For a full list of publications, see my googlescholar page or my CV (has clickable links).
I'm interested in algebraic structures on combinatorial objects, and their interplay with Markov chains. My current research themes are
New bases for combinatorial Hopf algebrasA combinatorial Hopf algebra is a graded vector space whose basis is indexed by a family of combinatorial objects, e.g. by trees or by permutations. There is a product operation encoding the combination of two objects into one, and a coproduct operation encoding the breaking of one object into two pieces. Because of certain rigidity theorems, operations that are different apriori often end up defining algebras that are abstractly isomorphic. In many cases, no explicit isomorphism is known, and constructing such isomorphisms is one motivation to define new bases. Also, rigidity theorems sometimes guarantee that a certain algebra is free or cofree, and again it is an interesting question to construct a basis that exhibits these properties explicitly.
Markov chains driven by linear operatorsThese chains are generalisations of random walks on groups and algebras - their transition matrices are the matrices for particular linear transformations. Studying the linear transformations then gives the convergence rates of each chain and the relationships between different chains.
Subalgebras of Solomon's descent algebraI construct new subalgebras of the descent algebra for symmetric and hyperoctahedral groups, and find their orthogonal idempotents.