Modular completions of false theta functions

模形式概括了经典三角函数的周期性；但是它们有更多的对称性。它们在许多领域发挥着核心作用，包括代数拓扑、算术几何、组合学、数论、表示理论和数学物理。这种情况由于模块化经常被破坏而变得复杂，然而，它并不总是先验地清楚以何种方式被破坏。在这个提议中，我们特别研究假函数。对于这些函数，引入了一个错误的符号因子，破坏了模块性。假函数有很长的历史，可以追溯到罗杰斯（在一维情况下）。已经做了几次尝试来理解这些函数，但不幸的是，他们失败了，因此假函数的模块化性质仍然未知。另一方面，假θ函数的历史很丰富，因为它们在许多情况下都会出现，因此对理解它们的要求很高。在这个提议中，我将展示如何在模块化世界中理解假函数。除了对数论本身的兴趣外，这将有应用（例如组合学，物理学和w -代数），我将在本提案中进行研究。

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.