SCHOLARLY INTERESTS

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.

My specific scholarly interests

• Cryptography
  Cryptography is the study of designing systems to send secret messages. Many of the current cryptosystems are based on interesting mathematics, especially number theory. I have developed courses which study the mathematics of cryptography and worked with several students on research projects in this area.

• 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 been studying some number theory connected to the Fujimoto approximation technique.

  Articles I have written that are related to this interest:

1. "Can Origami Compute The Order of Elements Mod n?", in submission.
2. "A Number Theoretic Application to the Fujimoto Approximation Technique", Origami^4: Proceedings of the Fourth International Meeting of Origami Science, Mathematics, and Education, AK Peters, Natick, MA, to appear.
3. "Generalizing Twist Boxes, with s-m belcastro", Origami^4: Proceedings of the Fourth International Meeting of Origami Science, Mathematics, and Education, AK Peters, Natick, MA, to appear.

• 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:
 
1. "The Matrix Connection: Fibonacci and Inductive Proof", with C. Miller, Mathematics Teacher, December 2005, Vol 99, No. 5, pp 328-333.
2. “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:

  1. "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.
  2. 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:

  1. This work appears in the article, “Siegel Modular Forms, L-functions, and Satake Parameters,” Journal of Number Theory 87, March 2001, pp. 15-20.