Research

Bianchi Modular Forms: 

Modular forms are defined as complex functions on the hyperbolic plane that have symmetry with respect to groups like SL(2, Z), two-by-two determinant one matrices with integer coefficients. Brilliant mathematicians like Birch, Manin, Mazur, Merel, and Cremona (and many others) have worked hard to find effective ways to compute them. I work on extending these techniques to other types of modular forms, such as Bianchi Modular Forms over imaginary quadratic fields that have a "bigger" class number.  I implemented an algorithm to compute Bianchi Modular forms over imaginary quadratic fields with higher-class numbers. We are working on the paper right now. 

In the future, I plan to extend these computations to more general number fields, higher rank, and higher-weight modular forms.

Papers:

1. [published] Perfect Forms over Imaginary Quadratic Fields, Kristen Scheckelhoff, Kalani Thalagoda, and Dan Yasaki, Advanced Studies: Euro-Tbilisi Mathematical Journal, Special Issue (9 -2021), pp. 33-46, https://arxiv.org/abs/2105.00593 

2. [paper in preparation] Bianchi Modular form over imaginary quadratic fields with arbitrary class group, John Cremona, Kalani Thalagoda, and Dan Yasaki, 2025+

Harmonic Maass Forms: 

Harmonic Maass forms are a generalization of classical modular forms. Instead of requiring the form to be holomorphic, we impose a vanishing condition with respect to a differential operator (the Hyperbolic Laplacian). Another way to generalize classical modular forms is to replace the weight by half and an integer. 

Why would we be interested in this?  

Broadly, number theorists are interested in integer or rational sequences and their generating functions.  One such sequence is the Hurwitz class number (which is related to the ideal class number of imaginary quadratic fields). The generating function for this was discovered by Zagier in 1975 to be *drumroll* a harmonic maass(holomorphic part) form of weight 3/2 (tada!). Now we can use the knowledge of harmonic Maass forms to study the sequence H(n). 

Here is a plot of the values of H(n). Here is the OIES sequence related to H(n): https://oeis.org/A058305 

The values of H(n) vary irratically. So instead of asking for values of H(n), we can ask questions about the distribution of numbers, one such example is the sums of H(n).  Here is a plot of the sum of Hurwitz class number. This seems to have a trend that is much cleaner. 

So, my collaborators (Beckwith, Gupta, Rolen, Diamatis) and I set out to find a formula for the sum of H(n). We were able to get a summation formula for the weighted summation of the Fourier coefficients for a wider class of  Harmonic Maass forms, like the H(n) generating function.  This allowed us to obtain more statistics about how H(n) is distributed. 

If we could push our weight to zero, we would get an actual sum. For the future, we plan to extend our theorem to get the weight-zero case. 

You can read our paper here:

[submitted] Summation formulas for Hurwitz class numbers and other mock modular coefficients, Olivia Beckwith, Nikolaos Diamantis, Rajat Gupta, Larry Rolen, and Kalani Thalagoda, 2025 (pdf)