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On Tuesday, February 5 (my birthday!), I gave a talk titled “A diagrammatic representation of the Temperley-Lieb algebra” in the NAU Department of Mathematics and Statistics Colloquium. Here is the abstract.

One aspect of my research involves trying to prove that certain associative algebras can be faithfully represented using “diagrams.” These diagrammatic representations are not only nice to look at, but they also help us recognize things about the original algebra that we may not otherwise have noticed. In this talk, we will introduce the diagram calculus for the Temperley-Lieb algebra of type $A$. This algebra, invented by Temperley and Lieb in 1971, is a certain finite dimensional associative algebra that arose in the context of statistical mechanics in physics. We will show that this algebra has dimension equal to the nth Catalan number and discuss its relationship to the symmetric group. If time permits, we will also briefly discuss the diagrammatic representation of the Temperley-Lieb algebra of type affine $C$.

And here are the slides.

Despite the fact that this was a 50-minute talk, it was intended to be an overview of one aspect of a long and complex story. The subject matter is intimately related to my PhD thesis, as well as a series of papers that I have written.

  • Ernst, D. C. (2010). Non-cancellable elements in type affine $C$ Coxeter groups. Int. Electron. J. Algebra, 8, 191–218. [arXiv]
  • Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley-Lieb algebra, I. J. Pure Appl. Alg. (to appear). [arXiv]
  • Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II. [arXiv]

In addition, there is (at least) a part III that goes with the last two papers that is in progress.

On Friday, February 1, I gave a 30-minute talk titled “The Stargate Switch” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for FAMUS is undergraduates. Here is the abstract for my talk.

An episode of Stargate SG-1 features a two-body mind-switching machine which will not work more than once on the same pair of bodies. The plot centers around two disjoint pairs of individuals who swap minds but subsequently wish they could reverse the process. We will discuss the mathematics behind the solution to this problem, as well as some generalizations.

This was my third FAMUS talk of the academic year, but my first of the semester. My first FAMUS talk was titled On an open problem of the symmetric group and my second talk was titled Euler’s characteristic, soccer balls, and golf balls.

In the 1999 episode “Holiday” from season 2 of Stargate SG-1, the character Ma’chello tricks Daniel into swapping minds with him. In an attempt to save Daniel, Jack and Teal’c accidentally swap minds, after which they then discover a limitation: the machine will not work more than once on the same pair of bodies. Physicist Samantha Carter saves the day by improvising a sequence of 4 switches that brings everyone back to normal.

In two recent papers, R. Evans and L. Huang from the University of California, San Diego, study the mathematics behind the Stargate Switch Problem, and generalize the solution to permutations involving $m$ pairs of bodies.

  • Evans, R., & Huang, L. (2012). The Stargate Switch. [arXiv]
  • Evans, R., & Huang, L. (2012). Mind switches in Futurama and Stargate. [arXiv]

The limitation of the machine in Stargate SG-1 is the same as the one suffered by the mind-switching machine in Futurama’s 2010 episode “The Prisoner of Benda”. However, the Stargate Switch, and Evans and Huang’s generalization, is a special case of the problem posed in Futurama. In the general case, we need to introduce two outsiders to solve the problem. But the Stargate Switch can be solved without the addition of outsiders (as long as we have at least 2 pairs of bodies).

The solution to the dilemma in Futurama is known as the Futurama Theorem and was proved by Ken Keeler, who is one of the show’s writers and has a PhD in applied mathematics. I’ve given a few talks about the Futurama Theorem and if you want to know more, check out my recent blog post located here.

As with my previous talks about the Futurama Theorem, I utilized deck.js to create HTML slides and I used MathJax to typeset all of mathematical notation. One advantage of this approach is that it allows you to view the slides directly in your web browser. You can view the slides by going here. To advance the slides, just use your arrow keys. Also, you can get an overview of the slides by typing “m” or go to a specific slide by typing “g”.

Several weeks ago I was asked to take part in the Project NExT Alternative Assessment Techniques panel discussion at the 2013 Joint Mathematics Meetings, which recently took place in San Diego, CA. I was extremely honored to be considered for the panel, but at the time I was not planning on attending the JMM, so I declined the invitation. A couple weeks later, it turned out that I was going to make it to the JMM after all. At about 11PM the night before I was going to fly to San Diego, I received an email from the organizers of the panel discussion indicating that one of the panelists was unable to make it and that they heard was going to be there. They asked if I could fill in at the last minute and I accepted.

Here is the abstract for the panel.

