Mathematics

Affine Lehmer codes for the affine symmetric group at k=2. From this paper. One of the great things about combinatorics is that it affords many chances to make pretty pictures!

As a mathematician, I do a combination of research and teaching, with a decent helping of computer programming on the side.

Research

My research area is probably a good place to start, since it heavily informs the way that I teach and the things that I program.  I work in an area called Algebraic Combinatorics.  Combinatorics is the study of (usually) discrete objects like permutations, combinations, graphs, and posets.  It has a number of sub-studies; enumerative combinatorics, for example, usually asks questions along the lines of: “How many graphs are there with property X?” or “How many permutations are there with property Y?”

Algebra, on the other hand, studies algebraic structures like groups and rings and monoids.  Essentially, this means sets with some notion of multiplication on them.  For example, the set of permutations can be thought of as functions, and function composition then gives a multiplication rule.  Under this rule, the permutations form what we call a group.

Algebraic combinatorics, then,  uses algebraic structures to say things about combinatorics.  Often it also goes the other way, and uses combinatorics to say things about algebra.  And then, inevitably, geometry gets involved, too.  That’s what makes it a very interesting area: Many of the major branches of mathematics come together in the study of the relevant objects, so one gets an appreciation of a whole wide range of mathematical topics.  Sometimes ideas from geometry push things forward, sometimes the combinatorics leads to new ideas.  But it’s hard to be good at all of these things, so there’s a great deal of collaboration.  Which is great.

Probably my first real brush with good combinatorics was at the Budapest Semesters in Mathematics, a program in Hungary which I attended for a year in 2002.  It’s a fantastic program, where I learned an incredible amount of good maths.  In fact, I’m still pretty convinced that the perspectives I picked up in Budapest really helped me sail through my first year of grad school a few years later.  I finished my undergraduate at the University of Oregon.  Admittedly, I moved there because Eugene seemed like a cool place to live, but when I arrived it turned out they had some really good mathematicians, too!  In my extra time, I took great political science courses with Prof. Ken deBevoise and worked at the Center for Appropriate Transport.  I worked three days a week as a bike messenger and two days a week as a teacher in their high school program, probably one of the most awesome jobs I’ve had.

I finished my PhD in the summer of 2010, at the University of California, Davis.  I had two thesis advisers, Anne Schilling and Nicolas M. Thiéry.  Nicolas was a professor at Paris Sud, but ended up visiting Davis for a total of a year and a half while I was a grad student; we ended up doing some work together, and eventually I spent four months with Nicolas in France, as well.

My thesis, Excursions into Algebra and Combinatorics at q=0 is available on the Arxiv.  It has a few different parts to it, but especially focuses on the 0-Hecke algebra of the symmetric group and crystal bases.  Both of these are examples of q-deformations of a well-known object, where there are interesting phenomena that arise when q goes to 0.  For the 0-Hecke algebra, I found a formula for a system of orthogonal idempotents, which solved a problem that had been open for about 30 years.  (And not just because it was boring!)  I also worked with Anne and Nicolas and our friend Florent Hivert on the representation theory of J-trivial monoids, of which the 0-Hecke algebra is an example.  Both of these papers show up (more-or-less) as chapters in the thesis.  Then there’s another chapter on some fun constructions with affine permutations and the affine 0-Hecke algebra, and another on an interesting problem concerning crystal bases.  But those two chapters aren’t published as papers (yet?).

Both Anne and Nicolas are instrumental developers for the Sage-Combinat project, which is a part of the Sage mathematics software.  Sage is available for free, is open source, and kind of glues together all of the best free and open-source math software that’s available.  It’s really useful for my research, and every once in a while I manage to submit some code to the project, too.  And that’s a decent part of my programming on a day-to-day basis: using Sage for my research and then polishing parts of the code I write to go into main Sage.

In fact, one of the things that led to the 0-Hecke paper was being able to “look farther” using the computer than people had been able to compute before.  Once I was able to construct a system of 32 idempotents for the n=6 case, I was able to apply some elbow grease and conjecture a general formula.  Then I spent about six months proving the thing actually worked.

After graduating, I worked on a postdoc with Nantel Bergeron and Mike Zabrocki at York University and the Fields Institute.  Here I got really into the study of k-Schur functions, which eventually led to this paper.  Toronto was a great deal of fun, and it was great having the space to really dive into the k-Schur literature and try to wrap my head around things.

Teaching

It’s dangerous to do so, but I think I’ll leave the teaching section for later….  For now, I’ll just leave it to an excerpt from an interview with Prof. Ken deBevoise, on his teaching philosophy:

“It’s more a philosophy of learning than of teaching. I don’t believe that I can actually teach anyone anything significant. Learning is generated by the learner from the various inputs that he or she experiences in daily life, which can include the classroom. I can facilitate that, but it is a process internal to the learner rather than an external one. Real learning is not done effectively by means of a lecturer sending out arrows of “knowledge” that somehow transfer it to a listener. Learning from the inside is very hard work. Thus I believe in getting students to work really hard.  There’s no magic to it, nor is there a short cut. The key, of course, is to create courses as well as the classroom atmosphere that will help energize the learner so that he or she will see it as a challenge to be met eagerly.

“Further, I am not particularly interested in whether a student learns a lot about any particular subject. If someone becomes an expert on Afghanistan or whatever substance, that’s great. But my main concern is that through the process of studying that stuff, the student will develop and internalize certain traits and habits that are necessary for success in life – things like work ethic, dependability, prioritization skills, pride in the quality of one’s work, refusal to tolerate anything less than doing one’s best, and the ability to do the best work possible on a consistent and sustained basis. It’s a very old-school approach.”

 

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