Network Science provides generic tools to model and analyse systems in a broad range of disciplines, including biology, computer science and sociology. Renaud Lambiotte (pictured) teaches a 4th year undergraduate course on Networks and we are making the whole course available via our YouTube Channel. The first lecture is below with seven more to follow over the next few weeks.

The course aims to provide an introduction to this interdisciplinary field of research, by integrating tools from graph theory, statistics and dynamical systems. Most of the topics to be considered are active modern research areas. This is a mathematical course, where we emphasise the inner working of the methods, but with real-world applications in mind. As a leitmotiv, we will explore the two-way relations between network structure and dynamics: how does network structure affect spreading dynamics, for instance epidemic spreading? And how can we use dynamical processes to uncover salient structures in a large network?

You can also read about Renaud and Michael T. Schaub's research into Modularity and Dynamics on Complex Networks (part of the Cambridge Elements series and available free until 4 January).

There is no doubt about it, Santa is a mathematician. He must be, having every year to calculate the best way to get presents to so many homes in such a short space of time.

So in homage to such genius, we have two Santa themed puzzles for you to solve. You can find the answers at the bottom of the page (but try not to look too quickly).

Puzzle the First Every year the elves mark Santa's birthday by baking a cake, and adding a candle in the shape of each digit of Santa's age. Being rather mathematical, these elves like to add up the values of the candles, and call the answer a Santa number. For example, the Santa number of 723 is 7 + 2 + 3 = 12.

What is the sum of the first 99 Santa numbers (the Santa numbers of 1, 2, 3, ..., 99)?

(a) 746 (b) 862 (c) 900 (d) 924

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Puzzle the Second One of Santa's elves has a tricky parcel to pack. Its shape is a hexagon, with all sides of equal length. The elf has placed the parcel inside a square of side length 1, as shown in the diagram. What is the length of one of the hexagon sides?

Puzzle the First - Solution Imagine the candles in a pile. 1 to 9 occur 10 times each as units digits, for example 1 is the last digit of 1, 11, 21, …, 91. And they occur 10 times each as tens digits, for example 1 is the first digit of 10, 11, 12, …, 19. So the total is \((1+2+\dotsb+9) \times 20 = 45 \times 20 = 900\).

So the answer is (c).

Puzzle the Second - Solution Say the hexagon has side length \( h \) (so certainly \( h < 1 \)). The right-angled triangle in the top right of the diagram has short sides both of length \( 1-h \), and hypotenuse of length \( h \) (it's a side of the hexagon), so, by Pythagoras's theorem, \( h^2 = (1-h)^2 + (1-h)^2 = 2 - 4h + 2h^2 \), so \( h^2 - 4h + 2 = 0 \). We can solve that using the quadratic formula, which shows that \( h = 2 \pm \sqrt{2} \). But we know that \( h < 1 \), so \( h = 2 - \sqrt{2} \).

So the answer is (b).

If you enjoyed these puzzles you will find many more like them on our Mathematics Admissions Test pages from where these two puzzles were taken and festively adapted by our very own Vicky Neale.

Sally, PA to the Head of Department, has the hardest of mathematical tasks in Oxford Mathematics, namely the herding of mathematicians. She also asks the toughest question of the year:

"Are we doing a Christmas card this year?"

Because, of course, Sally doesn't mean "are we"; no, she means "what are we", as in "what are we going to do for a Christmas card this year?"

The External Relations Manager, recipient of the emailed question, sighs. The 50th email of the day, hot on the heels of diary invites to committees a year in advance and emails from people claiming to have found a simpler proof of Fermat's Last Theorem. Another unwanted task surely? Or at least a difficult one to fulfil?

Actually, no. Not unwanted at all, and certainly not hard for one simple reason: mathematicians.

Mathematicians are often portrayed as inward-looking, communicating in a forbidden language, far removed from imagination and creativity. But of course such clichés are not just untrue but misunderstand the nature of mathematics and mathematicians. It is an intensely creative subject, requiring imagination and conjecture allied to descriptive precision; and all imbued with an instinctive desire to find not only the best but the most elegant answer. This was no tricky task because there was a whole department of creativity to draw on.

So we launched our annual competition. A small prize was offered in return for a mathematically themed card. November is a busy month (well, everyone says every month is busy but November seems to really take its toll), but nonetheless the answers came pouring in. Jane suggested Farey diagrams would make lovely baubles; Damon proposed beautiful patterns formed by colour coded plots of the absolute values of polynomials with all coefficients +/- 1 evaluated over C; Joo-Hyun went for an animation of the generation of a Christmas tree with a few parameterised surfaces; while Elle, a member of our student support team, went for a Penrose triangle Christmas tree. And there were many more.

