O-minimality, Point Counting and Functional Transcendence

Jacob Tsimerman, University of Toronto and Minerva Distinguished Visitor
Fine Hall 314

In-Person and Online Talk 

Register at: https://math.princeton.edu/minerva-2021

One of the main applications of O-minimality to Number Theory has been via the celebrated Pila-Wilkie theorem. This theorem says, in a precise sense, that transcendental sets  which are definable in an o-minimal structure contain `few' rational points. Quite unexpectedly, this result has played a huge role in the last 10 years in establishing transcendence theorems for a huge swath of functions, including e^x, modular forms, and even period integrals.  We will explain this theorem and the idea underlying its proof, and describe some of the functional transcendence applications, including the classical Ax-Schanuel theorem and its various incarnations.