Algebraic and geometric structures related to integrable systems

与可积系统相关的代数和几何结构

• 批准号: RGPIN-2014-05062
• 负责人: Odesski, Alexandre
• 金额: $ 1.02万
• 依托单位: Brock University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611414
• 关键词: Algebraic; geometric; structures; related; integrable

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.

数学的美丽和吸引力源于我们对现实的看法以及在不同社会化层面上的分析的深刻形象。那时，数学家的工作是用正式的代数结构来表达这种美丽。实际上，在形成代数结构时，我们数学家可以通过创建一种新的代数语言来捕获几何和物理学的图像，在该语言中讨论它们，并使它们可以访问我们的探索，分析和理解。 我的研究计划的目的是研究在所谓的综合模型理论中产生的代数结构。一个简单的例子是著名的korteweg – de vries方程，它是两个变量t（time）和x（空间坐标）的单个函数u（t，x）的部分微分方程。该方程式具有u_t = u_xxx+u_x的形式，其中索引t和x代表部分导数。 Korteweg – de Vries方程式尽管具有简单形式，但具有丰富而美丽的理论，其中包括有趣的代数结构，特定的解决方案（所谓的实体）以及与代数几何学到功能分析的数学领域的各种数学领域的联系。我建议在未来几年研究更复杂的整合模型。 该提案的第一部分致力于a（u）u_t+b（u）u_x+c（u）u_x+c（u）u_y = 0的偏微分方程的准线性系统，其中u（t，x，y）是矢量函数，a，b，c是基于u的a，b，c是矩阵。该形式的方程在流体动力学中很有用。这种可集成的系统也接受丰富的数学理论。许多数学领域（例如代数和差异几何形状）将受益于这种可集成系统的理论的发展。 我们还希望研究类似的无均匀系统，并具有两个独立变量T和X。一个典型的示例是一个两个方程组的系统，用于两个未知函数u（t，x）和表格的v（t，x）：u_t = v u_x+1/（u-v），v_t = u v_x+1/（v-u）。由于该系统接受许多新的和不寻常的属性，因此我坚信它的研究可能会显着丰富整体系统的整个理论。 其他研究将专门用于所谓的矩阵积分系统。这种系统的一个简单示例是通用的Euler顶部，它是一个普通的微分方程u_t = Cu^2-u^2c，其中u（t）是时间t和c的平方矩阵函数是常数矩阵。 该提案的最后一部分（但并非最不重要的一点）专门用于量子整合模型理论中产生的代数结构：即所谓的椭圆代数。要解释这个想法，请考虑三个变量x，y，z，而不是命令，但要受关系：xy-yx = z，yz-zy = x，zx-xz = y。众所周知，使用这些关系，任何单一（例如，zyxy）都可以独特地写作，作为有序单式的线性组合，例如xxyzzz。此陈述的证明并不难，并且基于x，y，z实际命令到线性项的观察。椭圆形代数理论涉及相似的关系，但仅以二次术语为例，例如xy-3yx = 5z^2，yz-3zy = 5x^2，zx-3xz = 5y^2。在这种情况下，关于有序单项的类似陈述也有效，但证据要困难得多。 椭圆代数在数学和数学物理学的各个分支中起着重要作用，包括代数几何，量子整合模型甚至同源代数。此外，与所谓的椭圆形代数的所谓半古典限制相关的一些结构在上面讨论的可集成微分方程的理论中很重要。 总而言之，拟议的研究致力于在现代数学物理学中产生的重要代数结构。