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Andrei Seymour-Howell works in computational number theory, particularly automorphic forms and trace formulas. His 2022 paper develops an algorithm to compute and rigorously verify Maass cusp forms for specified groups of squarefree level and trivial character. It combines a Selberg trace formula involving Hecke operators with linear algebra to isolate spectral information.^[ref]

The distinction between computation and certification is central. A decimal approximation to an eigenvalue suggests where to look; a rigorous verification must justify that the computed information corresponds to an actual mathematical object, with controlled error. As a schematic illustration, a proved enclosure $9.99\le \lambda \le 10.01$ conveys a different kind of information from a program merely printing 10.00. The numbers here illustrate the distinction and are not a reported eigenvalue. The paper applies its verified computations to gather evidence about further conjectures; verified examples still do not prove a conjecture quantified over all forms.^[ref]

A Maass cusp form is a nonzero Maass form with vanishing constant term of an automorphic form at every cusp. For weight zero on the modular group $\Gamma ={\mathrm{SL}}_2(\mathbb{Z})$, it is a smooth, $\Gamma$-invariant function on the upper half-plane, square-integrable on a fundamental domain with measure $dxdy/y^2$, satisfyingThe integral condition describes the cusp at infinity; other groups require the corresponding condition in coordinates at each cusp.^[ref]

Translation invariance gives a Fourier series. When $\lambda =1/4+r^2$, its decaying modes have the formwhere $K_{ir}$ is a Modified Bessel function of the second kind. The absent $n=0$ term expresses cuspidality. The Fourier coefficients must also satisfy the full automorphic symmetry.^[ref]

The constant function $f=1$ is a useful test: it satisfies the symmetry and has eigenvalue zero, but its horizontal average is $1$, so it is not a Maass cusp form. The extra condition isolates oscillating eigenfunctions connecting arithmetic symmetry with spectral geometry.