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Algebra, combinatorics, category theory
My research interests include topics in noncommutative algebra, category theory and algebraic combinatorics, with Hopf algebras and their generalizations appearing prominently. A goal of my past work has been to build a conceptual framework for the study of Hopf algebraic structures in combinatorics and to clarify its significance to concrete applications. Part of my current work is devoted to enlarging the scope of classical Hopf-Lie theory. The new theory includes hyperplane arrangements and idempotent semigroups as part of the fundamental data.
Bimonoids for hyperplane arrangements (with Swapneel Mahajan) Encyclopedia of Mathematics and its Applications 173 (2020), 850 pp. Cambridge University Press.
Topics in hyperplane arrangements (with Swapneel Mahajan) Mathematical Surveys and Monographs of the AMS 226 (2017), 611 pp. American Mathematical Society.
Hopf monoids and generalized permutahedra (with Federico Ardila) To appear in Memoirs of the American Mathematical Society, 113 pp.
Monoidal functors, species and Hopf algebras (with Swapneel Mahajan) CRM-AMS Monograph Series 29 (2010), lii+784 pp. American Mathematical Society.