Unremarkable ventures IV

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_gcx
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Thought I'd do this now since there's an influx, was going to wait until 19/20 was archived.

Welcome to le fifth GYG.

Modules
I'm a second year at the University of Warwick doing Maths, having transferred from Maths and Statistics last year. I wanted to take a very wide range of modules so my choices in the third year are maximised. I'm not sure what I'll want to do - geometry/topology seems likely but I want to keep the analysis pathway open by taking some analysis options. This year my core modules are overall 66 CATs, and I have to take optional modules to get up to 120. The second year is worth 20% if I stay on the MMath and 30% if I do the BSc, which I intend to.

My core modules for this year are:
  • MA244 - Analysis III [12 - 85% exam, 15% assignments]
  • MA251 - Algebra I: Advanced Linear Algebra [12 - 85% exam, 15% assignments]
  • MA259 - Multivariable Calculus [12 - 85% exam, 15% assignments]
  • MA213 - Second Year Essay [6 - 80% essay, 20% presentation]
  • MA249 - Algebra II: Groups and Rings [12 - 85% exam, 15% assignments]
  • MA260 - Norms, Metrics and Topologies [12 - 100% exam]
The optional modules I am taking are:
  • MA243 - Geometry [12 - 85% exam, 15% assignment]
  • MA250 - Introduction to Partial Differential Equations [12 - 100% exam]
  • MA257 - Introduction to Number Theory [12 - 85% exam, 15% assignments]
  • ST220 - Introduction to Mathematical Statistics [12 - 90% exam, 10% quizzes]
  • PX277 - Computational Physics [7.5 - 100% coursework]
  • CS262 - Logic and Verification [15 - 75% exam, 15% coursework, 10% in-class test]
Taking Computational Physics because I enjoyed Programming for Scientists last year. I'm taking CS262 just because it looks super interesting.


School Results
A-Level: A* Maths/Further Maths/Physics/EPQ. C History
STEP: 2, 2
GCSE: 3 9s, 4 A*s, 3 As, 1 B


University results
First Year: 90.5% average over content completed. (equating to about 45% of first year) Modules finished:
  • Mathematical Analysis - 96%
  • Sets and Numbers - 89%
  • Programming for Scientists - 87%
  • Mathematical Techniques - 81%
Second Year:
  • Computational Physics - (98 + 2/3)%


Aims
I would like strong 80s overall, particularly in Analysis III, NMT, PDEs, MV calc since whatever I go into will probably be closely linked to all/most of these 4, and stats because I've always been decent at it.

I am hoping to be able to apply for MASt Pure Maths (Part III) at Cambridge and OMMS at Oxford. My thought at the moment is to do a PhD and go into research but obviously this is subject to change.


Second Year Essay
I think I'll do my second year essay on applications of the Baire Category Theorem to the differentiability of functions f:\mathbb R \to \mathbb R. You know that not all continuous functions are differentiable everywhere, such as x\mapsto |x| at 0, and by taking copies of restrictions of this (the graph could look like VVVVVVV etc.) you can construct a function that is not differentiable at a countable infinity of points. (more specifically not at the apexes - but it's piecewise linear with no jump discontinuities so clearly continuous) Can there exist continuous functions that are not differentiable anywhere? Yes - an explicit example of this is the Weierstrass function. But we can prove non-constructively that such a function exists and in fact that "most" continuous functions are not differentiable anywhere using BCT.

The question I would move onto is the continuity of this derivative. We can certainly see that differentiable functions do not have to have everywhere continuous derivatives. Consider for example x \mapsto x^2 \sin(1/x) continuously extended to 0. (gluing many of copies of these close to 0 gives Volterra's function, which is an example of a function that is differentiable everywhere but with enough discontinuities to mean that this derivative isn't Riemann integrable) I then wondered if there exists a function whose derivative is not continuous anywhere. Turns out no, the derivative must be continuous at at least 1 point and has to be continuous on a dense set in \mathbb R. (ie. must be continuous at infinitely many points among other things)

This was motivated a lot Haskell Curry's answer to this MSE question: https://math.stackexchange.com/quest.../292380#292380.


