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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.