Recent Projects

Visual companions to the papers.

Final frame of the murmurations animation: two opposite point clouds in teal and purple following the dashed murmuration density.
Modular forms

Murmurations

Signed averages of Fourier coefficients of modular forms trace an explicit density: the main statement, and the universal limit shared by all weights.
Converse theorems

A twistless converse theorem

How the functional equations of twists by Ramanujan sums, which come from an Euler product, pin down a modular form, and where they fall short.