Nowadays, it seems that everyone is inundated with spam phone calls, which manifest as a spoofed phone number or name to hide the caller’s true identity. At best, answering such calls results in insistent pitches about things like insurance benefits we are missing out on or roofing inspections we are assured we need.

This isn’t terribly different from spam email, for which you might use software to automatically shunt offenders to a junk mailbox, or to /dev/null if you are a particularly frustrated Unix user. For telephone communication, an option is enrolling in ‘no-call lists’ that flag your number as off limits with respect to unsolicited calls from telemarketers and the like. Both of these approaches run the dual risk of allowing an actual spam call to pass through and of mistakenly rejecting a genuine call as spam. In statistics, these types of errors are commonly referred to as false negatives and false positives, respectively. The goal of any spam blocking approach is always to minimise both.

Another and seemingly very popular option is number blocking: the user vets a new call, and, if they deem the number to be spam, they then set their phone to reject all subsequent calls from that number. As successive spam calls are denied, the phone compiles its own increasingly large database of blocked numbers. Some time ago, my phone started to be inundated with spam calls. In reflexively resorting to number blocking, I found myself wondering about the success rate of this method. Aside from the obvious practical significance of the question, it is also mathematically interesting and can be investigated using the tools of combinatorial probability.

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Some two thousand years ago in ancient Greece, philosophers Aristotle and Zeno asked some interesting thought-provoking questions, including a case of what are today known as Zeno’s paradoxes. The most famous example is a race, known as Achilles and the tortoise. The setup is as follows:

Achilles and a tortoise are having a race, but (in the spirit of fairness) the tortoise is given a headstart. Then Achilles will definitely lose: he can never overtake the tortoise, since he must first reach the point where the tortoise started, so the tortoise must always hold a lead.

Since there are virtually infinitely many such points to be crossed, Achilles should not be able to reach the tortoise in finite time.

This argument is obviously flawed, and to see that we consider the point of view of the tortoise. From the tortoise’s perspective, the problem is equivalent to just Achilles heading towards it at the speed equal to the difference between their speeds in the first version of the problem.

Since $\text{distance} = \text{speed}\times\text{time}$, we can say that after time $t$, Achilles has travelled a distance equal to $v_At$ and the tortoise $v_Tt$. The distance between them is
\begin{equation*}
D-v_At+v_Tt = D – (v_A-v_T)t,
\end{equation*}
and so Achilles catches the tortoise—ie the distance between them is $0$—when the time, $t$, is equal to $D/(v_A-v_T)$.

There is another way to see this problem that satisfies better the purpose of this article and directly tackles the problem Aristotle posed. To get to where the tortoise was at the start of the race, Achilles is going to travel the distance $D$ in time $t_1 = D/v_A$. By that time the tortoise will have travelled a distance equal to
\begin{equation*}
D_1 = v_T t_1,
\end{equation*}
which is the new distance between them.

Travelling this distance will take Achilles time
\begin{equation*}
t_2 = \frac{D_1}{v_A} = \left(\frac{v_T}{v_A}\right)t_1.
\end{equation*}
Then the tortoise will have travelled a distance
\begin{equation*}
D_2 = v_Tt_2 = \left(\frac{v_T^2}{v_A}\right)t_1
\end{equation*}
and Achilles will cover this distance after time $t_3 = D_2/v_A = (v_T/v_A)^2t_1$.

Repeating this process $k$ times we notice that the distance between Achilles and the tortoise is
\begin{align*}
D_k &= v_T\left(\frac{v_T}{v_A}\right)^{k-1}t_1 \\ &= \left(\frac{v_T}{v_A}\right)^kD.
\end{align*}

