Undergraduate Thesis Defense - Matías Martres - December 20th, 2024

Matías Martres will defend his Undergraduate’s thesis in Mathematics on Friday, December 20th at 11:00 AM.

The defense will be held in the Seminar Room on the 14th floor of CMAT, Facultad de Ciencias

Títle: Representations of integers by quadratic forms

Abstract: A classical problem in number theory is determining which integers are representable as a sum of two squares. We present a solution that allows us to find r_2(n)—the number of ways that an integer n can be represented as a sum of two squares—by understanding that these values are linked to the coefficients of a modular form. We then work towards the generalisation of this result, proving, among other things, a theorem asserting that the theta function associated with a positive definite quadratic form in 2n variables is a modular form. This result is the core of this dissertation. We subsequently present formulae for the number of representations of an integer n as a sum of k squares. We conclude by outlining a possible application of our work: the use of theta functions to study lattices.