O-minimality and its applications

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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.