No-one asks me anymore what my favourite subject at uni was, but I have an answer prepared anyway: MATH4403, Partial Differential Equations. The course was chiefly concerned with maximum principles; in something that surprises me in hindsight, the advanced course in PDE’s did not introduce the concept of Sobolev spaces.
Instead, as my memory tells it, we spent week after week gradually generalising from Laplacians to eventually proving theorems like, Let L be a uniformly elliptic differential operator in D. If L[u] >= 0 in D, and if u attains a maximum in D, then u is constant.
I have to rely on memory, with some help from Protter and Weinberger, because I’ve lost my lecture notes for the subject. I have a big box with the notes for every other lecture course I did, but I must have lent out my notes to another student the next time MATH4403 ran, and we both forgot about it. It makes me a little sad to this day, almost 20 years later, because they were beautiful notes on a beautiful subject. Instead of writing on my usual loose A4 pages, I wrote into an exercise book, probably bought from a supermarket. It had a photo of a snowboarder on the front.
The course was taught by Jan Chabrowski, an old Polish associate professor who always stressed the third syllable in ‘derivative’. Jan would enter the room, pick up a piece of chalk, and start dictating the words and symbols he was writing on the blackboard. We dutifully copied the notes.
For the first ten weeks of that course, I understood every single concept by the time I’d finished writing it down. It was as though Jan had a direct link into my brain, communicating to me with perfect efficiency. I had never experienced the like of it before, and I never did so afterwards.1
In the final few weeks of the course, the material got too complicated for me to internalise it instantly. I guess that was partly because it was a fourth-year course in partial differential equations, and partly because the hyperbolic differential operators we were learning about don’t lend themselves to elegant maximum principles in the that elliptic and parabolic operators do.
In all the time I heard Jan speaking, I think he only ever made two jokes, neither of which would translate well outside a context in which you’ve spent months hearing only serious talk from him. I think he was my favourite lecturer.
My favourite moment from undergrad was in MATH4301, a subject nominally called ‘Advanced Algebra’, but which in 2005 was essentially a course on Galois Theory. It was a little away from my usual stomping ground, where numbers are real or complex, and functions are differentiable.2 But, like everyone else in that course, I knew that the general quintic equation couldn’t be solved with radicals, and I wanted to know why.
The lecturer Victor Scharaschkin promised an aesthetically pleasing treatment of rings, fields, groups, and so on. It was tough going – I needed help from the more algebraically-minded students get through some of the assignments – but I plugged away at it, and for several months I knew what a splitting field was.3
There’s little suspense in a course like this. The definition of a solvable group was introduced relatively early – it’s solvable if there’s a chain of normal subgroups down to {1} in which the quotients are all abelian – and we could deduce by linguistic means that a polynomial’s Galois group must be solvable if the polynomial is solvable by radicals.
Still it took some weeks to get there, building up the concepts, proving various intermediate theorems and lemmas. For example, the polynomial f(x) = x^2 + 1 cannot be factorised when working over the real numbers, but it can be written as (x + i)(x - i). We say that the complex numbers C are a splitting field for f, and that C is a field extension of the real numbers R.
A field automorphism is an isomorphism from the field to itself. Galois theory is concerned with automorphisms of a field extension which leave elements of the smaller field unchanged; for example, isomorphisms σ from C to C such that σ(t) = t for all real t. We call it an R-automorphism of C (in general, for a field extension L/K, we consider K-automorphisms of L).
Interestingly, there are only two of these automorphisms: by the linearity of field isomorphisms, we have σ(a + bi) = σ(a) + σ(bi) = a + bσ(i). But i^2 = -1, so σ(i^2) = σ(-1), which implies σ(i)^2 = -1, and hence σ(i) = ±i, and there are no other possibilities. It’s a suggestive hint that these automorphisms could be useful in characterising a polynomial.
In fact the K-automorphisms of L form a group under function composition – the Galois group. For the “practical” application of (not) solving the quintic by radicals, we take K as the rationals Q, and L a splitting field for the polynomial, essentially augmenting Q with whatever types of elements are needed to factorise the polynomial, just as we could augment R with i = sqrt(-1) to factorise x^2 + 1.
As we approached the end of week 12, we knew that we were close. When Victor completed the proof of the main result of the course – that a polynomial is solvable by radicals if and only if its Galois group is solvable – we gave a totally sincere round of applause. It was a unique moment in my education, and it remains a standout memory, all the students happy to work hard to try to understand a piece of humanity’s intellectual edifice, and appreciating the journey that the lecturer had taken us on.
The corollary that the quintic can’t be solved by radicals followed quickly from results proven earlier in the semester. Whatever elements are needed to adjoin Q to form a splitting field, they cannot all be derived from repeated root extractions.
These days I mostly work as part of the mining industry, because people will give you money if you can help dig up valuable rocks from the ground. Some of the problems I work on are intellectually interesting, in the realms of spatial statistics or algorithms, which is probably why I’m still here after almost 15 years.
But the aesthetics are so distasteful to me. Explosives, shovels, trucks, all very coarse compared to subharmonic functions or automorphism groups. I think we only saw six numbers in the whole of the MATH4403 lectures: 0, 1, 2, 3, 4, and 5.
I enjoyed my undergrad education immensely, learning things not for any direct applications but just because they were interesting. Sometimes I miss it.
I believe that it was due to a combination of Jan’s lecturing style and the subject matter. Jan doing the lecturing was not sufficient in and of itself for this magical transfer of mathematical understanding to occur – later I took Measure Theory from him, and my immediate reaction to sigma-algebras and measure spaces was utter bewilderment.
I’d spent a lot of time studying Lie algebras in mathematical physics, but these felt like they came from a different universe from pure maths.
For some years I’ve thought that a good joke would be a Twitter thread proving the insolubility of the quintic: “Guys. It’s time for some Galois theory.” But I’ve skimmed over Victor’s notes, and some notes online, and it would be far too much effort to re-learn it to the necessary level of detail.