But in the end, with the advice of our inestimable designers at William Joseph, we went for Marina Simonian's suggestion, a Fourier transform Christmas tree, feeling it would best lend itself to animation. And so began a volley of email exchanges between Marina and Stéphane at William Joseph as they worked up the idea. Was it all becoming too much? Marina gave a succinct answer:

"When you love maths, it's fine."

You can see the final version in the video below and below that, a short explanation from Marina. Thank you to everyone involved.

Emmanuel Breuillard, the Sadleirian Professor of Pure Mathematics in Cambridge, has been appointed to the Professorship of Pure Mathematics in Oxford starting on 1 January 2022. Held by Professor Roger Heath-Brown FRS from its inception in 1999 until his retirement in 2016, the Professorship of Pure Mathematics is one of the most prestigious statutory positions in Oxford. Professor Breuillard will be a fellow of Worcester College.

Emmanuel Breuillard is an exceptionally broad mathematician with diverse interests intersecting with those of half a dozen research groups here in the Mathematical Institute in Oxford including Geometry, Number Theory and Combinatorics. His many research accomplishments include uniform versions of the celebrated Tits alternative in group theory, structure theorems for approximate groups in various settings, new results about Bernoulli convolutions, and greater understanding of random polynomials with +/- 1 coefficients.

He won a European Mathematical Society Prize in 2012 and was an invited speaker at the International Congress of Mathematicians in 2014. We very much look forward to his arrival.

PROMYS Europe Connect 2021 saw a group of enthusiastic and high-achieving young mathematicians gather (online) in July and August for a four-week intensive summer programme designed to give them the experience of thinking deeply about mathematics in a community of similarly mathematically excited students and staff. In other circumstances, PROMYS Europe is a six-week residential programme in Oxford, organised by a partnership of PROMYS (Boston), Wadham College and the Mathematical Institute at the University of Oxford, and the Clay Mathematics Institute. Circumstances being what they were, the 2021 programme, PROMYS Europe Connect, was tailored for an online format, in which we sought to capture as much as possible of the essence of the traditional event.

PROMYS Europe Connect 2021 was attended by 28 students, 2 of whom were returning students who first took part in PROMYS Europe in 2019, and 9 undergraduate counsellors. Participants represented Austria, Bulgaria, Czech Republic, Germany, Hungary, Norway, Romania, Serbia, Spain, Sweden, Switzerland, the UK and Ukraine. Places are offered to students on the basis of their academic potential as demonstrated in their application; we waive some or all of the fee (which is already heavily subsidised) for students who would otherwise be unable to participate. The programme is funded and resourced by the PROMYS Europe partnership, and by further financial support from alumni of the University of Oxford and Wadham College, and from the Heilbronn Institute for Mathematical Research.

This year, as usual, the core of the programme was a Number Theory course, taught by Glenn Stevens (Boston University, founding Director of PROMYS) and Henry Cohn (Microsoft Research, MIT). Students are encouraged to discover as much as possible for themselves, through their work on daily problem sets and through careful individual mentoring by their counsellor.

Alongside this was a Group Theory course with a similar philosophy, taught by Vicky Neale (Oxford), primarily aimed at the returning students, but also well attended by the first years. The returning students also worked on a research project on elliptic curves. Alongside their own mathematical seminars on category theory and analytic number theory, the counsellors organised social activities to help the students get to know each other and to build the sense of the community that is so important to the programme. These were complemented by guest lectures, exposing students to a range of current mathematical research. Plans are already under way for summer 2022, when we expect to return to the in-person format.

Here’s what participants said after PROMYS Europe Connect 2021:

Thank you for organizing all this, I feel like I almost cannot grasp the impact that this programme has had on my life in these three years of participation. [Returning student]

The online experience was actually much more engaging than I had expected. [Student]

PROMYS Europe Connect has left me with so many fantastic memories. I have truly felt like I am part of an amazing community of mathematicians who are all so passionate about what they do. [Counsellor]

We can't interview all our undergraduate applicants in the time available, so to help us decide who to shortlist, we set the Oxford Mathematics Admissions Test (MAT) which all applicants for Maths, Computer Science, or joint honours courses must take.

Yesterday, 3 November, 5000 aspiring students from all around the world took the MAT for entrance to Oxford (and other universities). Here, courtesy of our admissions guru, James Munro, are the answers in 10 minutes. And remember, as they say before the football results on TV, if you don't want to know the answers, look away now.

James will also host a longer debrief of MAT 2021 in his weekly online podcast next week (11 November). Everyone is welcome.

