### Étale fundamental groups, section equivalence and "ét equivalence"

Linfoot Number Theory Seminar

15th June 2022, 11:00 am – 12:00 pm

Fry Building, Room 2.04

Lots of number theory problems can be rephrased in ways that look like algebraic topology. It turns out you can define a "fundamental group" of a variety over k that captures both the "fundamental group" (in terms of standard topology) that you'd expect, as well as some Galois theory. This fits into a natural short exact sequence, and sections of this short exact sequence are incredibly useful objects of study. As well as this, torsors are extremely natural geometric objects that pop up all the time in arithmetic geometry. In this talk, we'll define the étale fundamental group, define torsors, and then show that two points give us the same section if and only if they "look the same to all torsors". At various times, I'll also hint at how these ideas can be made into arguments about "higher homotopy" (though, both for the interest of time and the fact that this is still work in preparation, I won't be very exact about this). No knowledge about higher homotopy, torsors, or the word "étale" will be necessary.

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