Summing up all these distances we get how far Achilles has to move before catching the tortoise: if we call this $D_A$ it’s
\begin{equation*}
D_A =
\lim_{n\to\infty}\sum_{k=0}^{n}D_k =\sum_{k=0}^{\infty}D_k = D \sum_{k=0}^{\infty}\left(\frac{v_T}{v_A}\right)^k.
\end{equation*}
This is probably the simplest example of an infinite convergent sum. In particular, this is the simplest example of a class of sums called geometric series, which are sums of the form
\begin{equation*}
\sum_{k=0}^{n}a^k.
\end{equation*}
If $|a| < 1$, the sum tends to $(1-a)^{-1}$ as $n$ tends to $\infty$ and diverges otherwise, meaning that it either goes to $\pm\infty$, or a limiting value just doesn't exist. By 'doesn't exist', see for example what happens if $a=-1$: we get \begin{equation*} 1 - 1 + 1 - 1 + 1 - 1 + \cdots + (-1)^n \end{equation*} and the sum oscillates between $0$ (if $n$ is odd) and $1$ (if $n$ is even). In our case, $|a|=|v_T/v_A|<1$ so \begin{equation*} D_A = \frac{D}{1-v_T/v_A} = \frac{D v_A}{v_A-v_T}, \end{equation*} which, when divided by the speed of Achilles, $v_A$, gives exactly the time we found before. So Achilles and the tortoise will meet after Achilles has crossed a distance $D_A$ in time $t_A = D_A/v_A$. Thousands of years later, Leonhard Euler was thinking about evaluating the limit of \begin{equation*} \sum_{k=1}^{n}\frac{1}{k^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots + \frac{1}{n^2} \end{equation*} as $n\to\infty$, which is named as the Basel problem after Euler’s hometown. This sum is convergent and it equals $\mathrm{\pi}^2/6$, as Euler ended up proving in 1734. He was one of the first people to study formal sums—which we will try to define shortly—and concretely develop the related theory. In his 1760 work De seriebus divergentibus he says

Whenever an infinite series is obtained as the development of some closed
expression, it may be used in mathematical operations as the equivalent of
that expression, even for values of the variable for which the series diverges.

So let’s think about series which diverge. One way a series can diverge is simply by its terms getting bigger. One such example is the sum
\begin{equation*}
\sum_{k=1}^{n}k = 1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}{2},
\end{equation*}
the limit of which when $n\to\infty$ is, of course, infinite.

But now let’s think about the harmonic series,
\begin{equation*}
\sum_{k=1}^{n}\frac{1}{k} = 1 + \frac12 + \frac13 + \frac14 + \cdots + \frac1n.
\end{equation*}
This time, although the terms themselves get smaller and smaller, the series still diverges as $n\to\infty$. But we can still describe the sum and its behaviour. It turns out that
\begin{equation*}
\sum_{k=1}^{n}\frac1k = \ln(n)+\gamma + O(1/n),
\end{equation*}
where $\gamma$ is the Euler–Mascheroni constant, which approximately equals 0.5772, and $O(1/n)$ means ‘something no greater than a constant times $1/n$’. You can see the sum and its approximation here:

Historically, the development of the seemingly unconventional theory of divergent sums has been debatable, with Abel, who at some point made contributions to the area, once describing them as shameful, calling them “an invention of the devil”. Later contributions include works of Ramanujan and Hardy in the 20th century, about which more information can be found in the latter’s book, Divergent Series.

More recently, a video on the YouTube channel Numberphile was published, and attempted to deduce the ‘equality’
\begin{equation*}
1+2+3+4+\cdots=-\frac{1}{12}.
\end{equation*}
This video sparked great controversy, and indicates one of the dangers of dealing with divergent sums. One culprit here is the Riemann zeta function, which is defined for $\operatorname{Re}(s)>1$ as
\begin{equation*}
\zeta(s)=\sum_{k=1}^{\infty}\frac{1}{k^s}.
\end{equation*}
When functions are only defined on certain domains, it is sometimes possible to ‘analytically continue’ them outside of these original domains. Specifically at $-1$, doing so here gives $\zeta(-1)=-1/12$. The other culprit here is matrix summation—another method to give some value to divergent sums. By sheer (though neat) coincidence, these methods, such as the Cesáro summation method they use in the video, also give $-1/12$!

The main problem is this: at this point we no longer have an actual sum in the traditional sense.

Instead, we have a divergent sum which is formal, and by that, we mean that it is a symbol that denotes the addition of some quantities, regardless of whether it is convergent or not: it simply has the form of a sum.

These sums are not just naive mathematical inventions, instead, they show up in science and technology quite frequently and they can give us good approximations as they often emerge from standard manipulations, such as (as we’ll see) integration by parts.

Applications in physics can be found in the areas of quantum field theory and quantum electrodynamics. In fact, formal series derived from perturbation theory can give very accurate measurements of physical phenomena like the Stark effect and the Zeeman effect, which characterise changes in the spectral lines of atoms under the influence of an external magnetic and electric field respectively.

In 1952, Freeman Dyson gave an interesting physical explanation of the divergence of formal series in quantum electrodynamics, explaining it via the stability of the physical system versus the spontaneous, explosive birth of particles in a scenario where the corresponding series that describes it is convergent. Essentially he argues that divergence is, in some sense, inherent in these types of systems otherwise we would have systems in pathological states. His paper from that year in Physical Review contains more information.

