Stanford University Mathematics Department
Building 380, Rm. 381-M
Giving a talk on a boat
● My CV
● Zeta Functions of Curves with No Rational Points: We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.
● Symmetric Powers Do Not Stabilize: We discuss the stabilization of symmetric products Sym^n(X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h^0(X, \omega_X), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain "motivic zeta function" associated to a curve, and its Hodge polygon.
Talk Notes and Slides
● Motivic Analytic Number Theory: There are beautiful and unexpected connections between algebraic topology, number theory, and algebraic geometry, arising from the study of the configuration space of (not necessarily distinct) points on a variety. In particular there is a relationship between the Dold-Thom theorem, the analytic class number formula, and the "motivic stabilization of symmetric powers" conjecture of Ravi Vakil and Melanie Wood. I'll discuss several ideas and open conjectures surrounding these connections, and describe the proof of one of these conjectures--a Hodge-theoretic obstruction to the stabilization of symmetric powers--in the case of curves and algebraic surfaces. Everything in the talk is defined from scratch, and should be quite accessible. A video of a version of this talk may be found here.
● Icebreaker Talk: Sam Nolen has taken notes on a talk for somewhat general audiences I gave at the Geometry and Topology Berkeley and Stanford Icebreaker Conference; the linked document also contains notes on a talk by Jeremy Booher.
● Duality and Generalized Scanning: Sam Nolen has taken (very informal) notes on a (very informal) talk I gave at the XKCD seminar at Stanford, about duality and generalized scanning. The linked document also contains notes on a very interesting talk by Arnav Tripathy, about an etale-topological Dold-Thom theorem.
● Line Bundles on Plane Curves, September 27, 2012, Berkeley RTG Workshop: We give two descriptions of the geometry of the Picard stack of the universal plane curve of degree d, as well as the limiting geometry of cohomological loci on the Picard stack, e.g. special divisors.
● More Cotangent Complex Stuff, March 20, 2015: Notes on a talk I gave on the cotangent complex for the Stanford Number Theory Learning Seminar on perectoid spaces. I give several example applications: lifting local complete intersection curves, “uniformizing” Abelian varieties of good reduction (ongoing work), and constructing the Witt vectors in a ludicrous way.
● The Cotangent Complex, May 18, 2012, Michigan Derived Algebraic Geometry RTG Learning Workshop: Rough notes on a talk I gave constructing the cotangent complex for maps of simplicial rings (following Quillen).
● Notes, Introduction to the Landscape of Generalized Euler Characteristics, March 6, 2012, UC Berkeley Commutative Algebra and Algebraic Geometry Seminar: "This talk will be a gentle introduction to the Grothendieck ring of varieties (the "baby ring of motives") and its role in the search for computable invariants of varieties. This ring is important, mysterious, and in some ways pathological. I will describe various questions about the Grothendieck ring, motivated by arithmetic, topology, and geometry--in particular, an analogue of part of the Weil conjectures--as well as some of its pathologies. I'll answer some of these questions for curves and discuss how these answers manifest themes from arithmetic and topology. Time permitting, I'll discuss obstructions to the truth of related statements for surfaces. This talk should be a good introduction to the material in Ravi Vakil's talk during the following hour."
● The Music of the Spheres, November 2013, Stanford SUMO/Splash: A talk for undergraduates about the mathematics underlying various wrong theories of cosmology from the past two millenia. Based to some extent on Arthur Koestler's excellent book "The Sleepwalkers."
● Tiling Problems, April 13, 2011, Stanford SUMO: A talk for undergraduates about tiling problems.
Summaries of Ongoing Work
● Non-Abelian Lefschetz Hyperplane Theorems: This is a summary of my thesis work, comparing maps out of a projective variety to maps out of an ample divisor, where the target is a scheme or stack.
● Homotopical Enhancements of Cycle Class Maps: This is a discussion of work in progress on enhancing cycle class maps to intermediate Jacobians; the end goal is an analytic description of fibers of these maps.
● p-adic uniformization of curves and Abelian varities of good reduction: This is a discussion of work in progress on uniformizing p-adic curves and Abelian varities of good reduction. To appear soon.
● Student Algebraic Geometry Schedule, 2012
● Classics Reading in Algebraic Geometry, 2012-2013
● Hodge Theory Learning Seminar, 2011-2012. There was never a website for this seminar, unfortunately. That said, here is my introductory talk from the seminar. And here are the first and second talks on the cycle class map.
Expository Notes and Articles
● A Short Proof of the Tian-Todorov Theorem: I give a very short (4 pages!) proof of the Tian-Todorov theorem, that is, that Calabi-Yau varieties are unobstructed in characteristic zero. I think this proof is well-known, but I don’t know a great reference for it (if you do, please let me know). I also give two proofs of the T^1 lifting theorem.
● Local-Global Compatibility, Notes for number theory learning seminar, April 2014: Discusses relationships between the Langlands program for GL2, and in particular local-global compatibility, with the reduction types of modular curves.
● Etale Cohomology of Curves, Notes for etale cohomology learning seminar, October 2013: Computes the etale cohomology of smooth projective curves over an algebraically closed field--the notes are essentially self-contained, if slightly ridiculous in how they develop the theory of the Brauer group. Beware: there are a couple minor errors, which will hopefully be fixed soon.
