Symmetries: Algebra and Physics

对称在科学中很重要，尤其是在物理学中。可以精确求解的模型构成了许多理论理解的支柱，并且可以观察到，对称性的存在是这种系统的标志。对称是用代数在数学上描述的，由于这种联系，各种代数结构的发现常常导致理论物理学的进步。我的研究主题是由对称、代数、表示理论、特殊函数和物理模型组成的良性循环。我将确定量子多体系统的纠缠，设计与量子信息任务相关的模型，并着眼于特殊函数发展对称性数学。本方案由五个相关部分组成。这里有一些细节。1. 修运营商。标准Heun算子定义了具有四个正则奇点的微分方程；它出现在许多问题中，如广义顶或高丁磁体。我将在三对角化方法的帮助下，通过引入许多新的Heun型算子来超越目前的情况。Bethe ansatz将用于对角化这些算子，并将在物理学中得到应用。我将证明，从这些构造中可以得到Sklyanin型代数，它们是量子可积模型的中心。2. 纠缠。这是用熵来描述的量子理论的一个主要特征。我将使用信号处理中的时间和频带限制问题与费米子系统（图上）纠缠的表征之间的类比，以及Heun算子在前者上下文中所起的关键作用，以在后者中取得进展。在这个过程中，我将提出代数组合学中长期存在的问题。3. 部分复兴。这是一种现象，在不同的位置同时周期性地激发；在只有一个地点的情况下，它是完美转移。我将使用正交多项式来设计解析质量-弹簧链，它完全或近似地具有这个特征。这将涉及到执行量子信息任务的设备的主体工作。4. 理性的功能。对于Askey格式的每个正交多项式族，都有一个对其性质进行编码的代数。我将把Askey方案扩展到有理函数。这将涉及到广义特征值问题和引入元代数，包括与多项式相关的代数。将探讨ASEP模型的应用。5. Askey-Wilson代数，拓扑理论。我将得到有限群代数的量子重的反射矩阵，并确定它们在有边界的Kitaev量子纠错码中的作用。我将发展与Askey-Wilson多项式相关的代数的许多推广，并在包含链接不变量的chen - simons场论中找到它的实现。

Symmetries are important in Science and especially in Physics. Models that can be solved exactly form the backbone of much theoretical understanding and it is observed that the presence of symmetries is the hallmark of such systems. Symmetries are described mathematically by algebras and because of this connection, the discovery of various algebraic structures has most often led to advances in theoretical physics. My research bears on the topics of the virtuous circle composed by symmetries, algebra, representation theory, special functions and physical models. I will determine the entanglement of quantum many body systems, design models related to tasks in quantum information and develop the mathematics of symmetries with an eye to special functions. This program comprises five related parts. Here are some details. 1. Heun operators. The standard Heun operator defines the differential equation with four regular singularities; it arises in many problems such as generalized tops or Gaudin magnets. I will transcend the current picture by introducing many new operators of Heun type with the help of the tridiagonalization method. The Bethe ansatz will be used to diagonalize these operators and applications in physics will be developed. I will show that from these constructs one can obtain algebras of Sklyanin type that are central in quantum integrable models. 2. Entanglement. This is a chief feature of quantum theories which is described by entropies. I will use the analogy between the time and band limiting problem in signal processing and the characterization of the entanglement of systems of fermions (on graphs), together with the key role that Heun operators play in the former context, to make headways in the latter. In the process I will advance long-standing questions in algebraic combinatorics. 3. Fractional revival. This is the phenomenon where an excitation periodically resurges at different locations simultaneously; in the case of only one location, it is Perfect Transfer. I will use orthogonal polynomials to design analytic mass-spring chains that possess this feature exactly or approximately. This will relate to the body of work on devices performing quantum information tasks. 4. Rational functions. To each family of orthogonal polynomials of the Askey scheme there is an algebra that encodes their properties. I will extend the Askey scheme to rational functions. This will involve generalized eigenvalue problems and introducing meta-algebras encompassing the ones associated to polynomials. Applications to ASEP models will be explored. 5. Askey-Wilson algebras, topological theories. I will obtain the reflection matrices of the quantum double of the algebras of finite groups and determine their roles in the Kitaev quantum error correcting codes with boundaries. I will develop many generalizations of the algebra associated to the Askey-Wilson polynomials and find its realization in Chern-Simons field theories that embody links invariants.