Fourier analysis is a powerful tool in analysis. In the setting of abelian schemes, Fourier-Mukai transformation and the weight decomposition play a similar role. For degenerate abelian fibrations, the relative group structure disappears and understanding the intersection theory leads to many interesting questions, such as the \(P=W\) conjecture, \(\chi\)-independence phenomenon, and multiplicative splitting of the perverse filtration for the Beauville-Mukai system.

In this talk, I will connect Fourier transform between compactified Jacobians over the moduli space of stable curves and logarithmic Abel-Jacobi theory. As an application, I will compute the pushforward of monomials of divisor classes on compactified Jacobians via the twisted double ramification formula. Along the way, we will encounter instances of \(\chi\)-independence and the multiplicativity of perverse filtration for compactified Jacobians. This is a joint work with Samouil Molcho and Aaron Pixton.