Avi Zeff

I am a fourth-year PhD student at Columbia interested in number theory and arithmetic geometry, especially p-adic geometry and intersections on spaces of shtukas of various kinds. Many of my interests revolve around connections between number theory and various other fields, such as homotopy theory in relation to p-adic geometry, arithmetic applications of derived algebraic geometry, and the mysterious connections to physics appearing in the relative Langlands program. (Also, dead languages, music, and many other things.) My advisor is Chao Li.

Email: avizeff$@$math.columbia.edu.

Teaching

Spring 2024 I am a TA for Undergraduate Seminars II with Alisa Knizel, running a section on additive number theory.

Fall 2023 I taught Calculus 1 (section 10); you can find some of the materials here.

Spring 2023 I was a TA for Calculus 4 with Daniela De Silva.

Fall 2022 I taught Calculus 1 (section 10); you can find some of the materials here. (Note that some aspects of the course, both in form and content, vary from the Fall 2023 version.)

Spring 2022 I taught Calculus 2 (section 3); you can find some of the materials here.

Fall 2021 I was a TA for Calculus 3 with Inbar Klang.

Seminars

Fall 2022 I organized two seminars, one on prismatization and one on the p-adic Langlands program.

I organized a seminar in fall 2021 on the proof of the norm residue isomorphism theorem; the website can be found here.

Other than talks for the seminars linked above, I've given a number of talks in other learning seminars, notes for some of which can be found here.

Writing

Note: all of these are based on my limited understanding, and many are years old; all may (and some definitely do) contain at least some errors. Use at your own risk. (Also if you catch errors, or find one of the places where I have written something to the effect of ??? and moved on and have an explanation, please let me know!)

Expository writing:

• Introduction to "higher" math via number theory, for a general audience (i.e. essentially no background expected beyond at most some occasional calculus concepts). Note this is incomplete, and hasn't been updated in several years.
• Density of coprime values of polynomials, broadly also for a general audience
• (Some) perspectives on class field theory, for a mixed audience--most of the first section is targeted at a general audience, the second and third sections are targeted roughly at an undergraduate level, and the fourth section (on Clausen's K-theoretic Artin map) is at a graduate level

Notes I took for my own benefit, which may also be useful as expository, likely for a grad student audience:

• Fargues-Scholze's construction of local Langlands parameters
• Notes on Scholze's Emmy Noether lectures on the geometrization of real local Langlands
• Notes I wrote for a reading project with Chao Li on various conjectures and proofs related to special values of L-functions (all mistakes are my own):
  • Statement and outline of the proof of Gross-Zagier
  • The case N = 1: on singular moduli
  • A deformation-theoretic proof of a counting lemma
  • Notes on Howard-Yang's singular moduli
  • Counting supersingular curves with level structure via the Langlands-Kottwitz method, following Scholze
  • An illustration of the Siegel-Weil formula via some simple examples
• Some notes on Shimura varieties (global and local) and modular forms/L-functions I wrote to prepare for my qualifying exams, moved to a separate page since upon rereading them they are riddled with errors and generally messily done; use at your own risk.

Other writing:

• An article some friends and I wrote for Science for the People, also featured in Jacobin

Other resources

• An article by Denis Hirschfeldt on collective power in academia generally and mathematics specifically
• The Just Mathematics Collective and in particular their statement on mathematical collaboration with the NSA and the security state more broadly
• Library Genesis, a good source of textbooks if they cannot be found in other ways - knowledge should be free!
• Sci-Hub, similarly a good source of articles (ceterum Elsevier delendus est)
• researchseminars.org, a list and portal for many virtual seminars
• Wikipedia, perhaps the pinnacle of human accomplishment
• Burritos for the Hungry Mathematician, by Ed Morehouse

I have many opinions, but I am always soliciting more; if you have any good ones feel free to send them to me. (Responses not guaranteed.)