Algebraic theory of integrable systems. Representations of affine superalgebras and mock theta functions

偏微分方程是微积分中的方程，用于模拟物理世界中的现象，例如，当研究相互作用的水波的运动时。一个基本的问题是找到被波守恒的量，而这种寻找是研究可积系统的核心。经典的可积哈密顿系统，如KdV方程和非线性薛定谔系统，一直在波相互作用理论、等离子体物理和光纤光学等物理理论中发挥着基础性的作用。这个项目第一部分的主要目标是建立一个严格的哈密顿偏微分方程组的可积理论，基于受物理启发的代数结构。这一项目所发现的可积系统有望在各种物理现象的研究中发挥重要作用。这个项目的第二部分起源于拉马努扬在1920年代初写给G.H.哈代的最后一封信，S在信中写下了17个以前从未研究过的函数(他将这些函数命名为mock theta函数)。该项目的第二个方向是研究模拟theta函数与表示理论的联系。与第一个方向相关的一些有待探索的问题包括：(A)基于‘’非局部‘Poisson顶点代数’的概念，发展了非局部哈密顿结构理论和相应的可积系统理论；(B)计算不一定是拟常系数Poisson微分算子的变分Poisson上同调，发展了2013年的一篇论文的方法；(C)进一步发展广义Drinfeld-Sokolov族及其Dirac约化的理论，并建立它们与Kac-Wakimoto族的联系。与第二个方向相关的一些问题包括：(A)计算仿射李超代数的允许表示的特征标公式；(B)按照Zwegers的精神，为相应的修正的非全纯theta函数找到显式变换公式；(C)利用量子哈密顿约化，计算超共形代数新表示的模变换公式和融合规则。

Partial differential equations are equations from calculus that are used to model phenomena in the physical world, for example, when studying the motion of interacting water waves. A fundamental question is to find the quantities which are conserved by the waves, and this search lies at the heart of the study of integrable systems. The classical integrable Hamiltonian systems, like the KdV equation and the non-linear Schrodinger system, have been playing a fundamental role in physical theories, like the theory of wave interactions, plasma physics, and fiber optics, to name a few. The main goal of the first part of this project is to build up a rigorous theory of integrable systems of Hamiltonian partial differential equations, based on algebraic structures that were inspired by physics. It is expected that the integrable systems discovered as a result of this project will play an important role in the study of various physical phenomena. The second part of this project has its origins in the last letter that Ramanujan wrote to G. H. Hardy in the early 1920's, where he wrote down 17 functions that had never been studied before (and which he named mock theta functions). The second direction of the project is to study the connections of mock theta functions to representation theory. It is expected that this will lead to new types of mock theta functions and to applications in number theory.Some of the problems to be explored related to the first direction include the following: (a) develop a theory of non-local Hamiltonian structures and corresponding theory of integrable systems, based on the notion of a ''non-local'' Poisson vertex algebra; (b) compute the variational Poisson cohomology for not necessarily quasi-constant coefficient Poisson differential operators, developing the methods of a 2013 paper with De Sole; (c) develop further the theory of the generalized Drinfeld-Sokolov hierarchies and their Dirac reductions and establish their connection to the Kac-Wakimoto hierarchies. Some of the problems to be explored related to the second direction include the following: (a) compute character formulas for admissible representations of affine Lie superalgebras; (b) find explicit transformation formulas for the corresponding modified non-holomorphic theta-functions, in the spirit of Zwegers; (c) using quantum Hamiltonian reduction, compute modular transformation formulas and fusion rules of new representations of superconformal algebras.