Department of Mathematics, Columbia University
am a Ritt Assistant Professor at Columbia University.
I do number theory and algebraic geometry, especially non-archimedean
I can be reached at warner [at] math [dot] columbia [dot] edu or in office
512 in the math building.
My PhD advisor was Brian
Conrad at Stanford University. My thesis (on representability of
moduli in non-archimedean analytic geometry) is available on request, as it
is still undergoing minor changes.
Honors Math B, Spring 2021
Calculus III, Fall 2020
Honors Math A, Fall 2020
Analytic Number Theory, Spring 2020
Honors Math B, Spring 2020
Honors Math A, Fall 2019
Honors Math B, Spring 2019
Algebraic Number Theory, Fall 2018
Honors Math A, Fall 2018
Algebraic Topology, Spring 2018
Honors Math B, Spring 2018
Honors Math A, Fall 2017
TA for Math 21 (Calculus III), Spring 2016 (section
CA for Math 152 (Elementary number theory), Winter 2016
CA for Math 19 (Calculus I), Summer 2015
CA for Math 205C (Lie theory), Spring 2015
TA for Math 51 (Linear algebra), Winter 2015 (section
CA for Math 121 (Galois theory), Winter 2014
TA for Math 53 (ODEs), Fall 2013 (section
No teaching during 2012-2013 school year due to fellowships
For a while in grad school, I wrote fairly detailed typed notes for all
talks I gave. Use at your own risk.
property for adic spaces. Notes for a talk on the sheaf property for
the adic spectra corresponding to certain classes of Tate rings, mostly
following a preprint of Buzzard and Verberkmoes but in slightly more
generality (and containing a short discussion of the vanishing of higher
cohomology of the structure sheaf). Given at the Stanford Number Theory
uniformization of the stack of G-bundles. Notes for a talk in the
student "Lafforgue" seminar on function-field and geometric Langlands.
for and introduction to Jacquet-Langlands. Notes for a brief
motivating talk about the Langlands program and the text Automorphic
Forms on GL(2) of Jacquet and Langlands.
of Hecke operators. Notes for two talks on the derivation of the
formula for the traces of Hecke operators on cusp forms from the trace
formula for GL(2), joint with Z. Brady at the Stanford Number Theory
zeta functions. Notes for a talk in the Stanford student arithmetic
dynamics learning seminar about dynamical zeta functions, featuring a
theorem of A. Bridy on the transcendence of certain of
positive-characteristic dynamical zeta functions.
theory on bubbles. Slides for a talk on integral Apollonian circle
packings given to the Stanford Undergraduate Mathematical Organization
singular moduli: the analytic proof. An overview of the statement and
analytic proof of the Gross-Zagier formula at level one, following their
Crelle's Journal paper. Talk given to the Stanford analytic number theory
uniformization of curves. Why and how to uniformize the Tate curve and
Mumford curves in a rigid analytic setting, given as a talk to the Stanford
algebraic geometry student seminar.
the circle method. Waring's problem from the perspective of a
neophyte. Talk given to the Stanford analytic number theory student seminar.
of rigid analytic geometry. Notes for a talk given to the Stanford
algebraic geometry student seminar on rigid analytic geometry.
elementary operations according to Eichler. Notes for a talk given to
the Stanford graduate student seminar (KIDDIE) on modular forms and what to
do with them. Here
are the mentioned (computer-generated) calculations of relations between the
Eisenstein series for the full modular group, and here
are the corresponding sums-of-divisors equalities.
modular forms. A crash course in the classical theory of modular
forms, given as an informal talk.
crisis of mathematics. An expanded version of a talk given to high
school students as part of the SPLASH program.
sums. A note about polynomial-geometric sums.
approaches to Chow's theorem. Notes for a talk given to the Stanford
algebraic geometry student seminar on Chow's theorem on projective analytic
varieties. Discusses three very different proofs, due to Chow, Remmert and
Stein, and Serre.
what to do with them. Notes for a talk given to the Stanford graduate
student seminar (KIDDIE) on ultraproducts and an application to Szemerédi's
Ultraproducts and the foundations of higher order Fourier analysis.
Princeton undergraduate senior thesis, advised by Nicolas Templier. An
exposition of results of Balázs Szegedy in the field of higher order Fourier
analysis, starting from basic ultraproduct analysis and the theory of Loeb
analysis qualifying exam notes. Some possibly helpful notes for the
Stanford analysis qualifying exam, mostly distilled from Royden's Real
Analysis, third edition. No guarantees either to completeness or to
comprehensibility; they were written with review in mind and would be an
exceptionally poor introduction.