Euler’s motivation

Sometimes, such assignments of formal sums to finite values (constants or functions) can be useful. The fact that they sometimes diverge does not make much difference in the end, if certain conditions are met.

An example that follows Euler’s line of thought as described earlier emerges when trying to find an explicit formula for the function
\begin{equation*}
\operatorname{Ei}(x):=\int_{-\infty}^{x}\frac{\mathrm{e}^t}{t}\, \mathrm{d} t,
\end{equation*}
for which repeated integration by parts yields
\begin{align*}
\operatorname{Ei}(x) = \int_{-\infty}^x\frac{\mathrm{e}^t}{t}\, \mathrm{d} t =& \left[\frac{\mathrm{e}^t}{t}\right]_{-\infty}^{x}-\int_{-\infty}^{x}\mathrm{e}^{t}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{t}\right) \mathrm{d} t \\
=& \left[\frac{\mathrm{e}^x}{x} – 0\right] + \int_{-\infty}^{x}\frac{\mathrm{e}^{t}}{t^2} \mathrm{d} t \\
= \cdots =& \frac{\mathrm{e}^{x}}{x}\sum_{k=0}^{n-1}\frac{k!}{x^{k}}+O\left(\frac{\mathrm{e}^x}{x^{n+1}}\right),
\end{align*}
where we’ve been able to say $\mathrm{e}^x/x\to 0$ on the second line as $x\to-\infty$. Dividing through by $\mathrm{e}^x$, this allows us to say
\begin{align*}
\mathrm{e}^{-x}\operatorname{Ei}(x) &= \sum_{k=0}^{n-1}\frac{k!}{x^{k+1}}+O\left(\frac{1}{x^{n+1}}\right)\\
&\sim\sum_{k=0}^{\infty}\frac{k!}{x^{k+1}}\text{ as }x\to\infty.
\end{align*}
Now swap $x$ for $-1/x$ in this equation:
\begin{align*}
\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)&\sim\sum_{k=0}^{\infty}k!(-x)^{k+1}\text{ as }x\to0\\
&=-x+x^2-2x^3+6x^4+\cdots.
\end{align*}
As you can see below, this series now diverges as $x\to\infty$, but we still see convergence of the partial (truncated) sums as $x\to0$, even as we add more terms:

Euler noticed that $\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)$, in its original integral form, solves the equation
\begin{equation*}
x^2\frac{\mathrm{d}y}{\mathrm{d}x}+y=-x
\end{equation*}
(for $x\neq0$). Now here’s the thing: the formal sum
\begin{equation*}
\sum_{k=0}^{\infty}k!(-x)^{k+1},
\end{equation*}
to which $\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)$ is asymptotic as $x\to0$, also (formally) ‘solves’ the same equation for any $x$.

This solution is not unique, and in fact, adding any constant multiple of $\mathrm{e}^{1/x}$ to $\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)$ would still solve the equation; and the resulting solution would still be asymptotic to the same formal sum.

However, the coefficients of the powers of $x$ are unique. So there may be something in the formal sum that can give away the actual solution of the equation (which is often difficult to find via standard methods—unlike formal solutions that are easier to compute like the one above), at least up to some class of solutions and under certain conditions. In fact, this seems to actually be the case, at least for certain classes of formal sums—the ones that attain ‘at most’ factorial over power rate of divergence.

Solving a differential equation

To elaborate further, let’s consider one more example, the differential equation
\begin{equation}
-\frac{\mathrm{d}y}{\mathrm{d}x}+y = \frac{1}{x}, \quad \text{where } y(x)\to 0 \text{ as }x \to \infty.
\label{fs1}
\tag{*}
\end{equation}
Thinking about the boundary condition there, we could substitute in the (formal) sum of powers of $x$ which decay away as $x\to\infty$,
\begin{equation*}
y(x) = \sum_{k=0}^\infty a_k x^{-k-1} = \frac{a_0}{x} + \frac{a_1}{x^2} + \frac{a_2}{x^3} + \cdots.
\end{equation*}
Doing so, we get
\begin{align*}
-\frac{\mathrm{d}}{\mathrm{d}x}\left[\sum_{k=0}^{\infty}a_kx^{-k-1}\right]+\sum_{k=0}^{\infty}a_kx^{-k-1} &= \frac{1}{x}
\\ \implies \sum_{k=0}^{\infty}(k+1)a_kx^{-k-2}+\sum_{k=0}^{\infty}a_kx^{-k-1} &=
\frac{1}{x} \\
\implies a_0x^{-1}+\sum_{k=0}^{\infty}\big[(k+1)a_k+a_{k+1}\big] x^{-k-2} &=\frac{1}{x}.
\end{align*}
Then for our differential equation to be satisfied the coefficients have to satisfy
\begin{equation*}
a_0=1 \qquad \text{and}
\end{equation*}

