Oh you're doing maths? What do you do in a day, are you trying to find the next digit of pi?
Well no, that's almost entirely up to computer power at this point....
Are you solving equations all day?
Yes.... but probably not in the way you're thinking of....
Isn't all of maths already solved?
Far far from it....
I love talking about the maths but usually hate answering these questions. It's difficult to gage how much depth to go into without
I usually deflect. I love responding by saying I look for holes in things, and I may even recommend this Instagram account I follow.
But what does a mathematician do all day?
It's hard to put my finger on. Some are writing and running code to solve sometimes useful, sometimes (on the surface) very useless things. Others are playing around with abstract nonsense to find other abstract nonsense. Some of it is useful to every day life, other parts only useful in furthering our understanding of abstract concepts. In the end, you have some fascinating and miraculous pictures coming out of it.
In essence, researching maths is all about being creative and finding patterns and the maths itself is just a language to describe what we see when we do this. This is in stark contrast to what you are always told through school is the point of maths: memorise this formula, solve this equation for x, there is only one fixed way to do things. The thought of having a written equation where you need to actually interpret what to do is such a terrifying concept to so many. The fear of failure holds so many people back from pushing further.
Well, this ain't it chief.
In school we are taught to digest and spit out algorithms without gaining any intuition for what to do. I've seen many an adult scoff when seeing children learn about multiplication by drawing squares and rectangles instead of the tried and true "carry the 1" method. Both methods work. The latter is more concise, if lacking a clear picture of what is going on. The former though is how algebra was initially developed, and exploits a great geometric picture to show what is occurring. Multiplication is a mathematical language to concisely describe how to calculate the size (or "area", if you will) of something of interest.
In the same way, when I say "I look for holes in things", I am obscuring a lot of mathematical language that many people would immediately react to with fear. Formally, I could say that I use algebraic invariants such as homology, persistent homology and spectral sequences to study discretised geometric spaces. Translating to human language, "algebraic invariants" are abstract mathematical structures which characterise a particular object. In this case, the object(s) are "discretised geometric spaces" which is a fancy way of saying "points and data". Then tools like "(persistent) homology" are exactly the abstract nonsense which represents the notion of a hole in this shape. My research is guided by my intuitions from drawings shapes with different holes in them and my ability to imagine worlds where the rules of maths allow more things
But how does this work in practice? What am I doing day to day?
Well to answer that, we should really look at what you might want to research in maths - something I usually place into two distinct categories.
At the heart of it, maths is applied philosophy. We start with a set of rules and either see what is logically implied by those rules, or what happens if we relax them. There is a fascinating tangent we can get on here which looks at what the most fundamental and basic assumptions to make when studying maths and why or why not we should make them, but I digress.
There are many different ways to put this approach - of starting with a set of rules and exploring what is implied from there - into perspective, and these are not exclusive to the "all knowing genius mathematician"TM. I think Eugenia Cheng has really good examples of this in her book X+Y, where she looks at feminism from the perspective of a category theorist - looking at connections between people rather than the people themselves. In the book, she likes her abstractification to various analogies in maths. In primary school we are taught how to solve equations like "2+3=3+2=5" or "4x5=(2+2)x5=20". In high school we notice that these satisfy patterns, and we use algebra to state that "a+b=b+a" and "(a+b)c=ac+bc" where we can use letters as a placeholder for numbers. If we studying even further, we can ask even more questions, like "what happens if ab=/=ba?" In fact, this is what happens when a represents a reflection of a room and b represents a rotation of the room - if I rotate a square room by 90 degrees and then reflect it in a mirror, it can look completely different to first reflecting and then rotating.
To paraphrase a friend of mine, "the fact ab=/=ba is kind of like the fact you can't put underwear on over your trousers, unless you're Superman [in which case ab=ba]".
Another analogy I really like here comes from another friend of mine who studies topos theory. Topos theory is the height of abstract nonsense in maths, to the point where one often has no physical intuition for the mathematics beyond familiarity with the language and algebraic structure they deal with. Among other things, topos theory studies the fundamental logic which we use to inform maths. Things we take for granted in maths, like being able to pick something out of a list, are no longer allowed is some topos. It is up to a topos theorist to find what structures are present and what they look like in a different world.
