# Mathematical Sciences Research Institute

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# Seminar

DDC Junior Seminar: Monodromy groups in algebraic geometry September 15, 2020 (09:00 AM PDT - 10:00 AM PDT)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) Borys Kadets (University of Georgia)
Description

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

The seminar will feature research talks by the six postdoctoral scholars appointed to the Fall 2020 DDC program, along with talks by students and other pre-tenure researchers associated with this program.  Since seminar attendees will have disparate backgrounds, we plan that these talks will not be too advanced, nor will they assume substantial background knowledge.  Our postdocs include number theorists, model theorists, and computable structure theorists, and talks can be expected to span all of these areas.

Video

#### Monodromy Groups In Algebraic Geometry

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

To a finite unramified covering of nice topological spaces $f: X \to Y$ one can associate a finite permutation group $G$ -- the monodromy group of the covering.

This familiar topological construction has an avatar in algebraic geometry: to a finite unramified ("étale") morphism $\pi: X \to Y$ one associates a finite permutation group $\mathrm{Mon} \ \pi$.

When $X$ and $Y$ are varieties over the complex numbers $\mathrm{Mon}\ \pi$ is exactly the monodromy group of the topological covering $X(\mathbb{C}) \to Y(\mathbb{C})$; however, the algebraic definition makes sense for arbitrary ground fields, and, in fact, subsumes Galois theory: the Galois group of a field extension $L/K$ is the mondromy of the cover $\mathrm{Spec} L \to \mathrm{Spec} K$.

I will describe how these monodromy groups are defined and calculated in algebraic geometry through concrete examples, and show some applications to geometry of curves and Galois theory.