As part of the University of Oxford’s Black Academic Futures Scholarships, the Mathematical Institute and Pembroke College are delighted to invite talented UK Black or Mixed-Black students to apply for one fully funded postgraduate scholarship in 2022-2023 on one of the courses below:

DPhil in Mathematics or Centre for Doctoral Training in Mathematics of Random Systems.

So, the first term at university. And, more specifically, the first mathematical term at Oxford. What's in store? Well, our students' mathematical experience in their first term (and beyond) comprises two parts: lectures and tutorials. How do they work?

Lectures cover eight courses in the first term. These range from subjects such as Complex Numbers and Linear Algebra to an Introductory Calculus course. You can now watch an example, a Geometry lecture on Isometries, below. A full list of publicly available lectures can be found on our YouTube Channel.

Alongside lectures are tutorials where students, usually in pairs, meet with their tutor to go through the relevant course problem sheets. These tutorials provide the opportunity to spend time thinking and talking about the mathematics. You can watch an example - filmed in Trinity College in 2019 - below the lecture.

And of course there is the pleasure of meeting 200 other first term mathematicians like yourself and working with them on problems and sharing experience. Some of our first year students are sharing those experiences on our Twitter, Facebook and Instagram pages over the coming weeks.

"Historically, mathematics has been a largely male-dominated field, with women in mathematical academia consistently being underrepresented. A report in 2013 by the London Mathematical Society shows some progress has been made in increasing the participation of women in recent times, with the proportion of women pursuing an undergraduate degree in mathematics in the UK standing at around 42%.

However, when we look at the percentage undertaking a PhD this number drops to 19% and as we progress through the career stages of an academic, eventually we reach the disturbing statistic that only 6% of maths professors in the UK are women. This demonstrates what is sometimes referred to as a ‘leaky pipeline’ – one metaphor to describe the way in which women leave academia at a higher rate than men at every stage of a research career. The current female professors that make up that six percent have spent the last few decades working as mathematicians in a profession where they are largely outnumbered.

I wanted to speak to some women who chose to pursue a career in academia at a time when female mathematicians were few and far between, asking them to tell their stories, understand the challenges they have overcome and highlight the successes they have achieved. The common link between these women? They all started their journey into academia at Oxford in the 1980s, by undertaking a DPhil here at the Mathematical Institute.

The stories that follow come from interviews conducted with the three women: Sarah Rees, Frances Kirwan and Helen Byrne. Sarah Rees is a professor of pure mathematics at Newcastle, the first woman to be appointed to a permanent position in Newcastle’s Faculty of Science. Frances Kirwan is the Savilian Professor of Geometry at the Mathematical Institute here in Oxford, the first woman to hold this position since its creation in 1619. Helen Byrne is a professor of mathematical biology here at Oxford and was the recipient of the 2019 Society for Mathematical Biology Leah Edelstein-Keshet Prize. These women have gone on to have varied and successful careers as academics, providing real insight and new perspectives into their respective fields of mathematics, as well as helping the next generations of women succeed in maths.

The result of these interviews is this piece, describing their contrasting experiences and exploring issues such as: feeling and being treated differently; the importance of having inspiring figures and a strong community around you; the isolating world of research; the challenge of being a mother and a mathematician; what these women view as their biggest successes in life; and much more. The full piece can be found here."

Maddy Underwood

Maddy Underwood is an undergraduate at Worcester College. This article, and the longer piece, is the fruit of her Student Summer Research Project here in Oxford under the guidance of Mate Szabo. The Summer Research Projects aim to give our undergraduates a taste of the world of mathematical research.

Congratulations to Oxford Mathematics and Worcester College undergraduate Ellen who was a joint winner of the British Society for the History of Mathematics Undergraduate Essay Prize for her essay 'The "analysis" of a century: Influences on the etymological development of the word "analysis" in a mathematical context to 1750'.

Ellen says of her work: "I took the History of Maths module as I have always enjoyed hearing about how people and societies have thought about the concepts that we take for granted. I found that learning and exploring the original mathematical texts helped me to contextualise my place as an undergraduate in the overall mathematical story!

"My essay, which was adapted from the essay I submitted for my final coursework, explores the evolving meaning of the word ‘analysis’ in a mathematical context from Oughtred to Euler. It delves into themes including the geometric-analytic distinction and how the nature of mathematical texts, as well as their contents, has helped mathematical ideas to stick."

Ellen completed her degree this summer. Below, you can watch a lecture from the History of Mathematics course she took, one of the many undergraduate lectures we are making available to give an insight in to mathematical life in Oxford.