Author: alexyoucis

Attached below are notes written for two mentees I had in an independent study concerning the etale fundamental group. The goal was to motivate cohomology (in particular etale cohomology) via torsors and motivate torsors using the theory of twists. I think that the notes are well-intentioned and do genuniely have interesting didcatic value buried deep inside them. Unfortunately, they are long-winded, meandering, and overly self-indulgent. One day I intend to go back and tighten them up.

torsors

Exercises in etale cohomology

I have had the pleasure of helping to run a seminar on etale cohomology and, in the process, have been writing up questions for the participants to work on. In case it would be useful to any readers of my blog, I thought I’d include them here.

I will be continuing to edit this post with the most recent version of the exercises.

Please feel free to point out any errors and/or suggest any good problems!

Nov. 7th version of exercises

The Langlands conjecture and the cohomology of Shimura varieties

Below are some really extended notes that I’ve written about work I’ve done recently alone (in my thesis) and with a collaborator (A. Bertoloni Meli).

While the explanation of my work was the original goal of the notes, they have since evolved into a motivation for the Langlands program in terms of the cohomology of Shimura varieties, as well as explaining some directions that the relationships between Shimura varieties and Langlands has taken in the last few decades (including my own work).

I hope that it’s useful to any reader out there. Part I was mostly written with me, four years ago, in mind. So, in a perfect world someone out there will be in the same headspace as I was, in which case it will (hopefully) be enlightening.

In case you’re wondering the intended level for the reader is probably: 1-3 year graduate student with interest in number theory and/or arithmetic geometry. In particular, for Part I there is an assumption that the reader has some basic knowledge about: Lie groups, algebraic geometry, number theory (e.g. be comfortable with what a Galois representation is), algebraic group theory, and etale cohomology (although this can be black-boxed in the standard way–e.g. all one needs to know is the contents of Section 3 of this set of notes). Part II is mostly written as an introduction to a research topic, and so requires more background.

Enjoy!

PS, feel encouraged to point out any mistakes/improvements that you think are worth mentioning.

The Langlands conjecture and the cohomology of Shimura varieties

A fun (enough) talk

This is a rough transcription of a talk I gave to a class of algebraic number theory students at UC Berkeley with the goal of trying to understand how one might bring to bear modern techniques in number theory/geometry on some classical questions. I have essentially kept the format the same, while adding a bit of extra material (and adding in their responses to questions I asked).
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The Fontaine-Winterberger theorem: going full tilt

This is the first in a series of posts whose goal is quite ambitious. Namely, we will attempt to give an intuitive explanation of why the recent push of several prominent mathematicians (Fargues, Scholze, etc.) to ‘geometrize’ the ‘arithmetic’ local Langlands program is intuitively feasible (at least, why it seems intuitive to me!) and, more to the point, to understand some of the major objects/ideas necessary to discuss it.

The goal of this post, in particular, is to try and understand why perfectoid fields (of which perfectoid spaces, their more corporeal counterparts) are natural objects to consider. This is far from a historical account of perfectoid fields and tilting, of which I am far from knowledgable. Instead, this is more in the style of Chow’s excellent You Could Have Invented Spectral Sequences explaining how one might have arrived at the definition of perfectoid fields by ‘elementary considerations’.

This post is somewhat out of order. In some magical world where I actually planned out my posts, this would have been situated less anteriorly but, as we’re constantly reminded, we do not live in a perfect world!

Some motivation for p-adic Hodge theory

These are some notes that I wrote for a learning course at Berkeley–the goal being to understand the statement of the global Langlands conjecture.

The goal of the talk (that these notes were written for) was, specifically, to motivate $p$-adic Hodge theory with an eye, in particular, towards where it might be useful in understanding the statement of Langlands.

These are even less edited than usual, so I profusely apologize for any mistakes. As always, corrections/comments are very welcome!

Notes

Some notes on automorphic representations

I was asked to give a series of talks in an `automorphic representations’ learning seminar at Berkeley. Specifically, I was asked to talk about the local representation theory of $\text{GL}_2(\mathbb{Q}_p)$ and aspects of the global representation theory of $\text{GL}_2(\mathbb{A}_\mathbb{Q})$.

These are ongoing but I thought I’d post my notes for the first two talks. The first is on the representation theory of $\text{GL}_2(\mathbb{Q}_p)$ and the second is on the representation theory of $\text{GL}_2(\mathbb{A}_\mathbb{Q}^\infty)$ with a focus on the relationship to modular forms.

These notes are a bit on the ‘basic’ side, leaving most technical details/proofs to the more comprehensive texts. This could be useful for some people who just want to get an overview of the ideas involved.

Anyways, feedback welcome as always!

Represenations of GL_2(Q_p)

Representations of GL_2(A_Q^\infty)