Soggy in Milk

Are all undergraduate students flakes, or just the ones at my institution? Our final is tomorrow, and I've had several students make appointments with me over the last few days -- and show up 15-45 minutes late. Few things in the world irritate me more than missed appointments.

The Proof is in the Pudding

Doesn't that sound like an excuse that a grad student in mathematics might give his or her professor? "I don't have my homework today... the proof is in the pudding."

Finally, after a lot of effort, I proved (I think) that if integer matrices A and B satisfy A^2 = B^2 = -Id, then there is an integer matrix C such that CAC^-1 = B. We went over a similar procedure in class for rational matrices using Module Theory - make Q^2n into a Q[t]-module and jump through some hoops to decompose it as a product of Q[t] / (d1) x ... x Q[t] / (d2n) to come up with a nice basis for the transformation given by A (or B). Of course, the result follows for Q^2n using the Jordan Normal Form of a vector space endomorphism, but the point was we could get at it in a different and more general way using modules.

Meanwhile, preparation for the Algebra qualifier continues in earnest. It's amazing how instructive this preparation is - in looking at past qualifiers, I've run into a number of ideas and problems that I haven't necessarily had to deal with - such as solvable groups and diagram chasing - and so I've been able to expand my horizons as I also become more comfortable with useful tools I've seen before, such as the Sylow theorems and the minimal and characteristic polynomial of a matrix.

I've been thinking a bit today about hypergraphs and how various graph theoretic notions might generalize for them - but I'll post more later when it's percolated more.

Qual and Answer

My first qualifying exam is tomorrow! My chance to prove to the world - er, Northeastern - that I actually know Analysis. And I feel pretty good about it. I've been studying past qualifying exams for the past week with my officemate, Limin, and I've learned lots of little things that simply don't come up during the course itself. Some things I knew, but it didn't strike me how important they were - for instance, the fact that the Fundamental Theorem of Calculus requires that the integrand be continuous (this was inspired by a problem which asked for a function with a discontinuous derivative). At any rate, though I enjoy analysis somewhat, it'll be nice to file it away and focus on the subjects I care more about - primarily algebra and discrete math.

I'll leave you with a theorem whose proof I won't give here:

Euler's Metatheorem:
The concatenation of all of "Euler's Theorems" contains all of mathematics.