Research

In the United States, first-year university students enrolled in an entry-level calculus course for non-STEM majors often experience low engagement and motivation. These students potentially struggle with understanding calculus concepts due to a lack of fundamental mathematics concepts learned in high school. This paper explores implementing ChatGPT activities in one section of a University Summer Immersion course in New York and seven sections of a Calculus for Business course in the United Kingdom, both US degree programmes with non-STEM students learning calculus for the first time. We investigate how ChatGPT aids students in engaging with calculus applications and building their competence in calculus.

The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert modular forms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $\sigma$ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of $f||_k\sigma$

The aim of this article is to provide lower bounds for the p-adic valuation of local Whittaker newforms. We obtain these bounds by analysing the Fourier expansion of the local Whittaker newforms. 

This thesis studies modular forms from a classical and adelic viewpoint. We use this interplay to obtain results about the arithemtic of the Fourier coefficients of modular forms and their generalisations. 

In Chapter 2, we compute lower bounds for the $p$-adic valaution of local Whittaker newforms with non-trivial central character. We obtain these bounds by using the local Fourier analysis of these local Whittaker newforms and the $p$-adic properties of $\eps$-factors for $GL_1$.

In Chapter 3, we study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $\sigma$ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of $f||_k\sigma$. 

Chapters 2 and 3 are independent of each other and can be read in either order. In Chapter 4, we briefly discuss the relation between these two chapters and mention potential future work.