Dylan Pentland

Dylan Pentland

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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

Pages

Posts

Future Blog Post

less than 1 minute read

Published:

This post will show up by default. To disable scheduling of future posts, edit config.yml and set future: false.

Blog Post number 4

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 3

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 2

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 1

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

portfolio

publications

talks

Honda-Tate theory

Published:

I talked about classical Honda-Tate theory, as well as recent developments that allow us to describe the full category of abelian varieties over a finite field rather than just the isogeny category. Notes are here.

Shtukas

Published:

I talked about how cohomology of shtukas can be assembled into a sheaf on a certain stack of local systems. It is now known that this sheaf and the cohomologies can also be produced via categorical traces (in the unramified setting), which amounts to what is called the trace conjecture. Following Gaitsgory’s argument, I sketch how the trace conjecture is true in a certain toy model. Notes are here

The twistor line

Published:

I gave a talk on how the twistor line can be used to reformulate some statements in Hodge theory, presenting it in a form parallel to the role of the Fargues-Fontaine curve in p-adic Hodge theory. Notes are here.

The Beilinson Fiber Square

Published:

I gave a talk on the Beilinson fiber square, following the approach of Antieau-Mathew-Morrow-Nikolaus. I also talked on the applications of this result they give towards the p-adic variational Hodge conjecture, refining a result of Bloch-Esnault-Kerz. The notes can be found here.

The Almost Purity Theorem

Published:

I talked about the basics of adic spaces and gave a quick introduction to perfectoid spaces, closely following the approach in Scholze’s thesis. I then explain some of the main ideas in Scholze’s proof of Faltings’ almost purity theorem. Notes can be found here.

The arc-topology

Published:

I discussed the definition of the arc-topology on qcqs schemes, giving useful criteria for checking arc descent. I then explained arc and arc_p-descent of etale cohomology (which is what is used in the proof of the etale comparison theorem for prismatic cohomology). Notes can be found here.

The pro-etale topology

Published:

I defined the etale and pro-etale sites of rigid analytic spaces, and explained how when the base field contains the p-adic numbers that affinoid perfectoid objects form a basis for the pro-etale topology. I then went through some propositions about condensed pro-etale cohomology needed in this paper. Notes can be found here.

teaching

PROMYS

Summer program, Boston University, 2022

In Summer 2022, I was a head counselor at the PROMYS program at Boston University.

DRP

Directed reading, Harvard, 2023

In Spring 2023, I mentored a DRP project in etale cohomology, building up to computing the l-adic cohomology of a curve. In Fall 2023, I am mentoring a DRP project on p-adic Hodge theory.

Summer Tutorial: Groups and Trees

Undergraduate course, Harvard, 2023

In Summer 2023, I taught a course on Bass-Serre theory. The majority of the course followed Serre’s book “Trees”, with the end going into the construction of Ramanujan graphs from the Bruhat-Tits tree of SL2.