Research | Alicia Marino
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Research

Alicia Marino, PhD

My research is in number theory, with a particular interest in the study of quadratic forms. A classic problem in this area is to know when an integer can be written as sums of squares.

31 = 5^2 + 2^2 + 1^2 + 1^2

In this example, we say that 31 is the sum of four squares.

In 1770, Lagrange proved that in fact every nonnegative integer could be expressed as a sum of four squares. Similar to this example, there are many questions in the study of quadratic forms that are easy to state but in fact difficult to answer.

More specifically, I examine regularity conditions on quadratic forms in the context of a higher dimensional analog to the representation problem. My dissertation is a finiteness result for positive definite strictly k-regular integral quadratic forms.

Publications / Papers:

  • (With J. Liu) Strictly regular ternary Hermitian forms, J. Number Theory 168 (2016) 374-385.

  • (With W. K. Chan) A finiteness theorem for positive definite strictly n-regular quadratic forms, in preparation.

Undergraduate Research

I am currently working with an undergraduate student to explore cryptography while covering necessary elementary number theory background (we are working through Trappe and Washington’s Introduction to Cryptography with Coding Theory). In this course, we will cover the basics of modular arithmetic, the RSA Cryptosystem, the discrete logarithm problem, and more.

If you are an undergraduate interested in exploring number theory, cryptography, or anything related, please shoot me an e-mail. I’m sure we can find an interesting topic to dive into.