\begin{equation*}
(k+1)a_k+a_{k+1}=0\implies a_{k+1} = -(k+1)a_k
\end{equation*}
which recursively means that $a_k=(-1)^kk!$ and our formal sum solution is
\begin{equation*}
y(x) = \sum_{k=0}^{\infty}(-1)^k k!x^{-k-1}.
\end{equation*}
Now that we have a sum that solves the equation formally, we can obtain an actual solution assuming that it is asymptotic to the sum we found as $x\to\infty$ by using the repeated integration by parts result
\begin{equation*}
\int_{0}^{\infty}\mathrm{e}^{-xs} s^k \,\mathrm{d} s = k! x^{-k-1} \text{ for }x>0,
\end{equation*}
which implies
\begin{align*}
y(x) &= \sum_{k=0}^{\infty}(-1)^{k}k!x^{-k-1} \\
&= \sum_{k=0}^{\infty}(-1)^k\int_{0}^{\infty}\mathrm{e}^{-xs}s^k \,\mathrm{d} s \\
&= \int_{0}^{\infty}\mathrm{e}^{-xs}\sum_{k=0}^{\infty}(-1)^ks^k \,\mathrm{d} s.
\end{align*}
How is that helpful? Well, for $s:|s|<1$ we know that \begin{equation*} \sum_{k=0}^{\infty}(-1)^ks^{k} = 1-s+s^2+\cdots = \frac{1}{1+s}, \end{equation*} by the formula for geometric series for $a=-s$ from our discussion of Achilles and the tortoise. This is a nice function on the real line, having all the fine properties that we need in order to define \begin{equation*} y(x)=\int_{0}^{\infty}\frac{\mathrm{e}^{-xs}}{1+s}\,\mathrm{d}s, \end{equation*} which is the solution to our differential equation, \eqref{fs1}, we are looking for, and is also asymptotic to the formal sum $\sum_{k=0}^{\infty}(-1)^k k!x^{-k-1}$ as $x\to\infty$:

Notice that any linear combination of these formal sums will result from the same linear combination of the respective convergent (for $s:|s|<1$) series $1-s+s^2-s^3+\cdots$ inside the integral. In conclusion, it is possible to obtain a solution in closed form to a differential equation just by finding a formal power series to which the solution is asymptotic.

Not just reinventing the wheel

The aforementioned example is, of course, quite simple and trying to find a solution in the way we just described might look like we’re reinventing the wheel using modern-era technology. However, the true potential of the method described above can be seen in nonlinear equations, to which we generally cannot find solutions in standard ways. In my own research I used formal sums to study an equation with applications in fluid mechanics.

In one of the first talks I gave about this topic, I remember noticing several of my peers tilting their heads in distrust when I mentioned that the emerging sums are divergent. This reaction was almost expected and for obvious reasons. It took an hour-long talk and several questions later to convince them that the mathematics involved is genuine.

Controversial as it may sound, at first sight, this concept is even more realistic than imaginary numbers, which are simply symbols with properties that we just accept and use. The idea is that, although imaginary, these numbers can demonstrably give us, when interpreted properly, very real results such as solutions to differential equations like
\begin{equation*}
\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+y=0.
\end{equation*}
The same is true for formal sums too.

Why do we assign actual numbers to formal sums in the first place? Because they are sometimes easier to work with and can lead to interesting results (such as solutions to differential equations) if interpreted properly. The underlying mechanisms should be well-defined mathematical processes and well-understood in order to avoid any serious mistakes when working with such sums. An example of erroneous use of such sums is Henri Poincaré’s attempt to solve the three-body problem in order to win the King Oscar prize in 1889. He managed, however, in the next decade to spark the development of chaos theory. But that’s for another time.

When I talk about maths research, one thing I often take care to point out to budding mathematicians is to bear in mind just how cool it is that you could be the first person to discover a new thing, or prove an interesting theorem that nobody else has managed to prove before. And while the sheer joy of mathematical discovery and solving the puzzle should be motivation in itself for the future generation of mathematicians, I also make it very clear that it’s often the case that whoever discovers a thing also gets to pick a name for it.