● Picard Groups of Moduli Problems I and Picard Groups of Moduli Problems II, Notes for CRAG talks, February 2013: Describes stacks in the abstract, as well as quotient stacks more concretely, while largely avoiding the annoyance of fibered categories. Gives quotient stacks and the moduli stack of elliptic curves as examples, and computes their Picard groups, as in Mumford's paper "Picard Groups of Moduli Problems."
● Geometrizing Cohomology, Notes for XKCD talk: A very experimental and not very contentful talk about various ways one might think about higher-degree cohomology classes. Inspired to some extent by Pawel Gajer's paper Geometry of Deligne Cohomology.
● Yet Another Proof of the Fundamental Theorem of Algebra, A proof of the Fundamental Theorem of Algebra I came up with in 2010, which is to my knowledge original. (Please let me know if you've seen it before!) Written to be understandable by undergraduates.
● Variation of Hodge Structure, Notes for Number Theory Learning Seminar, January 18-February 1, 2013: Discusses Hodge Theory, Variation of Hodge Structure, and related topics; subsumes most of what's in my senior thesis.
● The Derived Category of Coherent Sheaves on P^n, Notes for SAGS Talk, October 10, 2012: Discusses Beilinson's explicit description of the derived category of coherent sheaves on projective space (essentially, the BGG correspondence). This is essentially an exposition of Beilinson's demurely-titled 1 page paere "Coherent Sheaves on P^n and Problems of Linear Algebra," though I spend a lot of time fleshing out details. The notes are hand-written.
● The Shimura-Taniyma Formula, Notes for Number Theory Learning Seminar Talk, May 10, 2012: An exposition of p-divisible groups and the Shimura-Taniyma Formula. Essentially: how does the Frobenius action on a CM Abelian variety over a local field depend on the CM type?
● Stable Homotopy Groups of Spheres via Ad Hoc Methods, Notes for student topology seminar, April 14, 2012: An computation of the first few stable homotopy groups of spheres via ad hoc methods--mostly using the Barratt-Priddy-Quillen theorem. Also gives a cute construction of stable homotopy classes using homology spheres, which can be used to give a generator of \pi_3^s.
● Rational Equivalence of 0-cycles, Notes for SAGS, April 11, 2012: An exposition of Mumford's description of the group of Chow 0-cycles on surfaces admitting a non-zero holomorphic 2-form.
● Zeroes of Integer Linear Recurrences, Very informal notes for a grad student seminar, February 8, 2012: Gives an informal exposition of the Mahler-Skolem-Lech theorem.
● Fulton's Trace Formula, Notes for SAGS Talk, January 23, 2012: An exposition of Fulton's trace formula in coherent cohomology, which counts the number of rational points on a projective variety over a finite field, mod p. The exposition follows Mustata and Fulton, but fills in some details.
● Automorphic Forms, Notes for automorphic forms learning seminar, January 16, 2012: An extremely quick and dirty introduction to the basic definitions in the theory of automorphic forms for GL(2).
● The Poincaré Lemma and de Rham Cohomology, The Harvard College Math Review, Vol 1. No. 2, Fall 2007: An expository account of differential forms and the Poincaré Lemma using modern methods, aimed at beginning undergraduates. Contains some minor errors and omissions (in the exterior power section), which I am attempting to get fixed in the online version.
● Introduction to Hodge-Type Structures, Harvard Undergraduate Senior Thesis, May 2010: An expository account of some Hodge Theory, concluding with a sketchy description of modern approaches (e.g. mixed Hodge modules, etc.)
● Prime Reciprocals and Primes in Arithmetic Progression, Harvard Junior Paper, May 2009: Gives some estimates on sums of prime reciprocals in certain residue classes; some of the arguments (e.g. the proofs of Propositions 5 and 6) are, to my knowledge, novel.
● Linear Independence over $Q$ and Topology, Note for MathOverflow, November 2010: Answers a MathOverflow question relating the topology of certain Riemann surfaces to the linear independence of certain numbers.
● Line Bundles on Projective Space, written for Dennis Gaitsgory's 2009-2010 Theory of Schemes course at Harvard; gives two proofs that the group of line bundles on projective space over a field is generated by the canonical bundle, and is isomorphic to the additive group of integers. That is, the only line bundles are those we know and love.
● The Hilbert Scheme of Points on a Surface, written for Dennis Gaitsgory's 2009-2010 Theory of Schemes course at Harvard; gives a very hands-off proof that the Hilbert scheme of points on a nice curve or surface is smooth and irreducible.
● A Categorical Construction of Ultrafilters, a short and extremely elementary paper written with Zachary Abel and Scott Kominers, answering a question of E. Rosinger in the negative. Published in the Rocky Mountain Journal of Mathematics.
● Some brief notes: A space for very short expository notes on varied subjects
● Photos of Magnets, some constructions made from Zen Magnets
● "Draft Obituary, Winston Wallace 1922-2008, by Joanna Lansom," a short story I wrote a while back
● Ohio, a poem about Ohio
● The Moth at Knossos, another old poem I wrote as an exercise in mythology
● On "Old Futurism," a (mostly) silly exercise in writing a manifesto