Grant: Discovery Grants Program - Individual

I am most interested in problems involving interplay between several branches of mathematics. I plan to work on problems involving Hopf algebras, linear algebra, operator theory, and algebraic combinatorics.******(1) I plan to continue working on open questions about 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. Collections in questions often posses an algebraic structure such as, for example, the structure of a semigroup, group, or an algebra. Sufficient conditions that guarantee triangularizability of semigroups can often be phrased in terms of the eigenvalues of matrices involved.******(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. ******(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, representation theory of symmetric groups, classical probability, to name a few) and opened numerous new avenues of possible research which we hope to pursue.