Talks
Averaging to Higher Depth Mock Modular Forms
Abstract: In recent years, mock modular forms with depth higher than one have started to make their appearance in physics and mathematics, finding applications in diverse contexts such as black hole counting, geometric invariants, and two dimensional conformal field theories. Mathematically, one aspect of higher depth mock modular forms that distinguishes them from their depth one counterparts is the absence of well-established Eisenstein/Poincaré series representations for (pure) mock modular forms at depth one, which allows one to recognize pure mock modular forms as averages of simpler objects over \(\mathrm{SL}_2(\mathbb{Z})\). In this talk, we will discuss how a certain type of double Eisenstein series can give rise to a similar picture for higher depth mock modular forms and connections that this leads to for the holomorphic parts of mock modular forms.
Abstract: In recent years, mock modular forms with depth higher than one have started to make their appearance in physics and mathematics, finding applications in diverse contexts such as black hole counting, geometric invariants, and two dimensional conformal field theories. Mathematically, one aspect of higher depth mock modular forms that distinguishes them from their depth one counterparts is the absence of well-established Eisenstein/Poincaré series representations for (pure) mock modular forms at depth one, which allows one to recognize pure mock modular forms as averages of simpler objects over \(\mathrm{SL}_2(\mathbb{Z})\). In this talk, we will discuss how a certain type of double Eisenstein series can give rise to a similar picture for higher depth mock modular forms and connections that this leads to for the holomorphic parts of mock modular forms.
Rademacher Expansions for False and Mock Modular Forms
Abstract: Holomorphic modular forms are tightly constrained by both holomorphy and modular symmetries. Going back to the seminal works of Hardy, Ramanujan, and Rademacher, we know that these properties can impose robust constraints on the Fourier coefficients of holomorphic modular forms, enabling the derivation of convergent, exact formulae for these coefficients at nonpositive weights. In this talk, we will review how such arguments can be extended to (mixed) mock or false modular forms, possibly at depths higher than one.
Abstract: Holomorphic modular forms are tightly constrained by both holomorphy and modular symmetries. Going back to the seminal works of Hardy, Ramanujan, and Rademacher, we know that these properties can impose robust constraints on the Fourier coefficients of holomorphic modular forms, enabling the derivation of convergent, exact formulae for these coefficients at nonpositive weights. In this talk, we will review how such arguments can be extended to (mixed) mock or false modular forms, possibly at depths higher than one.