I will talk more about the second year essay later in the thread.

Last edited by _gcx; 1 month ago
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absolutelysprout
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looking forward to reading this
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Sinnoh
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Very noice.

Hopefully there's enough of a challenge so that you enjoy the content more than last year but not so much that you don't meet the % you want. Essay topic looks pretty interesting
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Synergy~
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Oh, I didn't know you had a GYG! And a 90% is really impressive omds (knew u were a nerd) :eek:
Can I get a tag? (:
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_gcx
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(Original post by absolutelysprout)
looking forward to reading this
Hope it doesn't disappoint!
(Original post by Sinnoh)
Very noice.

Hopefully there's enough of a challenge so that you enjoy the content more than last year but not so much that you don't meet the % you want. Essay topic looks pretty interesting
This year I should be able to start on a bit of third year stuff which is where it gets very interesting. Particularly looking forward to doing Manifolds/Riemann Surfaces once I've covered enough background. Yeah, need to learn the theory behind it first though. (I finished norms, metrics and topologies over the holiday but Baire Category Theorem wasn't lectured this year so I don't actually know how messy the details might be)

(Original post by Synergy.x)
Oh, I didn't know you had a GYG! And a 90% is really impressive omds (knew u were a nerd) :eek:
Can I get a tag? (:
You haven't missed much and thanks.

Yeah sure!
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Starlight15
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So excited for your academic journey :woo:
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_gcx
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Oh I forgot to mention that there's going to be group projects. Exactly why I'm not entirely sure, I think it's to take weight off the exam in case they're online again. Not sure how big the groups are or any specifics - but I'm not sure I'd be anyone's first option to work with so I may just do them by myself if there's no presentation component. I'll put all the weightings in the OP.
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neko no basu
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hello warwick person

also a warwick person tag me pls


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did i just copy and paste your post from my gyg? yes, yes I did :rofl:
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_gcx
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(Original post by neko no basu)
hello warwick person

also a warwick person tag me pls


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did i just copy and paste your post from my gyg? yes, yes I did :rofl:
Will do
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Sinnoh
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oh btw the title says IV but your post says

Welcome to le fifth GYG.
:hmmm:
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_gcx
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(Original post by Sinnoh)
oh btw the title says IV but your post says



:hmmm:
I did one for GCSEs and hadn't yet adopted the UR title
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Sinnoh
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(Original post by _gcx)
I did one for GCSEs and hadn't yet adopted the UR title
ahhh
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Toastiekid
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your maths modules sound really cool:yep:
can i be tagged?
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Sinnoh
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(Original post by _gcx)
Oh I forgot to mention that there's going to be group projects. Exactly why I'm not entirely sure, I think it's to take weight off the exam in case they're online again. Not sure how big the groups are or any specifics - but I'm not sure I'd be anyone's first option to work with so I may just do them by myself if there's no presentation component. I'll put all the weightings in the OP.
Who wouldn't want a partner that does the whole thing themselves in their group?
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_gcx
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(Original post by Toastiekid)
your maths modules sound really cool:yep:
can i be tagged?
Yeah sure!

(Original post by Sinnoh)
Who wouldn't want a partner that does the whole thing themselves in their group?
Well if it's just like, groups of two I don't think I'm close enough to any straight maths people to be their first pick. If joint degree people also do these projects, then I possibly know one or two I could work with.
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_gcx
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Thought I'd briefly summarise my first term modules:
  • 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
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
Last edited by _gcx; 10 months ago
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Rohan77642
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Hi, can you tag me in this blog. I enjoyed your last blog
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_gcx
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(Original post by Rohan77642)
Hi, can you tag me in this blog. I enjoyed your last blog
I'll be sure to tag you!
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SilentSolitaire
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Logic and verification looks interesting and so do the analysis modules - are you a lot happier with the courses this year? The second year essay looks really cool; I've heard a bit about the Baire Category Theorem but I haven't studied much analysis or topology so I don't know a lot (hopefully I'll get time to read up on it at some point...). I think the first part is probably not too bad though? How long is the essay supposed to be?

Also can I be tagged this year?
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I AM GROOT 1
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:woo: here’s to another year :beer:
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