The pleasantness could not be understated. I tried to make the most out of it by walking around Oxford often. Above is a picture taken on the dam of the Farmoor Reservoir.

Most of the summer was spent on a summer project on applications of Lie groups to differential equations. It ended up being a very broad project and touched on basically every aspect of geometry (variational problems, non-Euclidean geometry, …) which I was quite fond of.

I went to Bristol for the Heilbronn Annual Conference which I do recommend. In terms of the city, I would recommend going to the top of Clifton Observatory (for free!), walking across the Suspension Bridge and hiking your way down following the River Avon. Noah’s was also a very nice restaraunt.

I also went to Cambridge’s postgraduate open days. I only had enough time to goto Gonville & Caius’s open day which I throughoughly recommend. Taken aback by just how similar Oxford and Cambridge is, I’m increasingly certain I want to go some place other than UK after graduation.

]]>**Claim:**

For integers \(m, n >0\),

\[\text{Orb}(\begin{bmatrix} m \\ n\end{bmatrix}) = \text{gcd}(m,n) \mathbb{Z}^2\]**Proof:**

We would emulate the Euclidean algorithm.

WLOG if \(m > n\), then by division algorithm there exists non-negative integers \(q, r\) with \(r < n\) such that \(m = qn + r\). I.e. \(\begin{bmatrix} 1 & -q \\ 0 & 1 \end{bmatrix} \begin{bmatrix} m \\ n\end{bmatrix} = \begin{bmatrix} r \\ n\end{bmatrix}\) and \(\begin{bmatrix} 0 & 1 \\ 1 & -q \end{bmatrix} \begin{bmatrix} m \\ n\end{bmatrix} = \begin{bmatrix} n \\ r\end{bmatrix}\)

Notice that both of square matrices above have determinant \(\pm 1\). Proceeding with the Euclidean algorithm we obtain some element \(T \in GL(2,\mathbb{Z})\) by composition such that \(T\begin{bmatrix} m \\ n\end{bmatrix} = \begin{bmatrix} gcd(m,n) \\ 0 \end{bmatrix}\). Hence \(\text{gcd}(m,n) \mathbb{Z}^2 \subset \text{Orb}(\begin{bmatrix} m \\ n\end{bmatrix})\).

For the other direction, for all integers \(a, b\) we have \(gcd(m,n) \vert am + bn\), so \(\text{Orb}(\begin{bmatrix} m \\ n\end{bmatrix}) \subset \text{gcd}(m,n) \mathbb{Z}^2\) \(\square\)

All the matrices used in the algorithm are shear matrices, and I wonder if there’s anything interesting going on geometrically.

The claim also supports defining \(gcd(0,0)\) as \(0\).

]]>for all real \(x,y , \lambda\). We call elements of the dual space linear functionals.

Given an linear functional \(f\), it’s clear that \(f(x) = f(1)\times x\) for all real \(u\). Similarly for all real numbers \(c\) we have \(f(x) = cx\) to be an linear functional. As such the dual space of \(\mathbb{R}\) is \(\{f: \mathbb{R} \to \mathbb{R} \text{ s.t. } f(x) := cx \;\vert\; c \in \mathbb{R}\}\).

That’s simple enough. How about the dual space of \(\mathbb{R}^2\)?

In essence, we seek all linear maps \(f:\mathbb{R}^2 \to \mathbb{R}\) such that

\[\begin{align*} f((u_1, u_2)+(v_1, v_2)) &= f((u_1, u_2)) + f((v_1,v_2)) \\ f(\lambda (u_1, u_2)) &= \lambda f((u_1,u_2)) \end{align*}\]for all real \(u_1, u_2, v_1, v_2, \lambda\).

Fundamentally we want to make use of the theory we’ve built up for the dual space of \(\mathbb{R}\). Given an linear functional \(f\), let’s try to see what happens if we set \(u_2\) and \(v_2\) to be zero.

\[\begin{align*} f((u_1, 0)+(v_1, 0)) &= f((u_1, 0)) + f((v_1,0)) \\ f(\lambda (u_1, 0)) &= \lambda f((u_1,0)) \end{align*}\]If we then define a new function \(g_1: \mathbb{R} \to \mathbb{R}\) by \(g_1(x) := f(x, 0)\), we observe that

\[\begin{align*} g_1(u_1 + v_1) &= g_1(u_1) + g_1(v_1) \\ g_1(\lambda u_1) &= \lambda g_1(u_1) \end{align*}\]for all real \(u_1, v_1, \lambda\).

