# research

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. You can read more about my current research interests here.

Furthermore, I am passionate about undergraduate mathematics education. Recently, I have become interested in inquiry-based learning and the Moore method for teaching mathematics. This educational paradigm has transformed my teaching.

## research and teaching statements

Below you will find PDF files for my CV, Teaching Statement, and Research Statements.

Here is a list of my current papers. To see the papers that I have posted on the arXiv, go here.

#### Publications

• D.C. Ernst. Diagram calculus for a type affine $C$ Temperley--Lieb algebra, I. J. Pure Appl. Alg. (to appear), 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. (to appear), 2011.
• D.C. Ernst. Non-cancellable elements in type affine $C$ Coxeter groups. Int. Electron. J. Algebra, 8:191-218, 2010. [arXiv:0910.0923]

#### Submitted/Preprints

• D.C. Ernst and A. Hodge. Within epsilon of independence: An attempt to produce independent proof-writers via an IBL approach in a real analysis course.
• D.C. Ernst. Diagram calculus for a type affine $C$ Temperley--Lieb algebra, II. [arXiv:1101.4215]

#### In Preparation

• D.C. Ernst, A. Hodge, A. Schultz. Collaborative peer review between two IBL number theory courses.
• J. Cormier, D.C. Ernst, Z. Goldenberg, J. Kelly, and C. Malbon. T-avoiding permutations in Coxeter groups of types $A$ and $B$.

#### Theses

• D.C. Ernst. A diagrammatic representation of an affine $C$ Temperley--Lieb algebra. PhD thesis, University of Colorado, 2008. [arXiv:0905.4457]
Advisor: Richard M. Green, University of Colorado
• D.C Ernst. Cell complexes for arrangements with group actions. Masters thesis, Northern Arizona University, 1997. [arXiv:0905.4434]
Advisor: Michael J. Falk, Northern Arizona University

#### Teaching Materials

• D.C. Ernst. IBL theorem sequence for an Introduction to Proof course. Available for free on GitHub.

## research with undergraduates

PSU students at the 2011 Hudson River Undergraduate Conference

The combinatorial nature of my research naturally lends itself to collaborations with undergraduates, and my goal is to incorporate students in my research as much as possible. I plan to recruit 1-3 students per academic year to investigate an original research project. If you are a PSU student interested in conducting research in mathematics, please come talk to me! Occasionally, there may be funding available to pay students to do research. Here is a list of my recent undergraduate research projects.

#### Fall 2011-Spring 2012

Exploration of T-avoiding elements in Coxeter groups of type $F$: Currently mentoring Ryan Cross, Katie Hills-Kimball, and Christie Quaranta on an original research project aimed at exploring the "T-avoiding" elements in Coxeter groups of type $F$. Students will present findings during at least one conference.

#### Fall 2010-Spring 2011

T-avoiding permutations in Coxeter groups of types $A$ and $B$: I mentored Joseph Cormier, Zachariah Goldenberg, Jessica Kelly, and Christopher Malbon on an original research project that classified of the "T-avoiding" permutations in Coxeter groups of types $A$ and $B$. The students made the following presentations:

We are currently in the progress of writing up our results with the intention of submitting an article for publication.

#### Spring 2010

Counting generators in Temperley--Lieb diagrams of types $A$ and $B$: Sarah Otis and Leal Rivanis obtained original results concerning Temperley--Lieb diagram algebras of types $A$ and $B$, which have a basis indexed by the fully commutative elements in Coxeter groups of types $A$ and $B$, respectively. We also obtained preliminary results in the type affine $C$ case. The students presented their work at the following conference:

## talks

Here are slides and abstracts for my recent talks.

## 2011 AMS special session

In the spring of 2011, Matt Macauley of Clemson University and I organized a special session on Combinatorics of Coxeter Groups at the 2011 AMS Spring Eastern Sectional Meeting on April 9-10, 2011 at the College of the Holy Cross in Worcester, MA. You can find additional information, included abstracts and slides for most of the talks by going here.

## collaborators

Here is a list of my recent collaborators.

• Richard M. Green, University of Colorado (Ph.D. advisor)
• Matt Macauley, Clemson University
• Angie Hodge, University of Nebraska at Omaha
• Andrew Schultz, Wellesley College
• Rob Beezer, University of Puget Sound (Sage collaborator)
• Ryan Crossugrad, Plymouth State University
• Katie Hills-Kimballugrad, Plymouth State University
• Christie Quarantaugrad, Plymouth State University
• Joseph Cormierugrad, Plymouth State University
• Zachariah Goldenbergugrad, Plymouth State University
• Jessica Kellyugrad, Plymouth State University
• Christopher Malbonugrad, Johns Hopkins University
• Sarah Otisugrad, (formerly) Plymouth State University
• Leal Rivanisgrad, University of New Hampshire
• Tom Boothbygrad, University of Washington
• Jeff Burkertugrad, Harvey Mudd
• Morgan Eichwaldugrad, (formerly) University of Montana