Since classroom assessment is used to determine a student’s level of mastery, how can we vary our methods of assessment to accurately reflect the diversity of ways that students learn and understand the material? Traditional methods of assessment, such as exams, quizzes, and homework, may not accurately and robustly measure some students’ understanding. In this panel, we will propose alternative methods and discuss the following questions:
– What assessments exist besides the traditional ones and how can I use them for my course?
– How can I determine the validity of an alternative assessment?
– How can I develop my own alternative assessments?
– How can alternative assessments help me evaluate the effectiveness of a non-traditional classroom?

It is worth pointing out that I’m not an assessment expert by any stretch of the imagination. Also, given that I had less than 48 hours to prepare amidst a pretty full schedule, I didn’t have a lot of time to come up with something new and creative for my talk. Inquiry-based learning (IBL) is one of my passions and I’ve given quite a few IBL-related talks in the past few months, so I decided to “twist” the ideas from some of my recent talks into a talk about assessment. In my talk, I propose implementing IBL not only as a pedagogical approach but also as an assessment strategy. This isn’t really a stretch since in my view, an effective IBL class is all assessment, all the time.

My fellow panelists included Theron Hitchman (University of Northern Iowa), Bonnie Gold (Monmouth University), and Victor Odafe (Bowling Green State University). Theron gave a talk on using Standards Based Assessment (you can find his slides here), Bonnie spoke on a variety of summative assessment techniques, and Victor shared his experience with oral assessment. It turns out that the person that I was filling for is mathematics education superstar Jo Boaler. Me filling in for her is ridiculous.

Here are the slides for my portion of the panel.

Thanks to the organizers of the panel (Cassie Williams (James Madison University), Jane Butterfield (University of Minnesota), John Peter (Utica College), and Robert Campbell (College of Saint Benedict and Saint John’s University)) for providing me with the opportunity to speak on the panel.

Image taken from http://theinfosphere.org/

On Tuesday, November 6, 2012 I gave a talk titled “The Futurama Theorem and some refinements” in the NAU Department of Mathematics and Statistics Colloquium. This was the fourth time that I’ve given a talk about the Futurama Theorem (also known as Keeler’s Theorem).

The Futurama Theorem is a theorem about the symmetric group that was developed for and proved in the episode “The Prisoner of Benda” for the TV show Futurama. The theorem was proved by show writer Ken Keeler, who has a PhD in applied mathematics from Harvard. During the episode, Professor Farnsworth and Amy invent a mind swapping machine and after swapping minds, they realize that the machine cannot be used on the same pair of bodies again. After several characters swap minds, they are confronted with the problem of putting everyone’s mind back where it belongs. The Futurama Theorem proves that regardless of how many mind swaps have been made, all minds can be restored to their original bodies using only two extra people.

As with my previous talks about the Futurama Theorem, I utilized deck.js to create HTML slides and I used MathJax to typeset all of mathematical notation. One advantage of this approach is that it allows you to view the slides directly in your web browser. You can view the slides by going here. To advance the slides, just use your arrow keys. Also, you can get an overview of the slides by typing “m” or go to a specific slide by typing “g”.

This most recent talk about the Futurama Theorem was very similar to previous versions. However, I did add a few new slides. In particular, I included a summary of some recent results by Evans, Huang, and Nguyen (University of California, San Diego) that provides some refinements of Keeler’s theorem. Their paper will appear in the American Mathematical Monthly, but you can also find it on the arXiv. Here is the abstract for the paper:

An episode of Futurama features a two-body mind-switching machine which will not work more than once on the same pair of bodies. After the Futurama community engages in a mind-switching spree, the question is asked, “Can the switching be undone so as to restore all minds to their original bodies?” Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two “outsiders.” We refine Keeler’s result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences of switches, each sequence effecting a cyclic mind-scrambling permutation in the symmetric group $S_n$. Finally, we give necessary and sufficient conditions on $m$ and $n$ for the identity permutation to be expressible as a product of m distinct transpositions in $S_n$.

If you are interested in comparing, you can find the slides for my previous talks about the Futurama Theorem by following the links below:

A couple hours before my talk, one of my colleagues asked if I minded if his son watched my talk via Skype. I suggested we try a Google+ Hangout and open it to anyone that was interested in watching. Having never done this before, I wasn’t exactly sure how it would work out. I sent out a quick announcement about the Hangout via G+, Twitter, and Facebook. The plan was to record the Hangout using Hangouts on Air, but we couldn’t get this set up in time. It looks like 13 people stopped by the Hangout, but it appears some of them were random Hangout lurkers. Also, it looks like there was an issue with the audio that we didn’t know about until the end.

