Expository Notes and Articles

Warning: some of these come from the summers I spent as a counselor at PROMYS, and so use non-standard notation: taking a quotient of a ring by a principal ideal is denoted by a subscript. In particular, Z_p is the integers modulo p, not the p-adic integers. Furthermore, the group of units in Z_p is denoted by U_p.

• A Gentle Introduction to the Langlands Program: Modular Forms, Elliptic Curves, Galois Representations The slides for a sequence of three talks at the 2022 NZMRI Summer Workshop Number Theory and Related Topics. They are aimed at post-grad students.
• The Isogeny Invariance of the BSD Conjecture Over Number Fields Notes for a talk in the 2015-2016 number theory learning seminar about the Birch and Swinnerton-Dyer Conjecture. The talk discussed why the constant term in the BSD conjecture is isogeny invariant. The individual pieces are not isogeny invariant, but there is no overall change. The proof makes substantial use of Tate's results about global Galois cohomology.
• Viewing Modular Forms as Automorphic Representations Notes for talks in a student reading group about automorphic forms that carefully show how to associate automorphic forms and representations to a classical modular form. Particular attention is paid to describing the local components of the automorphic representation. The talks were given after we had read the first ten sections of Jacquet and Langland's Automorphic Forms on GL(2), which influenced which background material was emphasized.
• The Trace Formula for Compact Quotients Notes for talks in the 2013-2014 number theory learning seminar about Jacquet Langlands. The talks discussed the easiest case of the trace formula, that of the unit group of a division algebra. After exploiting compactness to give a relatively simple proof, we explain how in the case of a quaternion algebra this recovers the Eichler trace formula, something easily computable.
• The Weil Conjecture and Analogues in Complex Geometry (2013) Notes for a talk about the Weil conjectures for the Geometry and Topology Berkeley and Stanford Icebreaker Conference. They state the Weil conjectures, describe the proof for elliptic curves, and then describe a complex analogue and Serre's proof of this using Hodge theory. This proof inspired the development of étale cohomology and Grothendieck's standard conjectures.
• The Cusps of Hilbert Modular Surfaces and Class Numbers (2013) Notes for a student algebraic geometry seminar about the how Hirzebruch compactified Hilbert Modular Surfaces and how the structure of the cusps relates to L-functions. This leads to some cute formulas for class numbers of imaginary quadratic fields in terms of continued fractions.
• The Brauer Group: Talk 1, Talk 2 (2013) Notes for CRAG talks about the Brauer group.
• Reciprocity Laws (2013) Notes for a KIDDIE talk about quadratic reciprocity and generalizations. Links the proof using Gauss sums with the standard algebraic number theory proof using the Frobenius, and then discusses cubic reciprocity, Eisenstein reciprocity, and the connection of quadratic reciprocity with class field theory.
• The Mordell-Weil Theorem for Elliptic Curves (2012) A proof of the Mordell-Weil theorem and an example of complete 2-descent, inspired by Tim Dokchitser's proof given in the Part III Elliptic Curves Course.
• Square Roots in Finite Fields and Quadratic Nonresidues (2012) Notes for a counselor seminar at PROMYS about how to compute square roots in finite fields. In addition to two algorithms, it describes Rabin encryption as well as the connection between finding quadratic nonresidues and finding square roots. It ends with a discussion of bounds on the smallest quadratic nonresidue.
• Constructions with Fractional Ideals (2012) Notes for a talk in the 2011-2012 number theory learning seminar leading up to the proof of the Main Theorem of Complex Multiplication for Abelian varieties.
• Transcendental Numbers (2011) Notes for a KIDDIE talk explaining the techniques, going back to Hermite, used to show numbers like e are transcendental, and how a reasonable mathematician would discover them.
• Cubic Reciprocity (2011) Notes for a PROMYS talk explaining cubic reciprocity, which is the mathematics behind the picture on the 2011 PROMYS t-shirt. It gives the standard proof using Gauss and Jacobi sums assuming only the most elementary number theory.
• Representations of the Symmetric Group through Young Tableau (2011) Notes for a PROMYS counselor seminar about representation theory. It illustrates all of the general theory previously discussed in the case of the symmetric group, constructing all the irreducible representations of the symmetric group, showing how to combinatorially evaluate their characters and how to induce and restrict representations.
• Constructing the Integers: Ordinal Numbers and Transfinite Arithmetic (2011) Notes for a PROMYS talk explaining how to use set theory to construct a model for the integers, and at the same time introducing the more general arithmetic of ordinal numbers.
• The Limits of Computation (2011) Notes for a PROMYS talk explaining the limits of computation with deterministic finite automata and with Turing machines.
• Elementary Problems in Number Theory (2011) A collection of number theory problems I've given to students or heard from other counselors at the PROMYS program. They are of wildly varying difficulty.
• The Class Number One Problem for Imaginary Quadratic Fields (2011) Part III essay proving there are 9 imaginary quadratic fields with class number one. It gives two approaches, one following Heegner's original proof using modular functions, the second by finding rational points on a modular curve of level 24, and finally explaining a comment of Serre that these are essentially the same argument.
• Continued Fractions (2010) An explanation of the elementary theory of continued fractions, plus connections to Pell's equation and real quadratic fields along with some less common applications. This grew out of my notes for a review of continued fractions at PROMYS.
• The Isoperimetric Inequality (2010) Notes for a PROMYS talk proving the isoperimetric inequality, that of all closed curves of a given length, the one enclosing the largest area is the circle. There are two proofs given, the first uses the concept of action and is due to Hurwitz, the second uses Minkowski sums.
• Intersection Theory in Algebraic Geometry (2010) Notes for a PROMYS talk introducing basic notions in algebraic geometry and intersection theory. It presents the calculations on the Grassmanian of lines in three space to show there are two lines intersecting four general lines in projective three space, and ten lines in projective three space secant to each of two general twisted cubic curves.
• The Spirit of Moonshine: Connections Between the Mathieu Groups and Modular Forms (2010) Harvard undergraduate senior thesis, advised by Benedict Gross. An explanation of the connection between cycle shapes of elements of the Mathieu group M₂₄ and modular forms that are products of the Dedekind eta function and for which the coefficients of the q-expansion are multiplicative. It also constructs an infinite dimensional graded virtual representation that in some sense explains this connection, analogously with the moonshine module connected with the monster group.
• The Circle Method, j Function, and Partitions (2010) Final paper for Harvard's Math 229, analytic number theory. An explanation of Rademacher's application of the circle method to obtain explicit formula for the q-expansion of the Dedekind eta function and the j function.
• The Poincaré-Birkhoff-Witt Theorem (2009) Notes for a talk from Harvard's Math 222, Lie groups and Lie algebras. A standard and unenlightening proof of the Poincaré-Birkhoff-Witt theorem.
• Number Theory in Cryptography (2008) Notes for a PROMYS talk about cryptography, in particular Rabin Encryption, Paillier Encryption, secret sharing, and zero knowledge proofs.
• Computability (2008) Notes for a PROMYS talk about Turing machines, the halting problem, and the arithmetic hierarchy.