Now that I’ve completed my second year here at Imperial College in
London I figured I’d update this thing again. Furthermore I’m doing it the lazy way. Since I like to keep my A-level maths teacher posted on how things are going I figured I’d mainly rip off the mail I sent him. So here we go:

This year we’ve had the following courses; Algebra II, Analysis II,
Probability and Statistics II, Numerical Analysis, Vector Calculus,
Differential Equations, Complex Analysis, Rings and Fields.

Algebra II was pretty straightforward as I had seen most of the group
theoretical part while working on my first year project (ie. normal
subgroups, even and odd permutations) and the part on vector spaces
was pretty all right too. So all in all it was one of the more
enjoyable courses.

Analysis II built on what we had done in Analysis I in the first year.
The idea of the course is to provide a rigorous foundation for the
integral and differential calculus. So there are things such as the
Intermediate Value Theorem, the Mean Value Theorem, etc. and the
course more or less ended with the Riemann integral. Next year I’ll
probably take Measure and Integration where we get to cover the
Lebesgue integral and other hopefully exciting measure theoretical
things.

Probability and Statistics II was not quite as boring as its
predecessor but still highly annoying. For some reason the stats
courses seem to have been designed for maximum confusion. While I had
a rough idea of what it was all about at A-level the lecturers have
now managed to make me lose touch with every aspect of the subject.
This course was especially amazing in the sense that it managed to
include the worst aspects of pure and methods courses. On one hand we
are given the rigorous definitions, eg. that the probability space has
to be a sigma algebra but we never build on these rigorous definitions
and instead are just expected to apply a bunch of methods without
really knowing why they work or why we would use them in the first
place. So this is quite likely the last stats course I’ll ever take.

Numerical Analysis was a pretty good course. It’s mainly about
orthogonality and we covered things such as the Gram-Schmidt
algorithm, Givens rotations and QR factorization.
Overall it felt a lot like a pure maths course as everything was
proved and we weren’t subjected to any kind of hand waving.

Vector Calculus and Differential Equations were definitely the two
worst courses this year. Mainly because the lectures and lecture notes
were completely useless. Which is kinda hard to believe considering
that most of the content of these courses is pretty straightforward.
So I essentially had to teach myself these courses out of books, so
far the only courses where I was forced to rely on books. Other than
that I didn’t find them particularly interesting. I have a tendency to
find methods courses mainly pointless as they only teach us methods
without understanding why they work or why we would want to use them
in the first place. Hence it is no surprise that we won’t choose any
methods courses next year.

Complex Analysis turned out to be better than I remembered once I read
through the entire notes at the end of the Easter holidays. The main
problem with that course was the amount of repetition. There is so
much material that we had covered before in Analysis I and II, so it
is no surprise that I stopped paying attention in lectures early on.
Seriously, I can’t believe we wasted the first three lectures on an
introduction of the complex numbers after we covered them in at least
three courses in the first year.

Rings and Fields was the conceptually most difficult course so far but
at the same time this makes it one of the best courses we had the
pleasure of taking. Unfortunately Rings and Modules isn’t running next
year so the only way we get to build on the material we’ve covered in
this course is by taking Galois Theory, Group Representation Theory
and Algebraic Number Theory next year. Though it’s a small price to
pay.

Overall the summer exams went all right, except for Probability and
Stats II which was several orders of magnitude harder than the past
papers we had seen, also most of the other exams were somewhat more
difficult than usual. Nevertheless I am confident that I have passed
all of them.

After the exams we’ve had about four weeks to work on a group project.
When choosing the three areas we’d like to work in I chose Algebra,
Numerical Analysis and Analysis, thinking I really had chosen three
pure courses, but as it turned out choosing Numerical Analysis meant I
ended up doing a highly applied project with some Matlab coding. The
project title was Computing Phase Transition Phenomena in Wetting
Problems. I know a lot more about wetting problems than before but I
can’t really pretend that I understand any of it. At least we won’t
have any group projects anymore. The group I was in was for the most
part great, but the topic was a bit of a downer. The next time we’ll
get to do a project will be the fourth year project that counts for
1/4 of the final year mark and since I’m mainly choosing pure courses
I guess it’s safe to say that I won’t be doing an applied project  then.

From next year on we don’t have any compulsory courses so my choices,
based on what courses are supposedly running next year, are:

Metric and Topological Spaces, Measure and Integration, Group Theory,
Elementary Number Theory, Functional Analysis, Galois Theory, Group
Representation Theory, Algebraic Number Theory. I might also be
checking out the lectures for the fluid dynamics courses after I’ve
seen some of the cool simulations the Applied Modelling and
Computation Group are coming up with.

Anyway, all in all it was a pretty good year, though I’m really
looking forward to the exciting courses I’m going to take next year.