The feedback that I received after the talk was that it went well, but I felt like I wasn’t at my best. The video camera made me more nervous than I anticipated, and trying to remember to stand where the video could see me was distracting. In addition, the light from the projector felt like a furnace.

Anyway, the Hangout was a cool experience and next time I’ll plan things out a little better. For example, as Vincent Knight suggested, instead of hosting a public Hangout, it would be better to have a G+ Circle of people that are interested in viewing and just open the Hangout to them. Thanks to Richard Green, Hugh Denoncourt, Luis Guzman and Barbara Boschmans Beaudrie for stopping by the Hangout.

On Friday, November 2, I gave a 30-minute talk titled “Euler’s characteristic, soccer balls, and golf balls” during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). This was my second FAMUS talk of the semester. You can read more about my first talk by going here. As the name of the seminar suggests, the target audience for the FAMUS talks are undergraduates. My last talk was well-received, but I wanted to discuss something a little “lighter.” Here is the abstract for my talk.

A typical soccer ball consists of 12 regular pentagons and 20 regular hexagons. There are also several golf balls on the market that have a mixture of pentagonal and hexagonal dimples. Both situations are examples of convex polyhedra. Loosely speaking, a polyhedron is a geometric solid in three dimensions with flat faces and straight edges. In this case, the faces are pentagons and hexagons. The adjective convex refers to the fact that a line segment joining any two points of the solid lies entirely inside or on the surface of the solid. For mathematicians, a natural question arises. Namely, what sorts of convex polyhedra can we build using only regular pentagons and regular hexagons? For example, is it possible to build a convex polyhedron using only regular pentagons? How about just hexagons? If we allow both, how many of each are possible? In this talk, we will explore these types questions by utilizing Euler’s characteristic formula for polyhedra, which establishes a relationship between the number of vertices, edges, and faces of a polyhedron.

And here are the slides.

I was inspired to start thinking about this topic while I was coaching my 4-year old’s soccer team this fall. As those of you with knowledge of the subject know, there’s so much more I could have said. However, 30 minutes isn’t a lot of time and I wanted to make sure that I discussed the proof of Euler’s characteristic slowly enough so that most of the audience could follow it.

I am currently overloaded with work, so I was planning to do a chalk talk and skip making any slides. However, the morning of the talk, I decided to make a couple slides that included pictures. Of course, as soon as I started dropping in images, I found myself adding text and before I knew it, I had slides for most of my talk. Only a few things didn’t make it in the slides. In particular, when I discussed the proof of Euler’s characteristic formula, I drew lots of pictures on the chalk board.

As a special treat, my mom, my mom’s husband, and both my sons were in the audience. I’m pretty sure this is the first time that my mom had ever seen me give a talk before and definitely the first time my kids had been to a math talk of any kind. It is customary for the FAMUS host, Jeff Rushall, to interview a faculty member after the talk. Typically, the speaker and the faculty member to be interviewed are not the same person, but this time, I did both. After Jeff asked his usual list of questions, the audience was allowed to ask me questions. It was a lot of fun.

Note: It just occurred to me that I cheated a little bit in a couple spots during my talk. See if you can figure out where.

Yesterday, I was part of a panel discussion about inquiry-based learning (IBL) at the Fall 2012 Indiana MAA Section Meeting. The other panelists included Robert Talbert (Grand Valley State University) and Mindi Capaldi (Valparaiso University). The panel was organized by the Indiana Section NExT, which is the Indiana version of the national Project NExT. Here is the abstract for the session:

We will discuss inquiry based learning, inverted classroom models, peer instruction, and other alternatives to lecture-based instruction. Panelists will give a brief intro of their experience in these areas, followed by an extended time of Q&A with the audience. This panel is open to all meeting participants.

You might be wondering how I ended up at the Indiana MAA Section Meeting. One of my Project NExT fellows, Lara Pudwell (Valparaiso University), sent me a message several weeks ago asking if I knew anyone near Indiana that would be interested in speaking on a panel about inquiry-based learning. I told her that I wasn’t anywhere near Indiana, but that I would love to be a part of the panel. Since I’m not swimming in travel money, I contacted Stan Yoshinobu (Cal Poly and Director of the Academy of Inquiry Based Learning) to see if the Visiting Speakers Bureau might be able to pay my way. Thankfully, my request for travel funding was approved. Woot! I’d like to thank AIBL and the Educational Advancement Foundation for funding these sorts of things.