This is exactly what it means for \(g_1\) to be an linear functional of \(\mathbb{R}\)! As such \(g_1(x) = c_1 x\) for some real \(c_1\). Similarly we could define \(g_2(y) = f(0,y)\) and we find out that \(g_2(y) = c_2 y\) for some real \(c_2\). Combining the two, we have

\[\begin{align*} f(u_1, 0) = g_1(u_1) = c_1 u_1 \\ f(0, u_2) = g_2(u_2) = c_2 u_2\end{align*}\]As such \(f(u_1, u_2) = c_1 u_1 + c_2 u_2\) by additivity. Similarly for any real numbers \(c_1\) and \(c_2\) we can see that \(f(u_1, u_2) = c_1 u_1 + c_2 u_2\) is an linear functional of \(\mathbb{R}^2\). We conclude that the dual space of \(\mathbb{R}^2\) is \(\{f: \mathbb{R}^2 \to \mathbb{R} \text{ s.t. } f(u_1, u_2) = c_1 u_1 + c_2 u_2 \;\vert\; c_1, c_2\in \mathbb{R}\}\).

Suppose we have a linear transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\). We know it could be represented by a 2 by 2 matrix

\[T\bigl(\begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\bigr) = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\]In other words we could write

\[\begin{align*} au_1 + bu_2 &= v_1 \\ cu_1 + du_2 &= v_2 \end{align*}\]but \(g_1(u_1, u_2) := au_1 + bu_2\) and \(g_2(u_1, u_2) := cu_1 + du_2\) are precisely linear functionals of \(\mathbb{R}^2\)! If we interepted \(h_1(v_1, v_2) = v_1\) and \(h_2(v_1, v_2) = v_2\) as linear functionals, then \(T\) could be interepted as a map that maps linear functionals \(h_1\) to \(g_1\) and \(h_2\) to \(g_2\). This is precisely how the dual map of \(T\) is defined.

Hopefully this makes dual spaces easier to grasp, in particular understanding why the dual basis is a natural and useful concept.

]]>The most important project, once the discussion turns to matters of professional and intellectual substance, is the articulation of shared values, for example, “we both believe in using research to change the world”, or “we both believe in using both qualitative and quantitative methods judiciously, without any a priori bias against either”. Shared values make for stronger professional bonds than shared ideas or shared interests alone. Don’t rush into this, but do keep the conversation focused on the concrete professional topics that will provide raw materials for it. On the other hand, if the conversation doesn’t seem to be going anywhere, that’s not your fault. Don’t force it. Don’t set enormous expectations for a single conversation. It’s a long-term process. Just say “nice chatting with you” in a pleasant way and let it go. If the interaction went well, you can end the conversation by saying, “do you have a business card?” in a mildly enthusiastic way (assuming you have one yourself); if they don’t have a card then shrug and let it go. If the interaction leaves you feeling bad, go get some fresh air, acknowledge the feelings, and be nice to yourself. Talk it out with someone if you need to. Then carry on.

It helps if you understand the structural reasons why graduate school can be so difficult. In passing through graduate school and joining the research community, you are making a transition from one social identity to another, and from one professional persona to another. In a sense you are becoming a new person. But you face an irreducible chicken-and-egg problem: you can’t do research without being a member of a research community, and you can’t be a member of a research community without doing research. This chicken-and-egg problem is typically at its worst in the middle phase of graduate school, after you finish your required coursework but before you narrow down a dissertation topic. During that middle period, the whole world can seem chaotic. All of your candidate topics will seem impossibly gigantic. It might feel like you are pretending to do research rather than really doing it. You might be seized by paranoia about people who will persecute you publicly as soon as you try to present your work. You might be seized by the delusion that someone else has already done your project. These are common feelings; understand that they result from the structural situation you are in, and not from your own personal failings or (necessarily) the failings of other people around you.

