First, my grad student, Kirsten Davis successfully defended her MS thesis titled "A cellular quotient of the Temperley-Lieb algebra of type D." I'm really proud of Kirsten and what she has produced. This is my first grad student, so I'm extra stoked! Thanks to Janet McShane and for serving on the committee. I'll have a another student defending in a couple weeks.
Second, , Brian Beaudrie, Roy St. Laurent, and I submitted a paper today on a study involving "flipping" second semester calculus.
A regular octagon has sixteen distance preserving symmetries. Eight of the symmetries are rotations, and the other eight are reflections.
This graphic by Jim Belk illustrates the symmetries of a regular octagon. It is not possible to illustrate symmetry in the way one would advertise a weight loss programme, because the “before" and “after” pictures would look exactly the same. However, if the word STOP is written on the octagon in order to break the symmetry, the result of applying a reflection or a rotation becomes obvious.
The top row of eight octagons shows the result of rotating the standard STOP sign anticlockwise (=counterclockwise) by multiples of 45°. The leftmost octagon in the second row is obtained from the standard STOP sign by reflecting in the horizontal axis. (It may be helpful to imagine that the sign is transparent, so that the word can be read from either side.) The rest of the second row is obtained by rotating the reflected version of the sign anticlockwise by multiples of 45°. This collection of sixteen signs illustrates all the symmetries of a regular octagon.
The inverted versions of the sign in the second row can all be obtained from the standard STOP sign by using a single reflection, rather than a combination of reflections and rotations. This is because as well as reflecting a regular octagon about the horizontal axis, one can reflect it (a) about any axis joining the midpoints of two opposite sides of the octagon or (b) about any axis joining two opposite corners of the octagon. There are four ways to do (a), and four ways to do (b), and applying these eight reflections to the standard STOP sign will produce the eight inverted signs in the second row.
Group theory is the branch of mathematics that deals with descriptions of symmetry like this one. A group can be thought of as a set of transformations that can be combined in various ways. The group that describes the symmetries of a regular octagon is called the dihedral group of order 16.
The transformations r and s, which are shown in the graphic, are two important elements of the dihedral group of order 16. The transformation r is “rotation anticlockwise by 45°”, and the transformation s is “reflection in the horizontal axis”. Each of these transformations can be inverted: r can be inverted by rotating clockwise by 45°, and s can be inverted simply by applying s again. The inverse transformation of r is denoted by r to the power –1.
The four stop signs at the bottom of the graphic show that the net effect of applying r followed by s is the same as the net effect of applying s followed by the inverse of r. However, the order in which the transformations are applied matters: applying s followed by r would have produced a different overall effect. Because the order in which transformations are applied is sometimes significant, mathematicians call the dihedral group of order 16 a nonabelian group. Because all of the symmetries can be obtained as combinations of the transformations r, s and their inverses, mathematicians say that the group is generated by r and s.
Picture credit: Jim Belk is an assistant professor at Bard College who has produced many illustrations for Wikipedia. You can see a gallery of his Wikipedia contributions here (http://en.wikipedia.org/wiki/User:Jim.belk/Image_Gallery).
The top part of this graphic appears on the Wikipedia entry on the dihedral group (http://en.wikipedia.org/wiki/Dihedral_group). The bottom part of the graphic is also on Wikipedia but is not currently used in an article there.
- Northern Arizona UniversityAssistant Professor, 2012 - present
- University of Colorado at BoulderGraduate Student/Teaching Assistant, 2003 - 2008
- Front Range Community CollegeMath Faculty, 2001 - 2003
- Northern Arizona UniversityGraduate Student/Instructor, 1997 - 2001
- Plymouth State UniversityAssistant Professor, 2008 - 2012
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. 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. See my scholarship page for more information.
Furthermore, I am passionate about mathematics education. In particular, I am interested in inquiry-based learning (IBL) and the Moore method for teaching mathematics. This educational paradigm has transformed my teaching. 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.
I am also interested in utilizing technology to enhance the teaching and learning of mathematics. Specifically, I choose free and open-source software and technologies when appropriate. For example, I have been incorporating Sage and GeoGebra into my teaching. Sage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface. For examples of a few of the cool things you can do with Sage, check this page. According to their webpage, GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. There are tons of awesome GeoGebra examples located here.
In addition to using free and open-source software, I am inspired by the recent open-source textbookmovement and I strongly believe that educators should choose free, open-source, or low cost textbooks when a viable alternative exists. For a list of open-source textbooks, go here and here.
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—and mathematics education—classroom. Our interest in and engagement with IBL color the column's content.
I also maintain a personal blog, which is part of the Booles' Rings network of academic home pages/blogs. On my blog, I typically write about topics related to mathematics, education, and technology. In addition, I occasionally post about my cycling, trailing running, and rock climbing adventures on my Elevation Gain blog.
Lastly, I am a husband and a father of two incredible sons. Oh, I enjoy drinking copious amounts of coffee, too.
- George Mason UniversityBS, Mathematics, 1993 - 1997
- Northern Arizona UniversityMS, Mathematics, 1997 - 2000
- University of Colorado at BoulderPhD, Mathematics, 2003 - 2008