题目: Valuation rings and mathematical logic
报告人： William Johnson博士后（复旦大学）
摘要： Model theory is the study of algebraic structures, such as groups and rings, through the lens of first-order logic. For any structure $M$ we can ask two related questions: Which first-order sentences are true in $M$? What subsets of $M^n$ are (internally) definable? The techniques of model theory answer these questions for many of the structures arising in algebra, such as the field of real numbers and the ring of algebraic integers. In the process, several notions of model-theoretic ``tameness'' emerge. For example, the real field and several related structures possess the ``o-minimal'' property, which ensures that definable sets have well-defined dimensions, compact definable sets can be triangulated, and definable functions are piecewise continuous. These tameness properties have been essential to the applications of model theory in algebraic geometry and additive combinatorics. The ``NIP'' property is a more general notion of tameness--the class of NIP structures includes the o-minimal structures as well as many of the structures arising naturally in algebra such as the field of p-adic numbers and the ring of formal power series. In recent years, a conjectural classification of NIP fields has emerged. Loosely, speaking, we expect almost all NIP fields to be henselian valued fields. In my talk, I will give an overview of model theory, discuss the NIP field conjectures, and state my partial results in the ``finite-dimensional'' NIP setting.