Harmonic analysis for resistance forms

The Harmonic Operator for Exterior Differential Forms.

Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms

This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few....

Morse Theory of Harmonic Forms

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature , if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use a theorem of E.Calabi C], characterizing 1-forms which are harmonic with r...

Binary Forms , Equiangularpolygons and Harmonic

Consider binary forms F having complex coeecients and discriminant D F 6 = 0. In a sequence of previous papers, the rst author studied the area A F of the planar region jF(x;y)j 1 deened by forms F of this type in connection with the enumeration of integer lattice points. In particular, the rst author showed that the GL 2 (R)-invariant quantity jD F j 1=n(n?1) A F is uniformly bounded when the ...

On Products of Harmonic Forms

We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and onl...