Research Interests

How can we tell if two social networks are different? How many ways are there to shuffle cards so cards are dealt in a certain order? How can clinical interns be assigned hospitals so both parties are satisfied? How are these questions answered precisely?

Solving these problems requires my counting abstract structures in concrete ways, and counting concrete structures in abstract ways, together known as algebraic combinatorics. 

Tech Specs

Initially I concentrated on descent algebras of Coxeter groups with a particular focus on their multiplicative structure, and the establishment of their basic algebraic structure and representation theory over fields of finite characteristic. Subsequently, I worked on Pieri operators on posets that link the diverse areas of Schubert calculus, combinatorics of polytopes, and P-partitions, amongst others. One particular discovery was the duality between peak functions and the cd-index. This led to studying ribbon Schur function equality, generalising this study to skew Schur function equality and applying the knowledge gleaned to the study of Schur positivity.
Most recently, I have been studying generalizations of Schur functions that reflect many of the properties displayed by classical Schur functions, such as Littlewood-Richardson rules, and their role in representation theory. I have also become interested in a generalization of the chromatic polynomial, known as the chromatic symmetric function.

Keywords and phrases combinatorial representation theory, graph theory, posets, symmetric functions and their generalizations, tableaux combinatorics.

Students in algebraic combinatorics

Those mentored in other subjects are not listed here.

Undergrad research: Kristin Shaw (2003, 2004, 2005), Chris Ryan (2005), Josh Fiddler (2006), Andrew Brown (2007), Farzin Barekat (2008), Jesseca Eng (2015), Ian Mak (2015), Foster Tom (2016), Adrian She (2017, 2018), Victor Wang (2020, 2021, 2022).

MSc: Matt Morin (2003-05), Vasu Tewari (2009-11), Caleb Cheek (2010-12), Foster Tom (2016-18), Ethan Shelburne (2021- ), Maria Esipova (2023- ).

PhD: Matt Morin (2005-2010), Vasu Tewari (2011-2015), Brian Chan (2016-2020).

Postdoc: Kevin Purbhoo (2006-07),  Eric Fusy (2008-09),  Kurt Luoto (2009-11),  Mahdad Khatirinejad (2010-11), Ed Richmond (2011-14), Julian Courtiel (2015-16), Samantha Dahlberg (2016-18), Orit Raz (2017-19), Farid Aliniaeifard (2019-)