My friend likes to compare a topos to an island. Some are big and others are small, otherwise from a distance they all look the same. When you look a little closer though, you notice that a variety of factors such as the specific climate inform what natural resources they have. Coconuts grow in Fiji, for instance, but you would be hard pressed to ever see one growing in the UK. On the other hand, both have islands with fresh water, food and places to build housing. The natural resources may look different for both, but using them in the appropriate way means you can build a civilisation with very similar rules. On the other hand, an island in the Arctic has very different natural resources and will be extremely sparsely populated by humans - instead it is where a polar bear can thrive. Every island is different, as is every topos in topos theory, but then a keen researcher can play around with their natural resources and other properties to see what the laws and civilisation on this island end up looking like.
In my own research I sometimes have a similar goal. The main project in my PhD thesis studies holes in spaces like crystals which have a high degree of symmetry. This started as a project in 2020 before I even began my PhD, motivated by the goal to infer structure in such large spaces from a small finite representation that loses much of the necessary information. It quickly became apparent that the case for low dimensional examples like crystals was comparatively easy to solve. What next? Celebrate a job well done? Well, no of course not! There was some low hanging fruit - how do I generalise this to more complicated higher dimensional examples that can model things like air bubbles and more? This has been a major focus of my last three years - by relaxing the restrictions on the object I opened up a whole can of worms where all sorts of crazy things were possible.
On the other hand, many mathematicians don't like researching this way, and prefer instead to start with a goal and determine how to solve it. This is neither better or worse than the former case, but it is the perfect complement, like Yin and Yang. The method of starting from a basic set of rules is powerful and generic, if at times somewhat aimless, whereas the method of solving a particular chosen problem provides intuition to inspire how to solve the problem, even if sometimes this may be misinformed. Some of the results are wonderful though.
Earlier this year, for instance, there was big news in the maths community when aperiodic monotiles were discovered. These are shapes which join together with copies of themselves to cover an entire plane, but so that no matter how this is done it is never periodic (unlike my work.... sigh). What was the motivation to study these? Well, tilings make for really pretty pictures, and model many things in nature. Most importantly, sets of two aperiodic tiles had been found, so why not make it a puzzle to narrow it down to one?!
It took one person playing around with shapes to find a candidate for an aperiodic monotile. Then it took a team of four to model this and check some preliminary computations to support the case. Then, it took the same group to spot patterns and use these to logically prove that what they were claiming was true. This is hard - there are infinitely many different combinations you have to check - but there are many techniques developed by mathematicians who have been stuck before to combat this. In this case, the researchers created substitution tilings, where every tile is replaced by smaller copies of itself. The picture above is of four of these tiles stuck together - something that I have on my bedroom wall.
Sometimes the goal can also be less tangible. In finding an aperiodic monotile, the researchers were able to answer a yes/no question to its existence. In a completely different field, like mathematical biology, this question is a lot more open ended. "What model reflects what we observe in this biological phenomenon?" "What do these cells look like?" These are often questions asked in this field and many other fields with an answer that has a lot of flexibility.
Quite famously (within my small field at least) last year, researchers were able to answer a question like this in neuroscience. While performing experiments on the medial entorhinal cortex of the brains neural system, neurobiologists were able to collect enough data to let data analysts infer things about the brain. They let topological data analysists at the data to answer the question of "what does this region of the brain look like?" The resulting analysis led to this great Nature paper which showed that the medial entorhinal cortex was in fact shaped like.... A DONUT.
So what am I doing all day as someone doing a PhD in maths? Well, lot's of things!
Sometimes I'm working with examples by drawing pictures or making abstract calculations to give myself an understanding of what patterns might emerge.
Other times I'm looking for how to rigorously and logically prove a result I suspect to be true. This is the part is where the equations arise amid constant questioning of "What X do I need to prove Y is true? Or is it altogether false?"
Sometimes I am writing code to automate large scale calculations to make them more accessible to compute for me and everyone else who may be interested.
Some days (although not as much as I should) I am reading other people's work. Sometimes it is to look for a particular result the author proved, other times it is to learn some advanced theory which I have not yet come across, and even then there are other times where I am less interested about the content than the approach and techniques used (I am quite proud to say that after my supervisor suggested I read this paper on percolation theory at the start of my PhD, I was able to adapt the idea behind the proof of their famous Theorem to prove my own (Theorem 4.3 in my first maths paper).
In all of this, I am trying to find connections between things and the intuition behind them, and very very very rarely locking myself away scribbling through pages and pages of paper to solve one long equation.
That's just not what maths is.
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