Teaching

The best way to learn mathematics is to do mathematics. So I don't just present concepts to students; I work to build each student's confidence in their personal problem-solving ability. That self-confidence inspires them to continue working on difficult exercises as well as independently seek out math which interests them. Similarly, students are more successful when they have role models with whom they can identify. So in my classes I highlight the accomplishments of diverse mathematicians, use language which makes no assumptions about the look or level of a mathematician, and think critically about how my identity informs my mathematical life. Courses whose instruction I have participated in are tabulated below.

I aspire that my teaching is consistent with Federico Ardila's axioms in Todos Cuentan: Cultivating Diversity in Combinatorics, which together constitute a "pressing call to action" for math educators:

1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

4. Every student deserves to be treated with dignity and respect.

Expository Writing

Below are expository notes I wrote to organize my understanding at the time. I collect them here so others may find them useful.

1. The Circle Method for Algebraic Number Fields
   pdf abstract

   In this short expository note, we motivate and describe an application of circle method over an algebraic number field K. In particular, we describe the Hasse principle – a fundamental local-global principle in number theory – and highlight through example how recent work of Schindler and Skorobogatov (SS14) employs the circle method to study the Hasse principle. Finally, we briefly introduce some ideas from algebraic geometry, and discuss the way in which SS14 characterizes an obstruction to the Hasse principle for a large family of varieties. We make an effort to phrase everything as accessibly as possibly; a reader with either background in the circle method or background in algebraic number fields – and possibly neither – will hopefully get something out of this note.

2. Motivation for the Étale Fundamental Group
   pdf abstract

   We motivate the need for a different notion of the fundamental group in the realm of algebraic geometry, one which is compatible with the Zariski topology. The precise construction of the algebro-geometric fundamental group -- the "étale fundamental group" -- is beyond the scope of this short note, so once motivated we precisely describe the way in which Galois theory manifests as a special case of the more general étale theory. Finally, we introduce sheaves, an object essential to defining the étale fundamental group and found throughout algebraic geometry and beyond.

3. \(\ell\)-adic Galois Representations
   pdf abstract

   In this paper, we introduce objects which are central to modern number theory: \(\ell\)-adic Galois representations. These special representations correspond with both highly-symmetric meromorphic functions on the upper half of the complex plane — so-called "automorphic forms" — and with the rational zero-sets of polynomials — so-called "algebraic varieties" over \(\mathbb{Q}\). In section 1, we motivate our study by describing how this correspondence proved Fermat's Last Theorem, one of the great triumphs of twentieth century mathematics. In section 2, we lay down a foundation of essential number theory. In section 3, we precisely define \(\ell\)-adic Galois representations and study some examples. And finally in section 4, we return to the big picture and discuss the vast frontier of current research in this area.

4. Modular Forms, Elliptic Curves, and their Connection to Fermat's Last Theorem
   (Undergraduate Thesis)
   pdf abstract

   Fermat's Last Theorem (FLT) states that for an integer \(n > 2\), the equation \(x^n + y^n = z^n\) has no integer solutions with \(xyz \neq 0\). This incredible statement eluded proof for over three-hundred years: in that time, mathematicians developed numerous tools which finally proved FLT in 1995. In this paper, we introduce some of the essential objects which enter the proof — especially modular forms, elliptic curves, and Galois representations — with an emphasis on precisely stating the Shimura-Taniyama Conjecture and explaining how its proof finally settled FLT. We offer proofs whenever they clarify a definition or elucidate an idea, but generally prefer examples and exposition which make concrete a truly beautiful body of mathematical theory.

As an undergraduate, I wrote two other short articles about graph theory which I share here as resources for others: Constructing Graceful Graphs by Extending Paths from Graceful Graphs and Graph Theory and Matrices.

Lastly, for a completely general audience (no math background needed!), I've also written three articles on the mysterious "perfect numbers": A strange definition of perfect, Perfectly even, and Perfectly odd.