The panel discussion was well-attended and it seemed to go very well. Each of the three panelist spoke for about 5-10 minutes and then the floor was opened to questions. The questions (during the session and later at lunch) covered a variety of topics, but as expected, people were interested in how to implement IBL in large classes and/or courses where coverage of a significant amount of content is a requirement. In my opinion, these are two of the biggest obstacles to adopting all sorts of effective and progressive teaching approaches. The obstacles are not insurmountable, but modifications (and compromises) of how I might run my upper-level proof-based classes must be made. I’ll try to write a post that addresses some potential strategies for dealing with large classes and the coverage issue.

Here are the slides for my portion of the panel discussion.

On Tuesday, October 16, Amy Rushall and I co-facilitated a Faculty Development Workshop at Northern Arizona University. The title of our workshop was “Designing Inquiry-Based Learning Experiences.” Here is the abstract for our session.

What is inquiry-based learning (IBL)? Why use IBL? How can you incorporate more IBL into the classes that you teach? In this talk, we will address all of these questions, as well as discuss a few different examples of what an IBL classroom might look like in practice.

The participants in the workshop, which included a cohort of visiting Chinese scholars, represented a wide variety of disciplines including math, chemistry, computer science, hotel and restaurant management, business, and more.

The purpose of my portion of the workshop was to introduce inquiry-based learning (IBL) in general terms and provide motivation for why teachers should consider implementing IBL in their classrooms. Below are my slides for the workshop.

Note: These slides are essentially a subset of slides for similar talks that I’ve given recently. In particular, see this blog post and this one.

Amy’s part of the workshop addressed more specific ways in which one might apply IBL techniques.

On Friday, I gave a talk titled “Inquiry-Based Learning: What, Why, and How?” at the ArizMATYC conference, which took place at Yavapai College in Prescott, AZ. ArizMATYC is the Arizona chapter of the American Mathematical Association of Two Year Colleges (AMATYC). I gave a nearly identical talk (with the same title) at the Mathematics Instructional Colloquium at the University of Arizona on October 2, 2012. Unlike last time, I did not get that many questions during my talk. It seemed well-received nonetheless. Here’s the abstract for the talk:

What is inquiry-based learning (IBL)? Why use IBL? How can you incorporate more IBL into the classes that you teach? In this talk, we will address all of these questions, as well as discuss a few different examples of what an IBL classroom might look like in practice.

And here are the slides.

Tomorrow, I will be co-facilitating a Faculty Development Workshop here at Northern Arizona University with Amy Rushall about implementing IBL. My portion of the workshop will include content from the slides above.

Yesterday, I gave a talk titled “Inquiry Based Learning: What, Why, and How?” at the University of Arizona Mathematics Instructional Colloquium. There weren’t too many people in attendance (about a dozen), but the talk seemed to be very well-received. Here’s the abstract for the talk:

What is inquiry-based learning (IBL)? Why use IBL? How can you incorporate more IBL into the classes that you teach? In this talk, we will address all of these questions, as well as discuss a few different examples of what an IBL classroom might look like in practice.

And here are the slides.

I knew that I wouldn’t get through all of the material, but I had enough prepared so that I could take the talk where the audience wanted to go. Surprisingly, I was able to discuss the content on all but the last two slides. I plan to give a similar talk at the 2012 ArizMATYC conference, which takes place at Yavapai College in Prescott, AZ.

Prior to yesterday’s talk, I was able to squeeze in a bike ride up Mount Lemmon.

On Friday, September 12, 2012, I gave a 25 minute talk titled “An open problem of the symmetric group” during NAU’s Friday Afternoon Mathematics Undergraduate Seminar (FAMUS). Here is the open problem that I discussed.

How many commutation classes does the longest element in the symmetric group have?

The main goal of the talk was to understand what this question is asking. The secondary goal was to illustrate that mathematics is a lively field with open questions and to provide an example of what research in mathematics looks like. Here’s the abstract.

Many people are often surprised to hear that mathematicians do research. What is mathematical research? Research in mathematics takes many forms, but one common theme is that the research seeks to answer an open question concerning some collection of mathematical objects. The goal of this talk will be to introduce you to one of the many open questions in mathematics: how many commutation classes does the longest element in the symmetric group have? This problem has been nicknamed “Heroin Hero” by my advisor in honor of a game from the TV show “South Park” in which the character Stan obsesses over chasing a dragon. We will review the basics of the symmetric group and introduce all of the necessary terminology, so that we can understand this problem.

Here are the slides.