Lessons from a first-year seminar 2014-02-10

Last term, I had the opportunity to run an interdisciplinary first-year seminar as part of Carleton's A&I program. The course I designed, “The Mathematics of Democracy”, focused on topics at the intersection of mathematics and political theory: social choice (the theory of voting mechanisms), representative apportionment, fair division, and such. Running this course was an incredibly rewarding experience for me, and generally seemed to be a positive one for my students as well.

Here's the nuts and bolts of how I planned the course, what I did during the term, and what I learned about how to make these courses successful (and how not to!).

Inverting the classroom, followup 2013-12-13

In a recent post, I wrote at length about my experiences inverting a section of Calculus I at Carleton College. Now that the term is over and I've had some time to process my experiences and the students' end-of-term evaluations, I want to follow up with some of my thoughts about what went well, what can be improved, and whether I'd do it again.

On interacting with LaTeX 2013-12-12

Warning: this post contains a rant.

Next term, I'll be running a workshop to help Carleton students get started using LaTeX. As part of this, I'm writing up guides for new users, designed to carry students from a state of zero knowledge to their first compiled document. Of course, this requires a functioning LaTeX environment, so I spent some time with the various distributions and editor/environments installed on the lab computers in the department.

After less than half an hour, I gave up and moved the whole project over to WriteLaTeX. The standard desktop LaTeX editing suites are an embarrassing, unusable mess.

What is a combinatorial species? II 2013-12-07

Last time, we saw how the language of category theory could inform our understanding of combinatorial structures. Specifically, we considered the "species of graphs" $\specname{G}$, which turns a set $A$ of labels into the set $\specname{G} [A]$ of graphs with vertex set $A$ and transforms a set bijection $\sigma: A \to B$ into the map $\specname{G} [\sigma]: \specname{G} [A] \to \specname{G} [B]$ which sends each graph to its relabeling under $\sigma$. Conceptually, this shifts our attention from individual graphs to the process by which graphs are assembled out of their vertex sets.

This time, I hope to explain why we might do such a crazy thing. The punchline is that, amazingly enough, this will actually help us count classes of graphs that would otherwise be totally inaccessible. To get there, though, we have some work to do.

What is a combinatorial species? I 2013-12-05

In enumerative combinatorics, and especially in enumerative graph theory, a rich array of high-tech tools have been developed to use algebra to count combinatorial structures. (If you aren't already familiar with the astounding feats that can be accomplished with generating functions, I highly recommend Wilf's excellent (and excellently named!) treatise generatingfunctionology. You will be amazed.)

Curiously, though, we find over and over again that, for a given class of combinatorial structures, the labeled and unlabeled counting problems—seemingly so similar in structure—in fact require wildly different techniques. (Indeed, in many cases, the unlabeled problems remained open for decades after the labeled problems were solved!) Perhaps even more unsettlingly, sometimes the same techniques do work on both sides of this divide.

In keeping with the spirit of twentieth-century mathematics, the way out of this mess turned out to be a unifying framework couched in the language of category theory—the theory of "combinatorial species", introduced by André Joyal in 1981. This theory significantly simplifies the conceptual underpinnings of structural enumerative combinatorics, but it also adds a lot of power—several open problems in graph enumeration have been resolved using these tools, and many challenging classical results can be rendered much more simply and elegantly with its help.

(Don't let the phrase "category theory" scare you off. It'll be gentle—I promise!)