Essay titles for week 11, 2019-20 version

This web page provides the list of essay titles for 2019-20 for submission at the end of week 11, with additional notes to help get you started.

You should

Read ahead in Gowers' book, think about the titles, and also read any other relevant materials you can find, e.g. on web pages. Make notes of anything you find that might be interesting or relevant.
Make a draft "essay plan" on one piece of paper. Include a short introduction on what you will be trying to say, space for definitions or explanation of the key terms, and a conclusion section.
Use your plan to inform yourself of what other reading/research you might want to do. Do further reading as necessary. If you find it difficult to think about or discuss any of these points in full generality, try to think of illustrative examples. Examples need not be complicated, in fact they are usually better if they are simple.
Start writing a draft essay. I suggest you start with putting down titles for sections from your plan, and then start to fill in details. You don't have to start writing at the very beginning in section 1 (but if definitions are going to be needed it obviously makes sense to start with these). Filling in details for the middle section(s) is often the best way to start. The opening section and final conclusions section are usually best written at the end.
By Friday of week 9, submit your draft to Canvas.
From this point onwards you will be able to see some other students' draft essays. Make constructive and helpful comments on them. You should expect to get constructive and helpful comments back, including ones from teachers on the module. I understand that student comments will be anonymous but teacher's and teachers assistants' comments are not anonymous.
By Friday week 11, submit your final essay for marking to Canvas. Make sure something is submitted as soon as you possibly can, even if it is late. We can only mark the essays that are actually on Canvas.

What are the possible benefits of a computer program that can check proofs?

One of the highlights of mathematics in the 20th century was the realisation that proofs can be made completely precise and proofs can be checked exactly. This can be (and is) carried out to the level of putting a large amount of mathematics on computers and having the computer check the validity of the proofs. There are a number of so-called "proof checkers" https://en.wikipedia.org/wiki/Automated_proof_checking used for research and testing purposes. Note a "proof checker" is not the same thing as a computer that can generate or find proofs. This is about programs checking proofs originally found by humans.

If you want to look at this, Gowers Chapter 3 is relevant, but you will probably have to search outside the book for more information on this. You might also want to explain in rough terms what a computer program that can check proofs does and why it is reasonable to assume that such a thing exists. (You may want to contrast this with the idea of a computer program that can generate proofs all by itself: this would be rather miraculous and it is not reasonable to assume that such a thing exists.) Of course there is plenty to say here just in mathematics, but there are many applications in Computer Science that have been proposed as well.

Be careful with this essay: computers may not ever be able to generate proofs in a reasonable way and it is not obvious that proof-checking will save human time. If it doesn't save time, what other advantages may there be?

Explain the axiomatic approach to the real numbers and the connection between the reals and the idea of limits.

You should take the point of view that the rational numbers are given and understood, and explain what extra rules are required to describe what real numbers "do". Gowers Chapter 4 is relevant.

The axioms for the reals are some "ordinary" axioms that also apply to the rationals, together with an extra axiom that says that the set of real numbers is "complete" i.e. is closed under the process of "taking limits". (Completeness here is not related to "completeness" in other contexts e.g. Gödel's incompleteness theorem.) There are a number of ways of expressing this last axiom - e.g. the idea of Cauchy sequences and the axiom that every Cauchy seqence has a limit, or the axiom that every unbounded nonempty set of reals has a least upper bound. These axioms for the completeness of the reals turn out to be equivalent.

As in an essay you may have done for week 6, it will help to keep the idea of axioms for the reals and the actual systrem of real numbers separate.

(The relationship between reals and rationals is interesting too: why is it that every real number with a finite decimal expansion is rational? How can you decide if a number is rational from looking at its decimal? Why is it that between any two real numbers there is a rational number? Why is every real number the limit of a sequence of rational numbers?)

Note that in his book Gowers implicitly thinks of real numbers as being decimal expansions (with special conventions on rounding, carry, and when it is that one real number is equal to another number). There are also further details on this on my web pages. There are other ways of thinking about real numbers and some of these turn out to be technically easier. You could explore the issues here. What method would you prefer to use to "define" the reals? What indeed is a real number? You will most likely find other ways of thinking about the real numbers if you look around on the web. What all these have in common is that they are rich enough to give you limits of sequences and this needs explaination.

How does the idea of limits make the calculus precise?

Newton and Leibniz introduced the ideas of the calculus (differentiation and integration) using the idea of infinitesimal number. This was criticised heavily (e.g. by Berkeley). What was the main idea and how did it work? What was the problem? Was the criticism valid?