In writing a dissertation, and especially when writing a talk about the dissertation research, one often encounters points that need to be stuck in the introduction or conclusion. Terms need to be defined, methodology needs to be explained, objections need to be anticipated, patterns need to be identified, distinctions need to be made, and unanswered questions need to be acknowledged and posed as problems for future work. Of course, everyone tries to assign these points to a suitable place when preparing an outline. But many students find that the points just keep coming, as if a volcano were continually erupting in the middle of the thesis, causing a disorderly mass of troublesome junk to flow out toward the edges. The sheer mass of this junk can be overwhelming, and it can seem as though the whole thesis is going to turn into a hypertrophied introduction and (to a lesser extent) conclusion, with the actual substance of the work left as an afterthought. You should plan for this process, and realize that it is crucial for the formation of your professional voice. What’s happening, believe it or not, is that your mind is reorganizing itself. You are integrating all of the many voices that will lay claim to your topic, and you are sorting out a conceptual framework for your research program that addresses all of those many voices in a coherent way. You may not think that you are engaging with other people’s voices, since the depths of thesis-writing are a very personal, even isolating process. But if you are at the point of writing a thesis then you have already done a great deal of reading, and so you are familiar with established patterns of thinking on many subjects. Those are the voices that you are integrating at this point of the process.

Now, many people do not get excited at the prospect of articulating research agendas and conversing with funding agencies. They do not see themselves as leaders, and they would rather stay in their labs and libraries doing their work. I say fine. It’s a free country. Nonetheless, you have to understand how these things work. Money for your research does not materialize from the clouds, and you don’t want to be stranded when the agenda-setting process strays away from the topics that interest you. Participating in the process, if only at a basic maintenance level, means that you retain a degree of control over your life, as well as an early-warning system that prevents you from getting stuck later on. But more fundamentally, as I have emphasized throughout, the networking process is good for your own thinking. Networking serves many functions, but the most important is as a process of collective cognition. When you talk to everyone and listen to their research agendas, and when you write all their agendas down in front of you and look for the emerging theme that brings order to them, you are stimulating the most crucial functionality of your mind: the largely unconscious ability to synthesize fragments into coherent wholes. Down deep, everyone has a drive toward wholeness. This is the force that makes you a more or less integrated human being and not a schizophrenic mess, and it is also the force by which like-minded individuals cohere their thinking and form movements that are intellectually and institutionally stronger than the separate individuals that make them up. In a sense, then, deliberately talking things through with everyone in your network simply amplifies a force toward wholeness that is already operating in everyone’s personality. The difference is that it’s now a force for the collective good, as well as your own.

]]>If the serious scholars don’t do their networking, then a vacuum opens up, and operators will seize the opportunity. That’s one more reason why serious scholars should build networks. Even so, the line between serious scholars and operators is not always clear, and as you get involved in intellectual leadership, you will definitely feel the temptation to operate. You will find yourself saying things because they mobilize people and not because you really believe them. And it’s hard to tell the difference, given that being socialized into a new profession inevitably means learning a new language. You will probably sound fake to yourself much of the time, as you learn how to speak this language, and so it’s easy to slip into manipulation instead of real leadership. From that kind of manipulation, it is a short step to the sorts of aggressive empire-building that I described above: encouraging others to talk your own language rather than coming up with their own. That is the deepest moral question that you will face as you engage in professional networking, and you might be surprised how quickly you have to face it.

Consider the transformation \(r: (x,y) \to (y,x)\). This is nothing more but reflecting along \(y=x\), and it’s certainly linear in the sense that \(r(u+v) = r(u) + r(v)\) for whatever coordinates \(u, v\) we throw at \(r\). We now embedd the real line inside the plane as the set \(\{(x, 1) \forall x \in \mathbb{R}\}\), i.e. the line \(y=1\). Under the reflection \(r\), we would map some point \((c,1)\) to \((1,c)\). Graphically, we are mapping the red points to the blue points as you could see below.

How does this have anything to do with \(f(x) = \frac{1}{x}\)? The trick is that for each blue point \((1,c)\), we draw a green line connecting it and the origin, and see where the green line intersects with the red line. We can see that that intersection point is exactly \((1/c, 1)\)!

Secretly, we’re using two similar triangles. One has vertices \((0,0), (0,1), (\frac{1}{c}, 1)\) and the other has vertices \((0,0), (0,c), (1,c)\).

As such we have sort of been able to map \(x\) to \(\frac{1}{x}\), but we cheated a little by going up one dimension and doing some projection at the end. As an exercise, could we make a similar construction for \(f(x) = 2/x\) by choosing a different transformation \(r\)? How about \(f(x) = x + 1\)? How about \(f(x) = 1 + \frac{1}{x}\)?

There’s another way of finding inverse points, often used in straighedge and compass construction, that uses circles and a pair of similar right-angled triangles. How is that method different from the above?

