Department of Mathematics, Columbia University

I am a Ritt Assistant Professor at Columbia University. I do number theory and algebraic geometry, especially non-archimedean analytic geometry.

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.

Teaching

At Columbia:

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

At Stanford:

TA for Math 21 (Calculus III), Spring 2016 (section notes here)

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 notes here)

CA for Math 121 (Galois theory), Winter 2014

TA for Math 53 (ODEs), Fall 2013 (section notes here)

No teaching during 2012-2013 school year due to fellowships

Expository notes

For a while in grad school, I wrote fairly detailed typed notes for all talks I gave. Use at your own risk.

The sheaf property for adic spaces (2015). 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 Learning Seminar.

The uniformization of the stack of G-bundles (2014). Notes for a talk in the student "Lafforgue" seminar on function-field and geometric Langlands.

Motivation for and introduction to Jacquet-Langlands (2014). Notes for a brief motivating talk about the Langlands program and the text

Traces of Hecke operators (2014). 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 Learning Seminar.

Dynamical zeta functions (2014). 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.

Number theory on bubbles (2014). Slides for a talk on integral Apollonian circle packings given to the Stanford Undergraduate Mathematical Organization (SUMO).

Gross-Zagier on singular moduli: the analytic proof (2014). 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 student seminar.

Introduction to the circle method (2014). Waring's problem from the perspective of a neophyte. Talk given to the Stanford analytic number theory student seminar.

Overview of rigid analytic geometry (2014). Notes for a talk given to the Stanford algebraic geometry student seminar on rigid analytic geometry.

The five elementary operations according to Eichler (2014). 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.

Classical modular forms (2013). A crash course in the classical theory of modular forms, given as an informal talk.

The foundational crisis of mathematics (2013). An expanded version of a talk given to high school students as part of the SPLASH program.

Polynomial-geometric sums (2013). A note about polynomial-geometric sums.

Three approaches to Chow's theorem (2012). 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.

Ultraproducts and what to do with them (2012). Notes for a talk given to the Stanford graduate student seminar (KIDDIE) on ultraproducts and an application to Szemerédi's theorem.

Ultraproducts and the foundations of higher order Fourier analysis (2012). 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 measures.

Real analysis qualifying exam notes (2012). 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.

Notes on several complex variables: the theorems of Remmert-Stein and Chow (2012). This and the following were presentations given, in reverse order but not sequentially, for an undergraduate class on the theory of several complex variables. A short presentation offering a sketch of the Remmert-Stein theorem and Chow's theorem on projective analytic varieties.

Notes on several complex variables: Fréchet sheaves, Cartan A and B (2012). A development of the theory of Fréchet sheaves with application to a proof of Cartan's Theorems A and B on Stein spaces.

Notes on several complex variables: amalgamation of syzygies (2012). Comprising two separate lectures, these notes carry out a proof of Cartan's Theorems A and B for simply connected bounded polydomains, assuming only a few basic facts (notably, the sheaf version of Hilbert's syzygy theorem and the vanishing of sheaf cohomology for analytic sheaves on polydiscs with terminating chains of syzygies).

Notes on several complex variables: sheaves (2012). These notes go over some of the basic properties of sheaves from an étalé space perspective.

Notes on modular forms: Hecke operators, oldforms and newforms (2011). This and the following were presentations for an undergraduate seminar on modular forms. These notes define the Hecke operators and prove a few basic facts about them.

Notes on modular forms: dimension formulas (2011). Using Riemann-Roch to calculate the dimensions of the spaces of modular forms and cusp forms for important congruence subgroups in terms of the numbers of elliptic points and cusps.

Notes on modular forms: elliptic points and cusps (2011). A few basic facts about elliptic points and cusps and the construction of the compactification of the modular curve.

Siegel's theorem and its cousins (2010). Notes written for an undergraduate seminar on elliptic curves (two separate talks).