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，在研究相互作用的水波运动时。一个基本问题是找到由波守恒的量，这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，一直在物理理论中扮演着重要的角色，如波相互作用理论、等离子体物理和光纤等。该项目第一部分的主要目标是建立一个严谨的哈密顿偏微分方程的可积系统理论，该理论基于受物理学启发的代数结构。预计该项目所发现的可积系统将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源于拉马努金在20世纪20年代初写给g·h·哈代的最后一封信，他在信中写下了17个以前从未被研究过的函数（他将其命名为模拟函数）。项目的第二个方向是研究模拟theta函数与表征理论的联系。预计这将导致新类型的模拟函数和数论中的应用。与第一个方向相关的一些有待探讨的问题包括：(a)基于“非局部”泊松顶点代数的概念，发展了非局部哈密顿结构理论和相应的可积系统理论；(b)计算不一定准常系数泊松微分算子的变分泊松上同调，发展了2013年与De Sole合著的一篇论文的方法；(c)进一步发展广义Drinfeld-Sokolov层次及其Dirac约简理论，并建立其与Kac-Wakimoto层次的联系。与第二个方向相关的一些有待探讨的问题包括：(a)计算仿射李超代数的可容许表示的特征公式；(b)根据Zwegers的精神，找到相应的修正非全纯函数的显式变换公式；(c)利用量子哈密顿约简，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.