To make all of this formal, one needs to study a bit of projective geometry. You can read more about linear fractional transformations from its wiki.

]]>The idea is that differential equations admit group actions that map solutions to other solutions. For example, if \(y=f(x)\) is a solution to \(\frac{d^2 y}{dx^2} = 0\). Then for any real constant \(\epsilon\), \(y=f(x) + \epsilon\) and \(y=e^{\epsilon} f(x)\) are going to be solutions as well. Intuitively it means that scaling and translations are symmetries of the differential equation.

This idea extends to more complicated differential equations such as the heat and wave equation, the symmetries of which uncover a lot of the fundamental properties about them.

]]>- Complex Mapping Visualizer: Visualising complex mappings
- Field Play: Visualising vector fields
- Mathematical operators and symbols in Unicode: Finding symbols in Unicode (that you can use in “plain text”)
- Detexify: Finding symbols in \(\LaTeX\)

Thanks to Isaac Li for telling me about Field Play.

]]>I decided to put my printed lecture notes into ring binders, which have been a game changer for me. It’s so nice to be able to read 2 pages on a flat table. The resolution of 2 A4 paper stack next to each other is certainly greater than almost all tablets available nowadays.

Compared to stapling, ring binders can handle a lot more pages, tearing happens a lot less often, you can add / change / reoder pages as you please, and you don’t have to deal with awkward angles when you try to read a few pages at once.

This term was exams season and mostly I had to stick to what I’ve learnt for the past year rather than coming up with new solutions. Setting up studying routines, walking often and setting boundaries to my working hours have helped keep my sanity. Obviously, it was very difficult sticking to boundaries during exam season but I think I’ve managed relatively well.

With the end of applied mathematics and college tutorials, it certainly feels like a stage is over in my university education.

]]>Speaking of symmetries, you may think of the rotational symmetries of a circle or the reflection symmetry of a rectangle. It turns out that symmetries like these greatly help us in simplifying, solving and understanding physical problems in the real world.

Starting from rotational symmetry, we can generalise the idea of symmetries. There are two components to every symmetry: An action changes the object from one perspective (a circle is rotated around its centre), yet the object stays the same from another perspective (it looks like it has the same shape). Under this generalisation, what other symmetries are there?

One example is the passing of time. If there’s an aspect of an object that stays the same as time passes, we say that the object exhibits a time symmetry. For example, imagine a spinning top on a desk. In an ideal world without friction, how fast the top spins remains the same whenever you measure it. As such, we say that the top’s spinning velocity exhibits a time symmetry.

Another example is changes in space. Many physical properties may be invariant under spatial changes. For example, the temperature in a room is almost the same wherever you place the thermometer (unless you place it near a heater!) The position changes, yet the temperature measured remains (roughly) the same. We call this spatial symmetry.

The abstract study of these symmetries is very much of interest to pure mathematicians, but as you would see they are also very useful in modelling real life and solving problems.

Pretend you’d like to know how far a cannon shoots out a cannonball. One way to do it is to model the cannonball flying through the air to figure out its path. Ignoring air resistance might give you too bad of a prediction, but considering the atomic interactions between the cannonball itself would be excessive.

As such, by making appropriate assumptions about the problem at the cost of realism, we’d increase the likelihood of getting a model that’s feasibly computable. A lot of these assumptions often come in the form of symmetries.

Going back to the cannonball, we could assume that the cannonball is spherical and the airflow around it is cylindrically symmetric. We could also assume that the gravitational force on the cannonball is the same regardless of how high it is. (i.e. we assume gravitational force is invariant under spatial transformations, even though gravity decreases the higher up you get)

In general, mathematical modelling is very much an art of balancing simplicity and realism. By choosing the right details to focus on and discarding the others, we’d extract the essence of the problem at hand in a timely fashion.

As we go down the atomic scale, more and more complicated symmetries exist in the subatomic particles and the forces between them, which require more and more mathematics to describe fully.

These symmetries, and their exploitations, shape much of the quantum physics we know today.

This was written to apply for an internship in Tom Rocks Maths. Thought I’d share it here.

I’ve been inspired by this wikipedia article on symmetries in physics. The discussion on mathematical modelling is also very much inspired by a talk given by Derek Moulton for the Invariants.

I’ve also been inspired by a very recent post on symmetries in string theory by Dewi Gould, a postgrad student at Oxford.

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