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分支，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，表示论和真实的解析几何。在过去的20年里，我们看到了一些非常令人兴奋的发展，在这一领域，包括皮拉的应用o-极小计数点的分析品种，Hrushovski的证明Manin-Mumford猜想的特征p和工作的Cluckers，Hales和Loeser在motivic积分上，他们用来转移Ngo在特征p中对朗兰兹纲领的基本引理的证明，特性0。所有上述工作要么使用理论的价值领域或有类比的理论价值领域。我自己的工作是继续发展价值域的模型理论的基础。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，以解决更多的问题，在其他领域的数学，其中涉及解析函数的一个有价值的领域。这需要深入了解这些结构的模型理论性质。在未来五年里，我将通过解决两大类项目来实现这一计划。第一类是研究具有限制解析函数的值域中的等价关系（即等价于可定义等价关系的等价关系）。基于我与合作者的早期工作，我们现在相当好地理解了代数值场论中的概念。继续合作，以解决更丰富的结构的有价值的领域与限制的分析功能。第二类项目是研究具有赋值和凸序的域。通过类比无序的情况下，我计划发展理论的剩余领域控制的代数和分析结构。这将对建立有序伯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. **