At least one way out of the problem was to introduce and use the idea of limit instead of infinitesimal number. How does this work?

Gowers Chapter 4 is again relevant. This is far too big a topic to give all the details, so you should aim to give an overview, something that might reasonably be read as an appendix to Gowers' book, or as an article in a popular science magazine perhaps. This goes beyond what was said in lectures, and is not ideal for a student not fully comfortable with calculus.

The idea is that there are two key concepts, integration and differentiation, and both are described using limits, integration being the sum of areas of rectangles covering a region as the size or width of the rectangles decreases, and the other is differentiation, which defined the "instantatious rate of growth" of a function analogous to the "speed at the precise moment when I passed that lamppost". To make this manageable you might want to restrict to just one of these concepts differentiation or integration, that would be fine, but discuss both if you prefer. (Ambitious students might be able to say what the Fundamental theorem of calculus is that connects these two.)

Can you give an idea of how the main concept(s) is(are) defined? The key to this essay to to explain precisely where limits arise in the definition, and then say something about it.

The construction of non-Euclidean geometries.

Gowers contains a lot of information about Euclidean geometry and spherical and hyperbolic geometry. You should take some time explaining what the questions are. There were plenty of mistaken "proofs" of Euclid's fifth postulate from the others, and you could discuss these, explaining why we know they are wrong. Or just explain one or more non-Euclidean geometries and why they are interesting, and how they were discovered, putting the matter into context, as well as giving a little more history. You will find much more online.

As a minimum requirement for this essay, I would expect you to present at least one non-euclidean geometry and be able to explain why it is non-euclidean. That means you will at some point have to say what Euclid's axioms are, and point to one of these axioms that is false in your example, and say why it is false. The ideas of abstraction come in too: to think of alternative geometries you will need to think abstractly and redefine what it means to be a "point" or a "line" in your geometry.

Fractals: explain the idea of "dimension of a fractal" by giving examples.

There are plenty of on-line sources, including of course wikipedia. The focus here is to explain what "fractal dimension" does (and what it does not do) through examples. Fractal dimension is of course an abstraction of ordinary dimension. You could also say what you think of this idea of "fractal dimension": is it a "true" notion of "dimension" or is it too abstract and not really much like ordinary dimension?

As a minimum requirement for this essay, I would expect you to to explain in rough terms the abstract idea of dimension that is relevant to fractals, present at least two examples, and explain with reasons what the dimensions of your examples are.

There is no such thing as the fourth, fifth, ..., dimension. Discuss.

For this essay I want you to focus on "dimension" as the number of coordinates in space or number of degrees of freedom. Do not consider fractals and fractional dimensions. (Note: if you want to talk about fractals, use the previous title, not this one.)

What does it mean to say "dimension" is the number of degrees of freedom? Is this a good definition or not? What degree of abstractiion is occuring here? Explain through giving examples, which need not be complicated. And what do you think of the statement in the title? Of course this essay is as much to do with the ideas of modelling and abstraction and the relationship of mathematics to the "real world" that we saw in Chapters 1 and 2, as much as it is about "dimension", and you might want to review the modelling process and explain that unusual dimensional spaces occur in mathematical models.

One interesting thing you could write about here are $n$ -dimensional versions of the tetrahedron or cube (etc.) for $n > 3$ .

The sentence in the title is intended to be somewhat provocative and you can take your own view on it. One approach would be to give examples of how $n$ -dimensional spaces (for $n > 3$ ) can be applied in science. You might regard such applications of $n$ -dimensional spaces as giving some sort of meaning to the "existence" of higher dimensional space in the real universe; an alternative view is that the word "dimension" can only be applied to mathematical models and does not apply at all to the "real universe".

If there was to be a single award of a Nobel prize for mathematics which mathematician (or small group of 2 or 3 mathematicians all working on the same topic) would you give it to, and why?

I would expect you to explain in lay terms this mathematician's contributions and why they are so important. The personal history of that person is much less important. Apart from that, you can consider anyone, but do argue your case carefully. The usual Nobel prize rules say that you must choose someone who is alive, but I will let you overlook this rule if you like.

However please note that the Nobel prize committee awards a prize to one person or a small group of up to about three people for a single achievement. So choosing one person for a large number of different achievements would not be suitable for this essay at all, and discussing more than one mathematical topic will take you away from what is required. Please stick to one single mathematical topic and one or more people that contributed to it. So as a minimum requirement for this essay you will need to explain one topic in mathematics and give a little history of the one (or two or three) person (people) that developed it, and also explain why the topic is of importance, e.g. for other parts of science or mathematics, or because of applications in the real world.