SCHOLARLY INTERESTS
My preferred flavor of mathematics is number
theory,
but I have also recently become interested in machine learning and data
science. Current number theory interests include cryptology and
communications security, and elliptic curves. I am especially
interested in computational projects involving SageMath and Python.
Past research interests include mathematics of origami (specifically
connections between number theory and geometry), modular forms, and
mathematics education.
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.
- 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 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 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.
- 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 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.