Hopf algebras, operator theory, and algebraic combinatorics

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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.