的基础。******我的研究计划的总体目标是增加模型理论的可能应用范围，以解决涉及值场上解析函数的其他数学领域的更多问题。这需要对这些结构的模型理论特性有深刻的理解。在接下来的五年里，我将通过处理两大类项目来推进这一计划。第一类是研究具有限制解析函数的值域中的虚（即可定义等价关系的商）。基于我和合作者的早期工作，我们现在对代数值场论中的虚量有了相当好的理解。为了用有限的分析函数处理更丰富的值域结构，合作还在继续。第二类项目是研究既有估值又有凸排序的领域。通过与非有序情况的类比，我计划发展代数结构和解析结构的剩余域控制理论。这将对创建有序伯科维奇空间的理论产生影响。**

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. **