Image by David Lowry-Duda (link)

Research

My research interests are in computational & analytic number theory. In general, I am interested in problems involving modular forms and their generalizations. I also enjoy studying and developing algorithms for computing modular forms (I love Programming!).

I did my thesis under the supervision of Dan Yasaki (UNCG) on developing techniques for computing a type of modular form called Bianchi modular forms (modular forms over imaginary quadratic fields) with special properties (non-trivial class groups).

For my postdoc, with my advisor, Olivia Beckwith (Tulane), I am working on understanding Harmonic Maass Forms. Primarily, I am interested in applying the theory of Harmonic Maass Forms to compute and understand the distribution of sequences, such as Hurwitz class numbers.

Bianchi Modular Forms

Modular forms are defined as complex functions on the hyperbolic plane that exhibit symmetry with respect to groups like SL(2, Z), which are two-by-two matrices with integer coefficients and a determinant of one. Brilliant mathematicians such as 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 currently working on the paper.

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

Papers:

[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
[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).

The left 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 irregularly. So, instead of asking for the values of H(n), we can ask questions about the distribution of numbers; one such example is the sum of H(n). The right plot shows the variation of the sum of the Hurwitz class numbers. This has a cleaner trend.

Values of H(n)

Sum of H(n)

My collaborators (Beckwith, Gupta, Rolen, Diamatis) and I set out to find a formula for the sum of H(n). We were able to derive a summation formula for the weighted summation of the Fourier coefficients for a wider class of Harmonic Maass forms, including 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.

The paper:

[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)

Side Hussle: Rethinking Number Theory Project

I am a part of the Rethinking Number Theory community. I met my fantastic collaborator, Kim Klinger-Logan, and Tian An Wong, through RNT. Collaborations are something I enjoy a lot, even when they require learning new techniques. This was my first exposure to analytic techniques, but my collaborators were patient. Here is a link to our paper:

[published] A Dedekind–Rademacher cocycle for Bianchi groups, Kim Klinger-Logan, Kalani Tha lagoda, Tian An Wong, Journal de Théorie des Nombres de Bordeaux, Vol. 37, No. 1(2025), pp. 285-298 (15 pages) https://www.jstor.org/stable/48827542

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