Unremarkable ventures IV

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Analysis III - split up about 50/50 real and complex analysis. The real side concentrates firstly on convergences of sequences (and similarly, series) of functions, introducing things like uniform continuity and convergence, and then Riemann integration. Slightly dry but very important. Reasonably confident with this stuff already, but may have to review proofs and the like. The complex side is infinitely more interesting. (I say this despite my planned essay being in real analysis) Complex functions (ie. functions $\mathbb C \supseteq U \to \mathbb C$) turn out to have much nicer properties than real functions, and far less annoying counterexamples than you'd find in real analysis. There are nice methods to calculate line integrals of complex functions in the complex plane and these can be used to deduce values of real integrals. Unfortunately a lot of that is omitted from this module - eg. the notion of poles, residue theorem, etc. which ruined my first planned essay idea. (I'd have to use too much of the essay covering this stuff I thought) I initially wanted to do it on Mittag-Leffler expansions (infinite partial fraction expansions, generalising the decompositions found for rational functions) and show them off a bit by using them to show $\sum n^{-2} = \pi^2/6$

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Algebra I - looks very dry. I didn't really like Linear Algebra in first year, some of the vector spaces stuff was fun but it really wasn't that interesting and I dreaded learning the proofs. Luckily exams were cancelled but I still have to deal with this module. It doesn't look very long and I've heard it's fairly straightforward. As dry as it is, the content is essential for doing most pure maths past second year. Never done anything in this course before. I've done some linear algebra incidentally as part of reading in geometry and stats over the summer though but nothing very heavy.

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Multivariable Calculus - Really the name is misleading - it sounds like a first course in vector calculus but that was what Geometry & Motion in first year was for. This is more like Multivariable Analysis, it basically seeks to generalise the analysis already done for functions in $\mathbb R$ to general functions $\mathbb R^m \to \mathbb R^n$. ie. first looking at sequences/limits/continuity of multivariable functions and then moving to their derivatives. Introduces some very powerful theorems like the Inverse Function Theorem & Implicit Function Theorem, and then finishes off with some discussion of (real) line integrals. Got about halfway through this in this holiday but I'll need a refresher.

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Geometry - I love the look of abstract geometry and can't wait to study things like manifolds and Riemann surfaces as I said earlier. But the Warwick second year course focuses more on specific examples of geometries, like spherical, hyperbolic, projective, etc. which seems fine too but I'm disappointed you don't look at many generalisations. Spherical trigonometry is a bit confusing at first (you see sines and cosines of lengths, but that's actually somewhat natural when you consider the relation between angles and arc lengths of sectors of circles, etc.) but it's neat.

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Statistics - The first part isn't much new, basically you're just reviewing stuff like a probability space, the concept of a random variable, moment generating functions, some common distributions. The new stuff starts with sequences of random variables and their convergence, then goes onto statistical inference with things like likelihood, hypothesis tests, confidence intervals, estimators. Stuff that's been seen before in passing, (eg. $n - 1$ vs $n$ in the formula for [the common estimate of] population variance is an example of eliminating bias in an estimator) in less formal contexts. Found it fairly straightforward so far and it should be a good one for high marks.

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Computational Physics - As expected, programming for physics in python. Think the emphasis is more on the programming than the physics though. Starts with some statistics, goes on to some random numbers, (generating a random variable using a specific distribution and stuff like that) stuff with Monte Carlo simulation, then some numerical calculus with numerical solutions to DEs and integration/differentiation. Not sure this is much different to what's covered in A-level but maybe there could be some curveballs, and again the focus is on implementing these things and applying them to problems.