You could take the boring option and name it after yourself (assuming your name is unusual enough that people would know it was actually you—who knows which Bernoulli was responsible for what?). But a much better option is to name your mathematical concept, object or theorem after something amusing, strange or otherwise silly.

In that spirit, here’s my roundup of mathematical things with silly names—you may have your own favourites, but these are a few I’ve been being amused by recently.

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Mathematicians are a diverse bunch. As a group, they come from different experiences and backgrounds; they have different hobbies and aspirations; different preferences for computing software; and, as we recently found out at Chalkdust HQ, wildly different opinions on how to pronounce ‘scone’.

One thing they do all have in common is a love of mathematics. We strongly believe that a mathematician is anybody who does maths and wants to define themselves as a mathematician. And mathematics itself is full of variety and diversity. People can study a whole universe of different things! These things can be entirely unrelated—or even cooler, they can seem entirely unrelated but actually end up having deep connections.

This is why we can find it difficult to describe exactly what mathematics is: because it encompasses so many different things. Maybe we should stick with saying that mathematics is ‘what mathematicians do’. But then…what do mathematicians do?

Fear not: Chalkdust has the answer! Although our definition of mathematician leaves room for all sorts of people, doing all sorts of things, day to day, we’ve decided to focus on those doing mathematics in a university setting, and give you an insight into the daily life of four people at different stages of their mathematical careers:

• Piper, an undergraduate student at Durham University,
• Jessica, a PhD student at RWTH Aachen, in Germany,
• Smitha, a postdoctoral researcher at the University of Liverpool,
• Helen, a professor of applied mathematics and the head of department for mathematics at UCL.

Piper Lane

I’m Piper, I like random walks on the theoretical beach and contemplating the viability of gravity being represented by a particle. I am a mathematics undergrad, having just completed my first year at Durham University.

Piper hard at work

My busiest and thus most exciting day of each week begins at 8am on a Thursday, with a cold shower and a hearty college breakfast. I am usually joined by fellow mathematician friends with whom I converse, and subsequently walk, to the maths and computer science department above the main science site for the first session of the day; a 9am calculus tutorial. I have found tutorials to be useful in helping consolidate the current content, as I am given the opportunity to collaborate with other students through new problems, with as much or as little guidance by the tutor as needed.

Afterwards, I have little time to make my way to the learning centre for calculus and linear algebra lectures, followed immediately by analysis and then linear algebra tutorials back atop the science site hill.

I find a space around the maths department to work for the next hour, often at a bench outside if the weather is nice, at which I can finally eat lunch while watching the designated pre-recorded probability videos for the week. Although there is a lot of content to cover per module, each lecturer provides detailed pre-recorded videos alongside a coherent set of lecture notes, as well as regular live lectures presented on Zoom and/or in person. This gives a range of options for students depending on their circumstances regarding Covid-19 and learning preferences.

Over time I have found that the best methods for my personal learning have been to annotate and edit the existing lecture notes during in-person lectures, and then watch the pre-recorded content videos in my own time afterwards should I fail to understand any part of the lecture.

Analysis I vibes

My last class of the day is a two-hour programming tutorial, during which students work through a practical sheet of ascending difficulty depending on the week. This is usually more relaxed due to existing programming experience accelerating my understanding.

Since my day ends around 5pm with limited time between sessions, I head straight back to college for dinner and a rendezvous with my friends. Evenings are often my favourite part of the day, as I still have some time to go for a walk with my friends around the city, enjoying the scenery and the challenge of navigating Durham in the dark. After returning to college, I like to order a toastie from the buttery—the gem of Trevelyan College—before winding down with a cup of tea and a good book.

Jessica Wang

Hi! I am a first-year maths PhD student at RWTH Aachen, Germany. I work in the field of symplectic topology, specifically on billiards. I did my bachelor’s and master’s degrees at Durham University, before coming to Aachen. On a typical day, I go to the office in the mornings and start working, which could be either studying new materials or doing research. Since I am still at the start of my PhD, I have so far spent most of my time here learning background knowledge.

Jessica fixing the mistakes on Wikipedia

On Tuesdays, all the members in our geometry and analysis chair have lunch together with our secretary (who deals with all the admin work in our chair) to discuss maths-related stuff such as organising or travelling to conferences, social events etc. The afternoons are pretty much the same as the mornings, except that I go to ballet classes two to three times a week during late afternoons. My supervisor and I would sometimes meet to discuss my progress.

