- Academic Writing
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.
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.
You can find a recent version of my curriculum vitae here.
- D.C. Ernst and N. Sieben. Impartial achievement and avoidance games for generating finite groups. Submitted to Int. J. Game Theory. [arXiv:1407.0784]
- D.C. Ernst. Diagram calculus for a type affine C Temperley-Lieb algebra, II. Submitted to J. Pure Appl. Alg. [arXiv:1101.4215]
- D.C. Ernst, A. Hodge, S. Yoshinobu. Doceamus: What Is Inquiry-Based Learning? Notices of the AMS 64(6), 2017. [ePrint]
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement games for generating generalized dihedral groups. Australas. J. Combin. 68(3), 371-384, 2017. [arXiv:1608.00259] [ePrint]
- D.C. Ernst, M. Hastings, and S. Salmon. Factorization of Temperley-Lieb diagrams. Involve 10(1), 89-108, 2017. [arXiv:1509.01241]
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial avoidance and achievement games for generating symmetric and alternating groups. Int. Electron. J. Algebra 20, 70-85, 2016. [arXiv:1508.03419] [ePrint]
- 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. Involve 9(4), 667-688, 2016. [arXiv:1503.08386]
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial avoidance games for generating finite groups. North-W. Eur. J. of Math. 2, 83-101, 2016. [arXiv:1506.07105] [ePrint]
- H. Denoncourt, D.C. Ernst, and D. Story. On the number of commutation classes of the longest element of the symmetric group. Open Problems in Mathematics 4, 2016. [arXiv:1602.08328] [ePrint]
- B. Beaudrie, D.C. Ernst, E. Kennedy, and R. St. Laurent. Inverted Pedagogy in Second Semester Calculus. PRIMUS 25(9-10), 992-906, 2015. [DOI:10.1080/10511970.2015.1031301]
- B. Love, A. Hodge, C. Corritore, and D.C. Ernst. Inquiry-Based Learning and the Flipped Classroom Model. PRIMUS 25(8), 745-762, 2015. [DOI:10.1080/10511970.2015.1046005]
- D.C. Ernst, M. Leingang, and R. Taylor. Facebook for Professional Educators: To Friend or Not to Friend? MAA FOCUS June/July 2015. [ePrint]
- 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]
- 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] [ePrint]
- D.C. Ernst and A. Hodge. Within \(\epsilon\) of Independence: An Attempt to Produce Independent Proof-Writers via IBL. In Beyond Lecture: Resources and Pedagogical Techniques for Enhancing the Teaching of Proof-Writing Across the Curriculum, R. Schwell, A. Steurer, & J.F. Vasquez (Eds.), MAA Notes, 2016.
- D.C. Ernst, A. Hodge, M. Jones, and S. Yoshinobu. The many faces of IBL. In STEM Education: An Overview of Contemporary Research, Trends, and Perspectives, E. Ostler (Ed.), 2015. Elkhorn, NE.
Conference Proceedings (Peer Reviewed)
- B. Beaudrie, D.C. Ernst, and B. Boschmans. Redesigning an Algebra for Precalculus Course. In Proceedings of World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education, T. Bastiaens & G. Marks (Eds.), 2013. Chesapeake, VA: AACE. [EdITLib]
- B. Beaudrie, B. Boschmans, and D.C. Ernst. First Semester Experiences in Implementing a Mathematics Emporium Model. In Proceedings of Society for Information Technology & Teacher Education International Conference, R. McBride & M. Searson (Eds.), 2013. Chesapeake, VA: AACE. [EdITLib]
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 Inquiry-Based Approach to Abstract Algebra. IBL course materials for an abstract algebra course that emphasizes visualization and incorporates technology. [Source]
- 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]
Online Columns and Blog Posts
- I am a co-editor/author for the online column Teaching Tidbits, which is sponsored by the Mathematical Association of America. Below are some of my posts.
- D.C. Ernst. Resources for active learning. Teaching Tidbits. To appear in Spring 2017.
- D.C. Ernst. Who generates the examples? Teaching Tidbits. Fall 2016. [Blog Post]
- I am a co-editor/author (joint with Angie Hodge) for Math Ed Matters, which is an online column sponsored by the MAA. The column explores topics and current events related to undergraduate mathematics education. Below is a selection of my posts.
- D.C. Ernst. Setting the Stage. Math Ed Matters. January 2015. [Blog Post]
- D.C. Ernst. The Twin Pillars of IBL. Math Ed Matters. January 2015. [Blog Post]
- D.C. Ernst. Fear is the mind-killer. Math Ed Matters. June 2014. [Blog Post]
- D.C. Ernst. Encouraging Students to Tinker. Math Ed Matters. August 2014. [Blog Post]
- D.C. Ernst, A. Hodge, and T.J. Hitchman. Engaging in Inquiry-Based Learning. Math Ed Matters. February 2014. [Blog Post]
- D.C. Ernst and A. Hodge. Math Ed Mania at the JMM. Math Ed Matters. January 2014. [Blog Post]
- D.C. Ernst and A. Hodge. The JMM: What’s Mathematics Education Got to Do with It? Math Ed Matters. December 2013. [Blog Post]
- D.C. Ernst. Give the Students the Colored Pen. Math Ed Matters. August 2013. [Blog Post]
- D.C. Ernst. Personality Matters? Math Ed Matters. July 2013. [Blog Post]
- D.C. Ernst. Grade School Utopia? Math Ed Matters. July 2013. [Blog Post]
- D.C. Ernst and A. Hodge. Try, Fail, Understand, Win. Math Ed Matters. June 2013. [Blog Post]
- D.C. Ernst. What the Heck Is IBL? Math Ed Matters. May 2013. [Blog Post]
- Teaching Calculus 1 with a Focus on Student Presentations. Discovering the Art of Mathematics. October 2015. [Blog Post]
- 4+1 interview with Dana Ernst. Casting Out Nines by R. Talbert. The Chronicle Blog Network. August 2013. [Blog Post]
- IBL Instructor Perspectives: Professor Dana Ernst. The IBL Blog by S. Yoshinobu. February 2012. [Blog Post]
- I also write sporadically on my personal blog. Topics focus on mathematics, teaching, and technology.
- 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]
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement games for generating nilpotent groups.
- D.C. Ernst and T.K. Petersen. The worst casino in Reno.
- D.C. Ernst and A. Lebovitz. Groups with cyclic subgroups.
- B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement games for generating finite groups.
- D.C. Ernst and B. Fox. Conjugacy classes of cyclically fully commutative elements in Coxeter groups of type $A$.
- D.C. Ernst and T. Laird. T-avoiding elements of Coxeter groups.