Started on a Maths degree

Moving in

On the 21st September 2014, I moved into my flat at Bristol university, to embark on a four-year degree course in Mathematics, from which I should obtain a Masters qualification at the end. Being my insurance choice (Cambridge fucked me over with STEP), I had a very limited choice of accommodation, and for the first one and a half weeks was put in a temporary share at a large catered hall in Stoke Bishop. The people there were great (most people are fine once you get to know them!), but I didn't much like the place - all institutional and the food was mediocre at best. Fortunately I managed to find a very nice place (once of the nicest available) to live for the year, which is situated right next to the Clifton Suspension Bridge, near Clifton village. It is not as close to the campus as some other flats are, but I have a bike so that is fine. The people in my new flat are really cool to be around and easy to make friends with, so I'm glad about that - plus, four or five of them are also mathematicians.

Societies

I have joined a few societies, including the Bristol University Gliding Club. I am a complete novice at gliding, but after going on a trial flight, I've decided to continue with it through this year at least - if the weather permits me to, anyway. Here are a few photos from my trial flight:

I also joined the Maths society, which is called Matrix. We haven't done anything yet, other than meet each others' "parents" and "siblings" on the first social. We have another social lined up for tomorrow, involving white T-shirts and pubs.

Course

So far the course has been very enjoyable, if a little esoteric. It makes you realise that whatever you may previously have thought was "maths", was really just writing numbers down on a page. Real maths starts now. (Doubtless I'll be saying that again when I get to third year.)

The course is split up into several modules. These are the ones we are starting with:

1. Analysis

This is perhaps the most esoteric and the hardest module. We start at the very basics, with definitions of just what numbers are. We go from considering the natural numbers (1, 2, 3, ...), to extending this to first the integers, then rationals, then the reals with the completeness axiom. We define what are called axioms; things that we can't prove (other than by relying on yet more axioms) and yet form the foundation stones of mathematics. An example of an axiom of the real numbers is that for all real a, b, we have that a + b is also real (this is known as closure under addition).

2. Foundations and Proof

We started out by learning about function mappings: how they can be surjective, injective or both (bijective), and about proper definitions of what a function is. We have since moved on to logic (truth tables). This module is a bit tricky at first, but simple once you get used to the language.

3. Probability

We recapped and extended our knowledge on combinatorics, permutations, events, sample spaces and etc. Not my favourite area of mathematics, though I can see it becoming more interesting later on.

4. Linear Algebra and Geometry

I think this is the one I find the most intuitive. We talk about vectors in Rn, the triangle inequality, complex numbers, subspaces and many other things.

5. Calculus

This is mainly about limits and derivatives, so far. It links in well with analysis.

6. Computational Mathematics

This was one of the most disappointing units for me. One of my favourite activities in general is programming, and combining that skill with mathematics is combining my two strongest interests. But on this module at least, we don't use any good programming languages and are instead forced to use MATLAB: an extremely expensive, proprietary and bloated system in my opinion. Maybe we'll have some better computing units in the next few years; who knows.

Conclusion

All in all, I am having a great time so far in Bristol and am very much looking forward to the next few years! It is a blessing in disguise that I am here instead of Cambridge: I have more time on my hands to enjoy myself in ways that don't necessarily involve pens, papers and wastepaper baskets. (Oh and by the way, those are the three essential tools of the mathematician: the most important being the last.)

Germany

I've just arrived back from a flying visit to Germany. My family had been staying there while I was at YRS2014, in a rural place off the northern Baltic coast, called Rügen. I flew out on the 6th to visit them, taking the slow train up from Berlin to Ostseebad Binz, the end of the line, through former GDR territory.

We were camping in a campsite situated a fair schlep from the nearest town, and there wasn't much in the way of things to do. Near where we were staying, there was a massive (and quite oppressive) "holiday camp" built by the Nazi Strength Through Joy program from 1936-39. The place in general had a brutal air to it, occupied as it was by two totalitarian empires within recent history.

Beach in Rügen
My phone took this.

The weather was good, apart from one rainy day, and the food was very good, especially the herring sandwiches and orange-flavoured ice cream. As I had not brought my bicycle on the plane, I had to hire one, which unfortunately broke down a couple of times (the chain broke in half).

On the 11th, we caught the train back down to Berlin, and stayed the night there in a hostel. I visited all the major attractions, including the Reichstag, Brandenburg gate and Holocaust memorial. Unfortunately I only had my phone on hand to take pictures, and its camera is not great.

Brandenburg Gate
Neoclassical architecture.

Holocaust Memorial
Imaginative and minimalist monument to indescribable crimes against humanity.

Reichstag
German parliament building (since 1999).

I also visited the remains of the Berlin wall, but forgot to take any pictures. For me, Berlin represents a historical front line between freedom and oppression. The Berlin wall clearly illustrates what the effect is of a country being occupied by imperialist powers with opposing ideas, and why liberal democracy is objectively a better system than Nazism or Communism.

2014 Festival of Code

From Monday the 28th July to Sunday the 3rd August, I took part in another Young Rewired State festival of code. This time our group - named "Flappyworks" as a portmanteau of flappybird and "works" - were stationed at the DigitasLBi center in Brick Lane, London (many thanks to them for providing an awesome center by the way!).

During the week leading up to Friday, we spent all day every day coding our project, which was a two-player tower defence game called Intersection, based on real-world data (police, government budgets etc.). The game is asymmetric, with one player as an attacker, and the other as a defender, with the attacker's objective being to take over the defender's city. We incorporated the Google Maps API version 3 and did many cool things using technologies such as WebSockets (using FirebaseIO's excellent wrapper), HTML5 and others.