Jessica strikes a pose

Apart from being in the office, I attend some seminars and lectures during term time which are closely related to my research. I find the lectures particularly useful since I came from a very different academic background (my master’s thesis was on representation theory of braid groups, which is pretty algebraic).

Lastly, and probably the most fun part of my job, is attending conferences; you get to listen to lots of interesting talks, know more people from your field, and travel a bit around the city where the conference is in!

Smitha Maretvadakethope

The difficulty of trying to capture a day in the life of a postdoc at any institution, whether it’s at the University of Liverpool in the department of mathematical sciences (where I work) or halfway across the world, is that in academia every day is different from the last. So, if you read this and think “that is not a representative sample, Smitha” I challenge you to tweet @chalkdustmag what a typical day looks like for you!

Smitha on loan to Cambridge

Picture this:

It is a rainy day and campus is littered with students and teaching staff navigating their hectic time tables. That’s when I arrive on the scene. I arrive in the department and have a friendly chat with the building manager and everyone’s favourite cleaning lady. A warm greeting makes every day brighter.

Once the computer is up and running, the first port of call is the staff common room, a key fixture for everyone’s caffeine and tea addictions. This includes a water cooler for all obligatory water cooler talk.

Hydrated and ready? Time for the real work.

Stage 1: Emails, emails, emails (Part 1).

Stage 2: Chat to office mates.

Stage 3: Actually do some work. Depending on the day this is likely to be meetings, analysis/debugging, reading papers, or academic writing. The most time-consuming.

Has it hit 12pm yet? If yes, it’s time to drop everything and huddle around for lunch. Postdocs from other groups might drop by, PhD students might drop by. Anyone is welcome.
1pm? Back to work.

Stage 4: Emails, emails, emails (Part 2).

Stage 5: Chat to colleagues about emails that are confusing/strange/newsworthy.

Stage 6: Attend/organise seminars/workshops and become a pro at all things Microsoft Teams and Zoom.

Stage 7: Repeat stages 2–3.

Is it past 4pm yet? If yes, move onto stage 8. If not, rinse and repeat stages 6–7 until it is.

Stage 8: Emails, emails, emails. (Part 3)

Stage 9: Turn off that computer and call it a day. A good work–life balance is important self care.

And… that’s it! If I’ve missed anything, please note that it’s not within the scope of this article.

Helen Wilson

My name is Helen Wilson. I’m a professor of applied mathematics and currently the head of department for UCL mathematics.

I have school-age children, so my working hours are unusual. I share the school runs with my husband: today I’m doing the morning one, so at 8.30 I’m walking across town from the school to the station. This is good ‘refocus’ time for me, when I move my head out of the family space and into planning my work day.

Helen fighting fires

I reach the office soon after 10 and deal with the most urgent emails (I get about 100 a day so I have to stay on top of them). I’ll be giving a tutorial at 11, so I print the example sheet and look over it, and check my whiteboard markers still work.

The tutorial is really enjoyable. The students are giving presentations, which they have worked on in their groups, so my role is very much in the background. Some students give an excellent display of how their question can be solved, and I need to do hardly anything. Others have struggled to find a way through, so I give small nudges to encourage them to work it out live at the board.

Back in my office, I log the tutorial attendance and eat lunch (I almost always eat at the desk these days) and plough through more emails. There’s a freedom of information request, which needs prompt action by law, so I get on with that one sharpish!

In the afternoon I work on teaching allocation for the next academic year. This is a hugely complex process, starting with gathering the views of the teaching staff, and ending with checking everyone’s happy with the changes.

Helen wears glasses now

Last thing, I have a Zoom meeting with my most senior PhD student. He’s had his viva and is working on the minor corrections, so he doesn’t need much from me—but it’s nice to check things are progressing well and also hear how he’s getting on with his new job.

I leave the office around 5 and get home at 6.30, only about half an hour after my husband gets in with the children. It’s all about them for the next few hours, but when they go to bed at about 9 we get some adult time. We eat and chat and then typically both do a bit more work. I save easy, unstressful jobs for this time of day—writing student references, approving expense claims, anything that takes time but doesn’t require serious decision-making.

Lettuce know!

What does a day in your life look like? If you spend a lot of time doing mathematics in a non-university setting, tell us what you do at @chalkdustmag or contact@chalkdustmagazine.com, and your story could be featured in a future issue!