expository notes

Expository notes and articles

Every once in a while I'll write a short expository note on something I think is interesting, either for a seminar or for myself.  I'm putting them up here in case someone else finds them useful.  Many of these notes are quite old (e.g. some were written when I was an undergrad) so buyer beware.

• 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 paper "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-Taniyama 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.

• 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 Mustaţă 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).

• 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.

Very Brief expository notes

Here are some very brief notes, usually answering a question someone asked me.  Many of these are quite old.

• A Serre Sylow problem, written after the question was posed to me by Jeremy Booher; the question deals with lifting Sylow subgroups of a quotient G/H to Sylow subgroups of G.

• Inversion is smooth. Often Lie Groups are defined as manifolds with a topological group structure, such that multiplication and inversion are smooth. This note shows that the smoothness of inversion follows from the rest of the definition.

• Dual Cantor-Schroeder-Bernstein for rings. Answers a question posed by Sam Lichtenstein; essentially, can a Noetherian ring be isomorphic to a proper quotient of itself?

• Notes on linear algebraic groups for quals. Gives techniques and examples for analyzing linear algebraic groups over finite fields, with an aim towards solving related problems on the old Stanford algebra quals.

• Closed subgroups of Lie groups are Lie subgroups. Shows that subgroups of Lie groups which are closed as topological spaces are closed Lie subgroups, i.e. they are closed submanifolds.

• Kernels of surjective bundle maps need not be bundles. Answers an offhand question of Ravi Vakil--is the kernel of a surjective map of (infinitely generated) locally free modules locally free? Though the answer is "yes" in the finitely generated case, it is "no" in general. I give a counterexample with *free* modules.

• Curves of genus at least two have finite automorphism groups. Shows that curves of genus at least two have finite automorphism groups, without using the representability of the Aut functor. This note is aimed at undergraduates who know a bit of algebraic geometry. Also shows that varieties with ample canonical bundle have finite automorphism group (with a mild restriction on the characteristic).