Maths Thoughts

A place to share thoughts that I think might spark curiosity—some technical, some intuitive, some philosophical.

Interesting Find of the Week (06/06/2025)

The interesting find for this week is the historic and beautifully conserved 'Twistor Newsletters'. For a bit of the history, when twistor theory had just emerged in the 1960s, these newsletters played a crucial role in disseminating ideas and fostering a community around this theorem. There was certainly a call for it, with many professors reaching out to those at Oxford, such as the renowned mathematician Roger Penrose, and asking what the state of current research was. Perhaps I am a particular fan of this, as I have an ongoing quest to collect handwritten notes of mathematicians from across various time periods, as there is something especially nice about preserving intricate details like handwriting. This marks just one of several artifcats I have managed to find online.

So without further adue, this is the resource: Twistor Newsletters

'There are many questions which fools can ask that wise men cannot answer.' - George Polya (a play on the original quote from Oscar Wilde)

On the Setting and Sitting of Mathematics Exams

Introduction

Early this morning, I sat in an exam room, ready to take my penultimate exam of my Master’s degree. In truth, this is likely the penultimate exam I will ever take. In the bustle of exam season, students scattered across the city, fatigued and nervous, it’s almost impossible not to reflect on the exam system itself. What intrigues me most is how our feelings towards exams vary by discipline: to the linguist or philosopher, an exam can feel like squeezing a creative process into rigid boxes; to the mathematician, it can be two hours of pure bookwork or a sweaty scramble to teach yourself something new under pressure.

What Is an Exam For?

Our view of exams mirrors what we believe their purpose to be. Is an exam:

  • A test of attention to lectures throughout the semester?
  • A challenge to see if you’ve read beyond the core material?
  • A proof you can solve new problems under time pressure?

To me, at least, an exam should do two things:

  1. Assess that students have met the learning goals laid out at the start of the course.
  2. Ensure the student has reached an appropriate level of mathematical maturity.

I don’t believe exams should trick students or demand author-level mastery; that’s just my view.

How to Set an Exam

A quick glance at online forums reveals many philosophies among lecturers and students alike:

  • Aim for a perfect grade curve to smooth out quirks in difficulty.
  • Craft artificially difficult exams to identify the 'best', and seperate the rest.
  • Design exams to deliberately fail anyone who hasn't attended lectures.

A balanced approach—used in Oxford mathematics exams—divides each 25-mark question roughly into 5–10 marks of bookwork, 10 marks of seen or similar material, and 5–10 marks of unseen questions. In theory, everyone can secure the foundational marks, most can tackle familiar problems, and some can stretch to the unseen. Any remaining ambiguities are smoothed out by grade boundaries. I think this is a fair system: too much focus on bookwork reduces exams to memory tests; seen material strikes a balance; and a small unseen segment rewards true understanding.

Example Exam Questions: Riemannian Geometry

1. Too Easy (Pure Bookwork)

  1. Define the Levi-Civita connection for a Riemannian manifold $(M, g)$.
  2. State the two properties that characterise it.
  3. Prove that if the Levi-Civita connection exists, then it is unique. Name the formula derived in the process.
  4. Define the different notions of curvature on a Riemannian manifold (Riemann, sectional, Ricci, scalar).
  5. Prove the symmetries of the Riemann curvature tensor.

2. Medium (Balanced)

Let $(M, g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$. On $\mathbb{R}^{2}$ with its standard metric, consider vector fields $X = \partial_{x}$ and $Y = x \partial_{y}$.

  1. Define an affine connection. Define the Levi-Civita connection and list its two defining properties. Briefly explain why one introduces the Levi-Civita connection.
  2. Prove that if the Levi-Civita connection exists, it is unique.
  3. Compute $\nabla_{X}Y$.
  4. Compute the Lie bracket $[X,Y]$, and verify whether $\nabla_{X}Y − \nabla_{Y}X=[X,Y]$. Explain how this reflects the torsion-free property.
  5. For any smooth function $f: M \rightarrow \mathbb{R}$ and vector fields $X, Y$ prove: $X(fY) = (Xf)Y + f \nabla_{X}Y$. Explain how this property characterises $\nabla$ as a connection.

3. Very Hard

Let $(M, g)$ be a connected, oriented, smooth Riemannian manifold with $\dim M \geq 3$.

  1. Let $X$ be a Killing vector field on $M$. Prove that the Lie derivative of the metric along $X$ vanishes, $\mathcal{L}_{X}g = 0$. Show that the divergence of a Killing field is zero. Conclude that the flow of $X$ preserves the Riemannian volume form.
  2. Let $M$ be simply-connected, compact, with irreducible holonomy $H \subset SO(n)$. Prove that any parallel tensor field on $M$ must be $H$-invariant. Argue that if $M$ admits a non-zero parallel vector field, the holonomy is reducible. What does this tell you about $M$ when it has full $SO(n)$ holonomy? Deduce that $H^{1}(M, \mathbb{R}) = 0$.
  3. Suppose $M$ is $7$-dimensional, simply connected, and admits a torsion-free $G_{2}$-structure. Prove that $M$ is Ricci-flat.
  4. Let $M = H_{3}$ be the $3$-dimensional Heisenberg group with a left-invariant Riemannian metric. Construct a non-trivial Killing vector field whose integral curves are not geodesics. Explain why this does not contradict intuition from constant curvature spaces.

Concluding Thoughts

And so, if there can be so much variability in what a lecturer considers appropriate, is an exam truly objective? Biases inevitably creep in, and students bear the stress year after year. Coursework and take-home exams offer alternatives, but AI and collaboration blur the lines of individual work. Perhaps a one or two page formula sheet strikes the fairest balance, valuing understanding over memorisation, but still allowing those who wish to put themslves the room to do so.

I hope that if I am to be a lecturer, and years have passed since I sat my last exam, I will remember how it felt to be sitting them, and equally the thoughts of those around me at the time. I don't think that the setting of an exam is something to be taken lightly, as for those who are taking them, it can affect one's stress for weeks, and discourage individuals from persuing further studies. I'm intersted in thinking about this deeper, and defining the differentiation between what is appropriate to set in problem sheets VS exams. I am also very interested in hearing views from individuals across a range of disiplines. It seems to me after thinking on this, that the examination process is more subjective than one would hope, and mistakes in judgement can easily fall through the gaps. Perhaps that is why having a system of examiners, who can all verify that the content and level is okay, from a variety of levels (E.G. PhD all the way up to professor), is but one way we can try to make the process a tad more objective. I would also like to revist this thought in the future, to see if my ideas change once I've been out of the exam system for some time!