My mathematical interests include algebraic number theory, Iwasawa theory, elliptic curves, continued fractions, and modular forms. Additionally, I am interested in the mathematical content of physics, music, art, and social justice.

    My PhD thesis focused on generalized Riemann-Hurwitz formulas for lambda-invariants of number fields first proven by Kida; there are two main applications: (1) explicit computations of lambda invariants for imaginary quadratic extensions of certain abelian number fields and (2) a criterion for a special case of Greenberg’s conjecture on the vanishing of lambda invariants.

    I have mentored several research projects for graduates and undergraduates. First, my PhD advisor William McCallum and I supervised an RTG (research tutorial group) project consisting of three graduate students who worked together on understanding how Heron triangles having a fixed area and perimeter can be realized as rational points on an elliptic curve. At UCSB, I supervised a group of five undergraduates in a FRAP (Faculty Research Assistance Program) project where we explored visualizations of the connection between symmetries of the icosahedron and Ramanujan’s famous continued fractions. I also wrote a research article “Rational Hyperbolic Triangles and Elliptic Curves” with an undergraduate Nicolas Brody as part of his College of Creative Studies senior thesis at UC Santa Barbara. I am currently working with two other UCSB students on their senior theses.

    Research Articles:

  1. 7.Rational Hyperbolic Triangles and Elliptic Curves (with Nicolas Brody)

    in revision, pdf

  1. 6.Using Continued Fractions to Compute Iwasawa Invariants of Imaginary Quadratic Number Fields

    Journal of Numbers, vol. 2014 (2014), 10 pages, pdf

  1. 5.Wendy’s Xenharmonic Keyboard

    submitted to Mathematics Magazine (MAA), pdf

  1. 4.An Alternative Approach to Kida and Ferrero's Computations of Iwasawa λ-Invariants

    Journal of Number Theory, 138 (2014), 84-96, pdf

  1. 3.Generalizations of Iwasawa's 'Riemann-Hurwitz' Formula for Cyclic p-Extensions of Number Fields

    International Journal of Number Theory, 10 (2014), 219-233, pdf

  1. 2.The Change in Lambda Invariants for Cyclic p-Extensions of Zp-Fields

    PhD Thesis, University of Arizona, ProQuest Dissertation Publishing, May 2012, pdf

  1. 1.Lehmer's Totient Problem and Carmichael Numbers in a PID

    Undergraduate Senior Thesis, pdf

    Expository Papers:

  1. 1.The Kummer Congruences, Divergent Series, and Nonstandard Analysis, pdf

  2. 2.A p-adic Interpolative Property of Iwasawa Lambda Invariants, pdf

  3. 3.Hilbert Functions, pdf

  4. 4.The Lower Algebraic K-Groups, pdf

  5. 5.Sheaf Cohomology, pdf

  6. 6.Bernoulli Numbers, pdf

  7. 7.Quadratic Reciprocity, pdf

  8. 8.Periodic Continued Fractions, pdf

    Slides and Posters:

  1. 1.Class Numbers, Continued Fractions, and the Hilbert Modular Group, pdf

  2. 2.A Brief Biography of Paul Erdős, pdf

  3. 3.Ramanujan’s Continued Fractions and the Icosahedron, website

  4. 4.Theory of Classical Modular Forms and Symbols, pdf

  5. 5.An Introduction to Iwasawa Theory, pdf

  6. 6.The Role of Continued Fractions in Rediscovering a Xenharmonic Tuning, pdf and aux files

  7. 7.Zeta Zeros and Quantum Energy Levels, pdf

  8. 8.Number Theoretic Analogs of The Riemann-Hurwitz Formula, pdf

  9. 9.Cyclic p-Extensions of Zp-Fields, pdf

  10. 10. (Special) Riemann-Hurwitz Formulas in Iwasawa Theory, pdf

  11. 11. Slumdog Millionaire: Srinivasa Ramanujan, pdf

  12. 12. Elliptic Curves Over Q, pdf

  13. 13. The Brumer-Stark Conjecture, scanned notes

  14. 14. Dynamics Over Number Fields, pdf

  15. 15. Iwasawa Theory of Elliptic Curves and BSD in Rank Zero, pdf

  16. 16. A Riemann-Hurwitz Formula for Number Fields, pdf

  17. 17. Brauer's Theorems and the Meromorphicity of L-functions, pdf

  18. 18. Voting Systems, Mass Murder, and the Enigma Machine, pdf

  19. 19. Making Math Count in the Community: Measuring Income Disparity, ppt

  20. 20. Smooth Manifolds and Minkowski Spacetime, pdf

  21. 21. Deriving Meaning: Math at Work in RHS, pdf

  22. 22. Base 18, Quaternions, Markov Chains, and Absurdity, pdf

  23. 23. Hyperbolic Geometry, Complex Periods, Stereoscopy, and 4D, pdf

  24. 24. How to Mathematize the World: Black Holes, Oil Spills, the Spread of AIDS, ..., pdf

  25. 25. A Discrete Nash Demand Game with Diagonal Punishment, pdf