Curtis D. Bennett

Partition Lattice q -Analogs Related to q -Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field Fq. Another member is constructed so as to mirror the partial order in the subspace lattice.

An Overview of the Scholarship of Teaching and Learning in Mathematics

Abstract This article provides an overview of the scholarship of teaching and learning (SoTL) in mathematics. It describes the origins of SoTL in higher education and distinguishes SoTL from good teaching, scholarly teaching, and mathematics education research. It includes a widely adopted taxonomy of SoTL questions and presents several examples of SoTL questions that have been investigated and made public. The heart of the article is a specific example of how a “teaching problem” can launch a SoTL investigation. The article also considers the value of SoTL to individual faculty, their departments, and their institutions. It closes with additional resources and suggestions for pursuing SoTL in mathematics.

A generalization of semimodular supersolvable lattices

Abstract Stanley (Algebra Universalis 2, 1972, 197–217) introduced the notion of a supersolvable lattice, L, in part to combinatorially explain the factorization of its characteristic polynomial over the integers when L is also semimodular. He did this by showing that the roots of the polynomial count certain sets of atoms of the lattice. In the present work we define an object called an atom decision tree. The class of semimodular lattices with atom decision trees strictly contains the class of supersolvable lattices, but their characteristic polynomials still factor for combinatorial reasons. We then apply this notion to prove the factorization of polynomials associated with various hyperplane arrangements having non-supersolvable lattices.

Partial orders generalizing the weak order on Coxeter groups

We define a new family of partial orders generalizing the weak order on Coxeter groups called T-orders, where T is a set of reflections determining the covers in this order. We show that the Grassmann and Lagrange orders on the Coxeter groups of type An and Bn introduced by Bergeron and Sottile are in fact T-orders. These partial orders were used to compute certain products in the cohomology ring of the flag manifolds associated to the complex Chevalley groups of these types. We exhibit T-orders generalizing these orders to partial orders for the Coxeter groups of type Dn, E6, and E7.

A simple definition for the universal Grassmannian order

In this paper, we provide a combinatorial definition of the Universal Grassmannian order (or the Grassmannian Bruhat order) of Bergeron and Sottile. This defines the order in terms of inversions, and thus the order can be viewed as a generalization of the weak order for Coxeter groups. Finally, we use this understanding of the order to analyze the generating function of the number of elements at rank n in this order.

Doing the Scholarship of Teaching and Learning in Mathematics

The Scholarship of Teaching and Learning (SoTL) movement encourages faculty to view teaching “problems” as invitations to conduct scholarly investigations. In this growing field of inquiry faculty bring their disciplinary knowledge and teaching experience to bear on questions of teaching and learning. They systematically gather evidence to develop and support their conclusions. The results are to be peer reviewed and made public for others to build on.

A remark on a theorem of J. Tits

Let G be a rank two Chevalley group and Γ be the corresponding Moufang polygon. J. Tits proved that G is the universal completion of the amalgam formed by three subgroups of G: the stabilizer P1 of a point a of Γ, the stabilizer P2 of a line ` incident with a, and the stabilizer N of an apartment A passing through a and `. We prove a slightly stronger result, in which the exact structure of N is not required. Our result can be used in conjunction with the “weak BN-pair” theorem of Delgado and Stellmacher in order to identify subgroups of finite groups generated by minimal parabolics.

Embeddings of Twin Trees

In this article we provide sufficient conditions for when a pair of trees having a semi-codistance function δ can be embedded in a twin tree with codistance function extending δ. We use these conditions to show that given a twin tree T of bidegree (d1,d2) there exists a twin tree of bidegree (e1,e2) containing T as a substructure as long as di≤ ei.

Twin trees and _{Λ}-gons

We define a natural generalization of generalized n-gons to the case of Λ-graphs (where Λ is a totally ordered abelian group and 0 < λ ∈ Λ). We term these objects λΛ-gons. We then show that twin trees as defined by Ronan and Tits can be viewed as (1, 0)Λ-gons, where Λ = Z × Z is ordered lexicographically. This allows us to then generalize twin trees to the case of Λ-trees. Finally, we give a free construction of λΛ-gons in the cases where Λ is discrete and has a subgroup of index 2 that does not contain the minimal element of Λ.

Effects of a Capstone Course on Future Teachers (and the Instructor): How a SoTL Project Changed a Career

In this chapter, I revisit my first scholarship of teaching and learning project as a 2000–2001 Carnegie Academy for the Scholarship of Teaching and Learning scholar. I describe my experience as a pure mathematician taking on a pedagogical research project and the effects of this project and of doing the scholarship of teaching and learning on my teaching and career. The project studied student development in a novel mathematics capstone course for future teachers. Student teams worked on semester-long mathematics research problems, while simultaneously completing a content-heavy course on how advanced mathematics informs the teaching of high school (and earlier) mathematical subjects. The course changed behaviors of the students by giving voice to students with prior negative classroom experiences. In addition, one student had a surprising change in attitude towards proof and its value to secondary mathematics teachers. Working through the context of the original study, I reflect on the effects of the course on the students and of the project on the next 15 years of my career.