**From Zariski-Nagata to local fundamental groups**

Hilbert’s Nullstellensatz gives a dictionary between algebra and geometry; e.g., solution sets

to polynomial equations over the complex numbers (varieties) translate to (radical) ideals in

polynomial rings. A classical theorem of Zarski-Nagata gives a deeper layer to this

correspondence: polynomial functions that vanish to certain order along a variety

correspond to a natural algebraic notion called symbolic powers.

In this talk, we will explain this theorem, and then pursue a couple of variations on this

theme. First, we will consider how the failure of this theorem over ambient spaces with

bends and corners allows to study the geometry of such spaces; in particular, we will give

bounds on size of local fundamental groups. Second, we will consider what happens when

we replace the complex numbers by the integers; we will show that “arithmetic differential

geometry” (in the sense of Buium) allows us to obtain a Zariski-Nagata theorem in this

setting.

Only a passing familiarity with polynomials and complex numbers is assumed.

This is based on joint projects with Holger Brenner, Alessandro De Stefani, Eloísa Grifo, Luis

Núñez-Betancourt, and Ilya Smirnov.