Abstract

Modular linear differential equations (MLDEs) have arisen in several different contexts in mathematics and mathematical physics in recent years.  Perhapsthe first example, which will be described briefly, was the second-order LDE        f”(z) – (k+1)/6 E_2(z) f’(z) + k(k+1)/12 E_2’(z) f(z) = 0 that arose in work of Kaneko and myself on supersingular j-invariants incharacteristi p, but more recently they have become important in thetheory of vertex operator algebras, where they often give non-trivial information about the properties (or sometimes non-existence) of VOAsof given type and given central charge.  The talk will describe recent jointwork with K. Nagatomo and Y. Sakai giving descriptions of all MLD Os (forthe full modular group and other lattices in SL(2,R) ) in terms of Rankin-Cohen brackets and in terms of quasimodular forms.  All notions, includingthe definitions of modular and quasimodular forms, will be reviewed.