*What is number theory?*

- At its simplest, number theory is, not surprisingly, the study of numbers! In particular, elementary number theory primarily studies properties of integers (...,-3,-2,-1,0,1,2,3,...). Sample topics include unique factorization of any integer into a product of primes, methods to determine if a number is prime, and exploring why various divisibility "tricks" work. An example of a divisibility "trick" is: to determine if a large number is divisible by 3, you can add the digits and see if that (smaller) number is divisible by 3. Modern developments involve combining number theory with other branches of mathematics especially abstract algebra and complex analysis.

*My specific scholarly interests*

- Mathematical Cryptology and Communications Security

- Cryptology is the study of how information can be exchanged
securely. Many of the current cryptosystems are based on interesting
mathematics, especially number theory. I have written an open access
cryptology textbook which features many interactive SageMath cells for
activities and exploration, have developed numerous courses which study
the mathematics and history of cryptology, and worked with many
undergraduate students on research projects (primarily expository) in
this area.

- Cryptology by Discovery: An Introduction to Conjecture and Proof, Open Educational Resource

Textbook, online version, August 2020.

- Elliptic Curves:

- I have taught several undergraduate seminars focusing on elliptic curves and their applications as well as worked with many undergraduate students on research projects (primarily expository).

- Origami

- Origami is the ancient Japanese art of folding paper. It turns out
there is some very interesting mathematics involved in how to make certain
oriami designs. I have studied number theory connected to the
Fujimoto approximation technique and the geometry of folding twist boxes.

- Articles I have written that are related to this interest:
- "Do the Twist! (on polygon-base boxes)," with s-m belcastro, College Mathematics Journal, November

2016, Vol 47, No 5, pp 341-345. - "Fujimoto, Number Theory, and a New Folding Technique, Origami^4: Proceedings of the Fourth

International Meeting of Origami Science, Mathematics, and Education, AK Peters, Natick, MA, 2009. - "Constructing Regular n-gonal Twist Boxes,” with s-m belcastro, Origami^4: Proceedings of the Fourth

International Meeting of Origami Science, Mathematics, and Education, AK Peters, Natick, MA, 2009.

- Fibonacci Numbers

- Fibonacci numbers and generalized Fibonacci numbers have many interesting properties. I have worked with several students on research projects in this area. I have also written several papers to help connect some of the interesting properties of Fibonacci numbers to the middle and secondary school curriculum.

- Articles I have written that are related to this interest:
- "The Matrix Connection: Fibonacci and Inductive Proof", with C. Miller, Mathematics Teacher, December 2005, Vol 99, No. 5, pp 328-333.
- “Fibonacci: Beautiful Patterns, Beautiful Math,” with C. Miller, Mathematics Teaching in the Middle School, January 2002, pp 298-305.

- Mathematics Education

- I am also interested in issues in mathematics education, primarily at
the collegiate level. My projects in mathematics education include studying
how students learn, in particular in my classroom, and working with pre-
and in-service teachers to connect advanced mathematics (especially number
theory) to topics in middle and high school curricula.

- Articles I have written that are related to this interest:

- "Visions of Self in the Act of Teaching: Using Personal Metaphors in Collaborative Study of Teaching Practices," with M. Heston, L. Fitzgerald, K. East, and C. Miller, Teaching and Learning: The Journal of Natural Inquiry & Reflective Practice, Summer 2002, Vol. 16, No. 3, pp 81-93.
- College Algebra with Applications: Math for Biology, with C. Miller, The AMATYC Review, Spring 2003, Vol. 24, No. 2, pp 15-22.

- Modular Forms

- Modular forms are special kinds of functions important in number theory. Modular forms became almost famous recently when Andrew Wiles announced his proof of Fermat's Last Theorem. For my Ph.D. I researched Siegel modular forms and their relationship to L-functions.

- Articles I have written that are related to this interest:

- Siegel Modular Forms, L-functions, and Satake Parameters,” Journal of Number Theory 87, March 2001, pp. 15-20.