O-minimality and its applications

O-极小结构是真实的数域的一种驯服的扩展，在这个意义上，每一个在这种结构中可定义的集合（即，一个可以用这种结构的语言描述的集合）都有许多连通分支。有许多应用o-极小的各个领域的数学和最近这样的应用被发现算术和代数几何，产生新的证明经典问题（马宁-芒福德猜想）和解决一些开放的情况下的安德烈-奥尔特猜想。这个拟议的研究项目打算采取的一些成分出现在这些最近的应用算术和扩展到更一般的设置（局部对称空间），希望这样的结果将能够解决其他类似的开放的情况下，类似的布局。拟议研究的一个重要特征是无限离散对象（例如）之间的相互作用。算术组或周期函数），这是不能定义的O-最小结构，和他们的痕迹可定义的集在这样的结构。模型论是数理逻辑的一个分支，它研究的是用一阶公式表示的数学结构中的可定义集合。在许多情况下，它允许一个消除野生现象，并提供一个严格的定义驯服的几何。 这种方法对于解决数论和算术几何等领域的问题很有用。本计画提出利用各种结构的简洁性来解答算术几何中的一些公开问题，算术几何是数学中一个非常重要的领域。

O-minimal structures are tame expansions of the field of the real numbers in the sense that every set which is definable in such a structure (namely, a set which can be described using the languauge of the structure) has finitely many connected components. There are many applications of o-minimality to various areas of mathematics and recently such applications were found to arithmetic and algebraic geometry, yielding new proofs to classical problems (the Manin-Mumford conjecture) and and solving some open cases of the Andre-Oort conjecture. This proposed research project intends to take some of the ingredients which appeared in these recent applications to arithmetic and extend them to a more general setting (locally symmetric spaces), with the hope that such results will enable to solve other open cases of similar conjectures. An important feature of the proposed research is the inter-play between infinite discrete objects (eg. arithmetics groups or periodic functions), which are not definable in o-minimal structures, and their trace on definable sets in such structures. The goal is to prove definability of classical periodic maps when restricted to appropriate definable sets, and then to explore the implications which this definability might have on the periodic sets and functions.Model theory, a branch of mathematical logic, studies sets definable in mathematical structures by first-order formulas. In many cases it allows one to eliminate wild phenomenas and to provide a rigorous definition of a tame geometry. This approach is useful for solving problems in areas such as number theory and arithmetic geometry. This project proposes to use tamness of various structures to answer some open problems in arithmetic geometry which is a very important area of mathematics.