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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 algebras

A 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 operators

These 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.

Summary tables:
Papers in reverse chronological order (of authorship):
• The Eigenvalues of Hyperoctahedral Descent Operators and Applications to Card-Shuffling (2021)

A preprint on arXiv.

• Markov Chains from Descent Operators on Combinatorial Hopf Algebras (2016)

A preprint on arXiv. A reader-friendly overview, known as "Card-Shuffling via Convolutions of Projections on Combinatorial Hopf Algebras" was an extended abstract and talk for FPSAC 2015 (last update: August 26, 2018).

• Lumpings of Algebraic Markov Chains arise from Subquotients (2018), in J. Theoret. Probab.

Update July 2018. A first version, released 2015, was entitled "A Hopf-Algebraic Lift of the Down-Up Markov chain on Partitions to Permutations".

• Hopf Algebras and Markov Chains (2014)

Revised version of my thesis. These Markov chains are driven by the coproduct-then-product operator.

• A Hopf-power Markov chain on compositions (2013)

An extended abstract and poster for FPSAC 2013. Superceded by section 6.2 of my thesis (item above).

• Hopf Algebras and Markov Chains: two examples and a theory (2014), with Persi Diaconis and Arun Ram, in J. Algebraic Combin.

Update Dec 2014. These Markov chains are driven by the coproduct-then-product operator.

### Subalgebras of Solomon's descent algebra

I construct new subalgebras of the descent algebra for symmetric and hyperoctahedral groups, and find their orthogonal idempotents.
• Subalgebras of Solomon's descent algebra based on alternating runs (2018), with Matthieu Josuat-Vergès, in J. Combin. Theory Ser. A