The actual festival of code took place from Friday to Sunday at Plymouth University. Everything about it was amazing, other than the coach journey down there, which was awful and took about seven hours. They provided plenty of food, and (unlike last time) good wifi and loads of sockets (not that I needed them with my new Lenovo Thinkpad laptop and its 14 hour battery life!).

Us setting off for Plymouth on Friday.

There were three rounds. Our group managed to pass the first, as well as the semi-finals, but narrowly lost out to an (apparently) amazing project by a German team in the "best example of code" category. During the final, which was staged in the Plymouth Pavilions, my live demo was interrupted by an eager member of the audience who decided it would be a good idea to join my session. It turned out it was a great idea, because they allowed me to demonstrate true two-player functionality and actually play the game, which was both interesting, fun and hopefully quite impressive!

You can check out how our presentation went by watching it here (I'm the one operating the computer):

We managed to get some nice live twitter feedback, which was good:

All in all, it was an awesome experience, and I'll have to go back next year as a mentor (as I'll be 19 then).

Dunwich Dynamo

On the 12th of July at 8 pm, I set off on a 120 mile, overnight cycle journey with about 2,000 other riders, called the Dunwich Dynamo. The route took us from London Fields in Hackney, up to a small village called Dunwich, on the Suffolk coast. It was by far the longest cycle journey I had ever done non-stop (I cycled the Thames in 2008, but that was over five days). I found the experience of riding together with so many other people to be quite energising, and didn't become tired until the last 20 or so miles - by that point we were all very spread out anyway. I went with some family friends who then gave me a lift in their car back to Diss, where I caught the train home.

Route
This took a long time.

The route was fairly flat the whole way, being an overall down-hill, although there were moments near the end where any sort of non-negative incline was a struggle. The weather was a bit iffy. Although the wind was (thankfully) behind us for virtually the entire time, it did rain a couple of times and I had to get off and put my coat on. Cycling with a coat on is very sweaty, regardless of the temperature outside.

This was my first really long cycle journey, and as such I over-prepared. I took a lycra cycling vest, padded shorts and standard trainers (I would have liked clipped cycling shoes, as they give you extra force while pedaling). I also borrowed my sister's Cannondale road bike with racing handlebars - my city bike would not have cut it, with its hub gears and non-streamlined posture.

Food-wise, I took five home-made chocolate flapjacks and six sandwiches (three peanut-butter, three salmon), of which I ate two of the former and one-and-a-half of the latter. I was surprised at how little hunger I felt for the entire journey, but I did make sure to take small bites regularly out of whatever I had in my back-pocket. It is also very important to keep hydrated, so I kept my bottle open on my frame holder and took regular sips of water. I also took some energy gels, but did not get any use out of them so cannot say whether they are any good. I also had a "protein recovery" milkshake when I arrived; these are supposed to prevent exercise-induced muscle atrophy.

One thing I would definitely change if I were to do this trip again is the quantity of weight on my back-rack. I took two panniers, and these were a significant burden towards the end. I could easily have stored all the food in the same pannier as my coat and water.

Other than that, I enjoyed the trip enormously and would not hesitate before doing it again next year!

You can read more about the Dunwich Dynamo at Southwark Cyclists' page.

Cambridge STEP Easter School

During the past week, I attended an Easter school set by Cambridge university for maths applicants from not-so well off backgrounds (like my own, apparently), ostensibly in order to help them with preparation for STEP exams this summer, but (in my case at least) mainly to have fun doing maths and talking to current undergraduate students there. It took place in Churchill college, as well as the Centre for Mathematical Sciences nearby (a strange, parabolic building complex dug into the ground). About a hundred other applicants were there, from various different colleges within Cambridge, and it was great fun. Each day consisted of three lectures, with "question solving sessions" sandwiched in-between, where we went into rooms with our mentor groups and discussed mathematics - sometimes directly from the lecture, but other times weird and advanced pure mathematics - redefining, or rigorously proving results which seem totally obvious to a naive person like myself - such as the fact that if you have a continuous function running from ^`(x_0,y_0)`^ to ^`(x_1,y_1)`^ where ^`y_0y_1<0`^, then the graph of the function will cross the ^`x`^ axis at a point between ^`x_0`^ and ^`x_1`^. (This may not be obvious from my convoluted description, but if you were to picture it in your head or draw a graph, it should seem obvious. But to a mathematician, apparently it requires rigorous proof all the same!). Or conversely, disproving results which may seem true to the uninitiated but nevertheless are not, or are only true given certain axioms which have no real foundations.

As a taste of university life, it was of course quite short, but nonetheless enjoyable, and now I'm more determined than ever to pass my offer by gaining at least two firsts at STEP II and III. This doesn't seem quite as unachievable as it did about six months ago, as long as I work hard and don't become complacent. We had a mini-test in which I managed to get two questions out fully (sans a special, irrelevant case of ^`k=0`^ in a differential equation) within three quarters of an hour. It seems like the Tripos is very difficult, but equivalently rewarding.

Matrix Calculator

Matrices are one of my favourite topics in mathematics. They are very useful for things such as 3D vector space manipulation and transformation, or indeed anything spatial-related - and of course can be transferred to the realm of algebra. A trivial example is the application of a two-by-two matrix to solve simultaneous equations in ^`x`^ and ^`y`^. Suppose

$$ax + by = \alpha$$
and $$cx + dy = \beta$$

Then:

$$\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\alpha\\ \beta\end{matrix}\right)$$

So, matrices are useful. However, they are awful to manipulate by hand, due to a large amount of arithmetic. So I have created an application to do it for me:

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