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Akihiro Munemasa connects finite geometry, quadratic forms and algebraic combinatorics. His author-written notes The geometry of orthogonal groups over finite fields by Akihiro Munemasa develop geometric objects and associated graphs from quadratic forms over finite fields. The preface makes the purpose explicit: the geometry supplies nontrivial examples for studying graph symmetries.^[ref]

For a concrete comparison over ${\mathbb{F}}_3$, consider $Q_1(x,y)=xy$ and $Q_2(x,y)=x^2+y^2$. The first vanishes on both coordinate axes, giving nonzero isotropic vectors. The second vanishes only at $(0,0)$: a nonzero square is $1$, and $1+1=2\ne 0$ in ${\mathbb{F}}_3$. Thus equal-dimensional nondegenerate quadratic forms can have different Witt indices, here one and zero. The existence or absence of totally isotropic subspaces changes the incidence geometry built from the form. Munemasa's notes turn these algebraic distinctions into explicit geometric and combinatorial constructions.^[ref]

Igor Pak works in discrete mathematics, including enumerative combinatorics, algebraic combinatorics, polyhedral geometry, and probability and computation on groups. His research page presents these as interconnected subjects and lists lectures on convex polytopes and combinatorial inequalities.^[ref] A common theme is to explain a numerical counting identity through the structure of the objects being counted, such as a bijection, rather than relying only on algebraic manipulation. Polyhedral methods add geometric constraints to those discrete enumeration questions.

Homepage: www.math.ucla.edu/~pak/

John R. Stembridge works on algebraic combinatorics, representation theory, Coxeter groups and symmetric functions. His Michigan profile also identifies computational methods in algebra as a focus.^[ref]

With Richard P. Stanley, he studied immanants of matrices associated with the Jacobi–Trudi identity. An immanant replaces the sign character in a determinant by another symmetric group character. Their paper proves special cases of stronger positivity conjectures and connects the calculation with cycle structure of permutations having restricted positions.^[ref]

Richard P. Stanley works in enumerative combinatorics and algebraic combinatorics, including connections among partially ordered sets, generating functions and symmetric functions. His Enumerative Combinatorics, Volume 1 (second-edition draft) by Richard Stanley develops systematic methods for counting combinatorial structures. His later Catalan-number monograph organizes many different counting interpretations of the Catalan numbers, showing how the same sequence can encode different discrete objects.^[ref][ref]

Homepage: math.mit.edu/~rstan/