This is the first in a series of 5 posts whose goal is to briefly introduce rigid geometry with a focus on providing a big picture between the interactions between different perspectives of rigid geometry.
Objects with no rational models
The goal of this post, which is just a bit of fun between more serious posts/projects, is to discuss some examples of algebro-geometric objects over which have no models over smaller subfields and explain how moduli theory can help clarify their discovery in certain situations.
Punctured disks and punctured curves and cohomology, oh my! (Part III: the affine curve case)
Punctured disks and punctured curves and cohomology, oh my! (Part II: cohomology with supports)
Punctured disks and punctured curves and cohomology, oh my! (Part I: the proper curve case)
This is the first in a series of posts whose goal is to compute the cohomology of the (several times) punctured closed -adic disk as well as the cohomology of smooth (and some non-smooth) algebraic curves.
Weird example: finite maps don’t preserve projectivity
In this post we discuss a weird example of a finite map of varieties which doesn’t preserve projectivity.
Classifying one dimensional groups (II)
In this post we classify one-dimensional connected group varieties of dimension .
A smattering of representation theory
Attached below are notes that I wrote for a seminar at Berkeley.
The goal of the notes was to understand some of the representation theory surrounding Scholze’s paper on the cohomology of the Lubin–Tate tower. In particular I, Koji Shimizu, DongGyu Lim, and Sander Mack-Crane were/are interested in understanding whether there is a function field analogue of this paper.
In particular, it has an eye towards modular (i.e. mod ) representations of -adic groups. So, it discusses some of the classical theory of representations of -adic groups from a categorical perspective which serves one better in the modular representation setting. It also discusses the fascinating theorem of Kazhdan relating Hecke algebras for and where and are soemthing like the tilts of and .
A ‘brief’ discussion about torsors
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.
Please feel free to leave any constructive comments!
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!