Hopf algebras, operator theory, and algebraic combinatorics

我最感兴趣的是涉及几个数学分支之间相互作用的问题。有很多这样的问题涉及到霍普夫代数、线性代数和组合学。下面我将提到一些。(1)我计划继续研究关于作用于巴拿赫空间（有限维或无限维）的算子集合的开放问题。我将集中讨论不仅可以应用解析技术，还可以应用代数、几何和组合技术的问题。许多这样的问题涉及到对矩阵集合的同时约化性、三角化性和传递型性质的研究。如果一个n × n矩阵的集合可以通过同时相似变换成为上三角矩阵的集合，那么它就是可三角化的。问题中的集合通常具有代数结构，例如，半群的结构（如果两个矩阵属于集合，则它们的乘积也必须属于集合）。保证半群可三角化的充分条件通常可以用所涉及的矩阵的特征值来表述。最近已经证明，在许多情况下，一个精确的条件（例如，如果AB-BA的所有特征值，对于某些矩阵半群中的所有矩阵A，B，都是0，那么所讨论的半群是可三角化的）可以用一个近似的版本来代替（对于某些矩阵半群，只要假设AB-BA的所有特征值都很小就足以保证可三角化）。Hopf代数是在数学和物理学的许多领域中出现的代数结构。Hopf代数的例子包括群代数、李代数的泛包络、李群上的代表函数代数、代数群的坐标代数、对称和拟对称函数代数以及量子群。我计划研究Hopf代数的代数结构，以便对它们进行分类，构建新的有趣的例子，并将它们应用于代数，分析和组合学中的问题。我还计划从组合的角度学习Hopf代数。在组合学中，它们通常用于对离散结构的组装和拆卸进行编码。我对代数、分析和组合学的兴趣结合在一起的一个领域是自由概率。自由概率论是研究非可交换随机变量的一种数学理论。自由独立是经典独立概念的类比，它与代数的自由积有关。它在算子代数理论和随机矩阵理论中有许多应用（随机矩阵理论变得非常重要，因为它们在物理和工程中的应用）。在随机矩阵理论中，一个典型的问题是这样的：“对于由某个已知随机过程产生的矩阵的特征值，我们能说些什么？”如果矩阵足够大，并且可以分解为易于理解的矩阵的和或乘积，那么自由概率技术可以帮助回答此类问题。在与a . Nica的合作中，我们构造了一个组合Hopf代数，它可以用于研究非交换概率空间中k元组的联合分布。这种方法暴露了与其他数学分支（对角谐波，对称群的表示理论，经典概率，仅举几例）的许多联系，并开辟了许多我们希望追求的可能研究的新途径。

I am most interested in problems involving interplay between several branches of mathematics. There is an abundance of such problems involving Hopf algebras, linear algebra, and combinatorics. Below I mention a few.(1) I plan to continue working on open questions ragarding collections of operators acting on a Banach space (of either finite or infinite dimension). I will focus on questions where not only analytic, but also algebraic, geometric, and combinatorial techniques can be applied. Many of such problems involve the study of simultaneous reducibility, triangularizability, and transitivity-type properties of collections of matrices. A collection of n-by-n matrices is said to be triangularizable if it can be transformed by a simultaneous similarity to become a collection of upper triangular matrices. Collections in questions often posses an algebraic structure such as, for example, the structure of a semigroup (if two matrices belong to the collection then their product must also belong). Sufficient conditions that guarantee triangularizability of semigroups can often be phrased in terms of the eigenvalues of matrices involved. Recently it has been shown that in many cases an exact condition (for example, if all eigenvalues of AB-BA, for all matrices A,B in some semigroup of matrices, are 0, then the semigroup in question is triangularizable) can be replaced by an approximate version (for certain semigroups of matrices it is enough to assume that all eigenvalues of AB-BA are small to ensure triangularizability). (2) Hopf algebras are algebraic structures arising in numerous parts of mathematics, as well as in physics. Examples of Hopf algebras include group algebras, universal envelopes of Lie algebras, algebras of representative functions on Lie groups, coordinate algebras of algebraic groups, algebras of symmetric and quasi-symmetric functions, and quantum groups. I plan to study the algebraic structure of Hopf algebras in order to classify them, construct new interesting examples, and apply them to problems in algebra, analysis, and combinatorics. I also plan to study Hopf algebras from combinatorial point of view. In combinatorics they are usually used to encode assembly and disassembly of discrete structures.(3) One of the areas where my interests in algebra, analysis, and combinatorics are combined is free probability. Free probability is a mathematical theory that studies non-commuting random variables. The free independence is an analogue of the classical notion of independence, and is connected with free products of algebras. It has many applications in the theory of operator algebras as well as in the theory of random matrices (which have become very important because of, among other things, their applications in physics and engineering). A typical type of question one asks in the random matrix theory is as follows: "What can we say about the eigenvalues of a matrix that resulted from some known random process?" If matrices are sufficiently large and can be decomposed as sums or products of well understood matrices, then free probability techniques can help answer such questions. In joint work with A. Nica we have constructed a combinatorial Hopf algebra which can be used to study joint distributions of k-tuples in a noncommutative probability space. This approach exposed numerous connections with other branches of mathematics (diagonal harmonics, represention theory of symmetric groups, classical probability, to name a few) and opened numerous new avenues of possible research which we hope to pursue.