Modular completions of false theta functions

模形式推广了经典的三角函数，因为它们是周期性的;然而它们有更多的对称性。它们在许多领域中发挥着核心作用，包括代数拓扑学、算术几何学、组合学、数论、表示论和数学物理学。这种情况是复杂的，因为模块化往往被打破，但它并不总是先验清楚的方式。在本提案中，我们特别研究了假theta函数。对于这些函数，引入了一个错误的sgn因子，破坏了模块性。假theta函数有很长的历史，可以追溯到罗杰斯（在一维情况下）。人们曾多次尝试去理解这些函数，但不幸的是，他们失败了，因此假theta函数的模块性属性仍然未知。另一方面，关于假θ函数有着丰富的历史，因为它们发生在许多环境中，因此对理解它们有很高的要求。在这个提议中，我将展示如何在模块化世界中理解错误的theta函数。除了它本身对数论的兴趣之外，这还将有应用（例如组合数学、物理学和W-代数），正如我将在本提案中研究的那样。

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.