My primary research interests are in the interplay between combinatorics and algebraic structures. More specifically, I study the combinatorics of Coxeter groups and their associated Hecke algebras, Kazhdan-Lusztig theory, generalized Temperley-Lieb algebras, diagram algebras, and heaps of pieces. By employing combinatorial tools such as diagram algebras and heaps of pieces, one can gain insight into algebraic structures associated to Coxeter groups, and, conversely, the corresponding structure theory can often lead to surprising combinatorial results.
More recently, my research has expanded into combinatorial game theory (joint with Nandor Sieben and Bret Benesh). In particular, our research has focused on avoidance and achievement games involving finite groups.
The combinatorial nature of my research naturally lends itself to collaborations with undergraduate students, and my goal is to incorporate undergraduates in my research as much as possible. If you are an NAU student interested in conducting research in mathematics, please come talk to me! Occasionally, there may be funding available to pay students to do research.
My interests also include the scholarship of teaching and learning (SoTL) with a focus on inquiry-based learning (IBL) as an approach to teaching/exploring mathematics. I am currently a Special Projects Coordinator for the Academy of Inquiry-Based Learning and a mentor for several new IBL practitioners. Moreover, I actively give talks and organize workshops on the benefits of IBL as well as the nuts and bolts of how to implement this approach in the mathematics classroom.
Below is a list of my published articles and theses.
- B. Love, A. Hodge, C. Corritore, and D.C. Ernst. Inquiry-Based Learning and the Flipped Classroom Model. Accepted to PRIMUS.
- D.C. Ernst, M. Leingang, and R. Taylor. Facebook for Professional Educators: To Friend or Not to Friend? MAA FOCUS, June/July 2015, pages 6-7. [ePrint]
- B. Beaudrie, D.C. Ernst, E. Kennedy, and R. St. Laurent. Inverted Pedagogy in Second Semester Calculus. Accepted to PRIMUS.
- D.C. Ernst, A. Hodge, M. Jones, and S. Yoshinobu. The Many Faces of IBL. Accepted as book chapter.
- D.C. Ernst, A. Hodge, and A. Schultz. Enhancing Proof Writing via Cross-Institutional Peer Review. PRIMUS 25(2), 121-130, 2015. [DOI:10.1080/10511970.2014.921652]
- B. Beaudrie, D.C. Ernst, and B. Boschmans. Redesigning an Algebra for Precalculus Course. In T. Bastiaens & G. Marks (Eds.), Proceedings of World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education, 2013. Chesapeake, VA: AACE. [EdITLib]
- B. Beaudrie, B. Boschmans, and D.C. Ernst. First Semester Experiences in Implementing a Mathematics Emporium Model. In R. McBride & M. Searson (Eds.), Proceedings of Society for Information Technology & Teacher Education International Conference, 2013. Chesapeake, VA: AACE. [EdITLib]
- D.C. Ernst. Diagram calculus for a type affine $C$ Temperley-Lieb algebra, I. J. Pure Appl. Alg. 216(11), 2012. [arXiv:0910.0925]
- T. Boothby, J. Burkert, M. Eichwald, D.C. Ernst, R.M. Green, and M. Macauley. On the Cyclically Fully Commutative Elements of Coxeter Groups. J. Algebraic Combin. 36(1), 2012. [arXiv:1202.6657]
- D.C. Ernst. Non-cancellable elements in type affine $C$ Coxeter groups, Int. Electron. J. Algebra 8, 2010. [arXiv:0910.0923]
- D.C. Ernst. A diagrammatic representation of an affine $C$ Temperley–Lieb algebra, PhD Thesis, University of Colorado, 2008. [arXiv:0905.4457]
- D.C. Ernst. Cell Complexes for Arrangements with Group Actions, MS Thesis, Northern Arizona University, 2000. [arXiv:0905.4434]
Below is a list of papers that have been submitted, but not yet accepted for publication.
- N. Diefenderfer, D.C. Ernst, M. Hastings, L.N. Heath, H. Prawzinsky, B. Preston, J. Rushall, E. White, A. Whittemore. Prime Vertex Labelings of Several Families of Graphs. Submitted to Involve. [arXiv:1503.08386]
- D.C. Ernst and N. Sieben. Impartial achievement and avoidance games for generating finite groups. Submitted to International Journal of Game Theory. [arXiv:1407.0784]
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial avoidance games for generating finite groups. Submitted to Glasgow Mathematical Journal. [arXiv:1506.07105]
- D.C. Ernst and A. Hodge. Within $\epsilon$ of Independence: An Attempt to Produce Independent Proof-Writers via IBL.
- D.C. Ernst. Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II. [arXiv:1101.4215]
The following is a list of papers that are in progress.
- D.C. Ernst and T. Laird. T-avoiding elements of Coxeter groups.
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial avoidance and achievement games for generating symmetric and alternating groups.
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement games for generating finite groups.
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement and avoidance games for generating generalized dihedral groups.
- D.C. Ernst, M. Hastings, and S. Salmon. Factorization of Temperley-Lieb diagrams.
- D.C. Ernst and D. Jacobson. Optimizing Solutions of $2\times 2$ Spinpossible boards.
Open-Source Course Materials
Below is a list of course materials that I have written to be used with an inquiry-based learning (IBL) approach.
- D.C. Ernst. An Introduction to Proof via Inquiry-Based Learning. IBL course materials for an introduction to proof course. The first half of the notes are an adaptation of notes written by Stan Yoshinobu and Matthew Jones. [Source on GitHub]
- D.C. Ernst. An Inquiry-Based Approach to Abstract Algebra. IBL course materials for an abstract algebra course that emphasizes visualization and incorporates technology. Very much a work in progress. [Source on GitHub]
- D.C. Ernst, N.Sieben. MAT 220 Problem Collection. MAT 220 is an introductory course in mathematical reasoning in multi-step problems across different areas of mathematics. The goal is to use elementary mathematical tools to solve more complex problems in already familiar areas of study instead of teaching new mathematical tools that are used in straightforward one-step exercises. The focus is on problem solving and solution writing.
In addition to the posts that I write for my personal blog, Angie Hodge and I are coauthors for Math Ed Matters, which is a (roughly) monthly column sponsored by the Mathematical Association of America. The column explores topics and current events related to undergraduate mathematics education. Posts aim to inspire, provoke deep thought, and provide ideas for the mathematics classroom. Our interest in and engagement with IBL color the column’s content.
Here are some online interviews that I have given.