Algebraic and geometric structures related to integrable systems

数学的美丽和吸引力根植于我们对现实的感知和不同复杂程度的分析所产生的深刻的几何和物理图像。数学家的工作就是用形式代数结构来表达这种美。事实上，在形成代数结构的过程中，我们数学家可以通过创造一种新的代数语言来捕捉几何和物理的图像，用这种语言来讨论它们，并使它们易于我们的探索、分析和理解。我的研究计划的目的是研究所谓的可积模型理论中出现的代数结构。一个简单的例子就是著名的Korteweg-de Vries方程它是一个包含两个变量t（时间）和x（空间坐标）的单一函数u（t,x）的偏微分方程。这个方程的形式是u_t=u_xxx+ u_x其中t和x表示偏导数。Korteweg-de Vries方程，尽管其形式简单，却拥有丰富而美丽的理论，其中包括有趣的代数结构，特解（所谓的孤子），并与从代数几何到泛函分析的各个数学领域联系在一起。我建议在未来几年研究更复杂的可积模型。建议的第一部分致力于拟线性系统的偏微分方程的形式为A(u)u_t+B(u)u_x+C(u)u_y=0，其中u（t,x,y）是一个向量函数，A， B， C是依赖于u的矩阵。这种形式的方程在流体力学中是有用的。这样的可积系统也有丰富的数学理论。许多数学领域（如代数和微分几何）将受益于这种可积系统理论的发展。我们还希望研究具有两个自变量t和x的非齐次类似系统。一个典型的例子是两个未知函数u（t,x）和v（t,x）的两个方程的系统，其形式为：u_t=v u_x+1/(u-v), v_t=u v_x+1/(v-u)。由于这个系统有许多新的和不寻常的性质，我相信它的研究具有极大地丰富整个可积系统理论的潜力。其他的研究将致力于所谓的矩阵可积系统。这种系统的一个简单例子是广义欧拉顶，它是一个常微分方程U_t=CU^2-U^2C，其中U(t)是时间t的方阵函数，C是常数矩阵。提案的最后（但并非最不重要）部分致力于量子可积模型理论中出现的代数结构：即所谓的椭圆代数。为了解释这个想法，考虑三个变量x， y, z，它们不交换，但受制于关系：xy-yx=z, yz-zy=x, zx-xz=y。众所周知，使用这些关系，任何单项式（比如zyxy）都可以以一种独特的方式写成有序单项式（比如xxyzzz）的线性组合。这个命题的证明并不难，它基于观察到x y z实际上可以交换成线性项。椭圆代数理论处理类似的关系，但只处理二次项，例如xy-3yx=5z^2, yz-3zy=5x^2, zx-3xz=5y^2。关于有序单项式的类似陈述在这种情况下也是有效的，但证明要困难得多。椭圆代数在数学和数学物理的各个分支，包括代数几何、量子可积模型甚至同调代数中起着重要的作用。此外，与椭圆代数的所谓半经典极限有关的一些结构在上面讨论的可积微分方程理论中是重要的。综上所述，本研究致力于现代数学物理中出现的重要代数结构。

The beauty and attraction of mathematics is rooted in profound images of geometry and physics coming from our perception of reality and it’s analysis at different levels of sophistication. The job of mathematicians then is to express this beauty in terms of formal algebraic structures. Indeed, in forming algebraic structures we, mathematicians, can capture images of geometry and physics by creating a new, algebraic language in which to discuss them and make them accessible to our exploration, analysis and comprehension. The aim of my research program is to investigate algebraic structures arising in the theory of so-called integrable models. A simple example is the famous Korteweg–de Vries equation which is a partial differential equation for a single function u(t,x) of two variables t (time) and x (spatial coordinate). This equation has a form u_t=u_xxx+u u_x where indexes t and x stand for partial derivatives. The Korteweg–de Vries equation, in spite of its simple form, possess a rich and beautiful theory that includes interesting algebraic structures, particular solutions (the so-called solitons) and links with various fields of mathematics from algebraic geometry to functional analysis. I am proposing to study more complicated integrable models over the next few years. The first part of the proposal is devoted to quasi-linear systems of partial differential equations of the form A(u)u_t+B(u)u_x+C(u)u_y=0 where u(t,x,y) is a vector function and A, B, C are matrices depending on u. Equations of this form are useful in hydrodynamics. Such integrable systems also admit a rich mathematical theory. Many fields of mathematics (such as algebraic and differential geometry) will benefit from the development of a theory of such integrable systems. We also wish to study similar systems that are non-homogeneous and have two independent variables t and x. A typical example is a system of two equations for two unknown functions u(t,x) and v(t,x) of the form: u_t=v u_x+1/(u-v), v_t=u v_x+1/(v-u). Because this system admits many new and unusual properties, I am convinced that it's study has the potential of significantly enriching the whole theory of integrable systems. Other studies will be devoted to the so-called matrix integrable systems. A simple example of such system is the generalized Euler top which is an ordinary differential equation U_t=CU^2-U^2C where U(t) is a square matrix function of time t and C is a constant matrix. The last (but not least) part of the proposal is dedicated to algebraic structures arising in the theory of quantum integrable models: namely, the so-called elliptic algebras. To explain the idea, consider three variables x, y, z which do not commute but are subject to relations: xy-yx=z, yz-zy=x, zx-xz=y. It is well known that using these relations any monomial (say, zyxy) can be written in a unique way as a linear combination of ordered monomials such as xxyzzz. A proof of this statement is not hard and based on the observation that x, y, z actually commute up to linear terms. The theory of elliptic algebras deals with similar relations but with quadratic terms only, for example xy-3yx=5z^2, yz-3zy=5x^2, zx-3xz=5y^2. The similar statement about ordered monomials is also valid in this case but the proof is much harder. Elliptic algebras play a significant role in various branches of mathematics and mathematical physics including algebraic geometry, quantum integrable models and even homological algebra. Moreover, some structures connected with the so-called semi-classical limits of elliptic algebras are important in the theory of integrable differential equations discussed above. To summarize, the proposed research is devoted to important algebraic structures arising in modern mathematical physics.