Dylan Johnson About Me Research Teaching Expository Writing CV

About Me

I am a mathematician at Duke University, where I am a first-year PhD student. I completed my undergraduate at the University of Utah.

When I'm not doing math, I enjoy rock climbing, cooking, reading, and playing video/board games.

Pronouns: he/him, they/them
Email:
Office: Phytotron 110

me

Research

My early research interests lean toward algebraic number theory.

Teaching

The best way to learn mathematics is to do mathematics. As such, I don't just present concepts to students; I also 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. For this reason, I try to 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.

Organization My Role Course Term Comments
Duke University Grader Math 404: Mathematical Cryptography Spring 2021
Duke University Grader Math 305S: Number Theory Seminar Fall 2020
University of Utah Teaching Assistant Math 1220: Calculus II Spring 2020 Course website
University of Utah Help Session Tutor Math 3210/3220: Foundations of Analysis Fall 2018 — Spring 2020

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. \(\ell\)-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.
  2. 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.

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.