Bethe ansatz, Hecke algebras, and asymptotics

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目的目的是研究量子可积模型的交换Hamilton代数，称为Bethe代数，以及它们在表示论和代数几何中的关系。主要的兴趣点是与李代数gl_N相关的可积模型的Bethe代数，如Gaudin模型和XXX-型模型。最近的发展表明，这些Bethe代数代数的Calogero-Moser空间和交叉点的Schubert周期在格拉斯曼函数代数的密切联系。其中一个预期的目标是更深入地了解这一重要的联系，并将其扩展到其他代数的几何性质。另一个目标是建立直接联系的Bethe代数的问题与Cherednik代数和双仿射Hecke代数。这种联系的存在是非常合理的，因为后者代数与贝特代数的相同几何对象密切相关。该项目的另一个方向是利用准经典量子化的思想来获得Bethe代数的大特征值和相应特征向量的渐近性。量子可积系统理论中的主要问题是寻找交换运动积分的联合本征值和本征向量。Bethe anomaly起源于一套聪明的技术技巧来执行这项任务。最终，它已经发展成为各种工具，将这个问题与许多领域联系起来，例如偏微分方程的变量分离，Fuchsian微分方程，差分方程，舒伯特微积分。这一互动证明是非常非常富有成果的，并为所有相关主题产生了许多重要成果。该项目将有助于这一领域的进一步发展。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The aim of the project is to study algebras of commuting Hamiltonians for quantum integrable models, called Bethe algebras, and their relations in representation theory and algebraic geometry. The main points of interest are Bethe algebras for integrable models associated with the Lie algebra gl_N, such as the Gaudin model and the XXX-type model. Recent developments show a close connection of those Bethe algebras to algebras of functions on the Calogero-Moser space and intersections of Schubert cycles in Grassmannians. One of the intended goals is to understand deeper this important connection and to expand it to other algebras of geometric nature. Another goal is to establish direct links of the Bethe algebras in question with Cherednik algebras and double affine Hecke algebras. The existence of such links is very plausible because the latter algebras are closely related to the same geometric objects as Bethe algebras. One more direction of the project is to employ ideas of quasiclassical quantization to obtain asymptotics of large eigenvalues and the corresponding eigenvectors of Bethe algebras. Hopefully, those asymptotics can be formulated in terms of solutions of classical integrable systems.The main question in the theory of quantum integrable systems is to find joint eigenvalues and eigenvectors of commuting integrals of motion. The Bethe ansatz originated as a set of clever technical tricks to perform this task. Eventually, it has developed into a variety of tools connecting this problem to many areas such as separation of variables for PDEs, Fuchsian differential equations, difference equations, Schubert calculus. This interaction turned out to be very extremely fruitful and yielded many important results for all the subjects involved. The project will contribute to further development of this area.