Modular completions of false theta functions

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Modular forms generalize classical trigonometric functions as they are periodic; however they have more symmetries. They play a central role in many areas including algebraic topology, arithmetic geometry, combinatorics, number theory, representation theory, and mathematical physics. The situation is complicated by the fact that often modularity is broken, it is however not alway a priori clear in which way. In this proposal we in particular investigate false theta functions. For these functions, a wrong sgn-factor is introduced which destroys modularity. False theta functions have a long history, going back to Rogers (in the one-dimensional case). Several attempts have been made to understand these functions, but unfortunately, they failed and thus the modularity properties of false theta functions remain unknown. On the other hand, there is a rich history on false theta functions as they occur in many settings and there is thus high demand to understand them. In this proposal I will show how false theta functions can be understood in a modular world. Besides its own interest for number theory, this will have application (for example to combinatorics, physics, and W-algebras) as I will investigate in this proposal.