Friday, 22 January 2021

Maths Makes A Difference - a webinar for Year 12 students about how maths contributes to society - 2 February, 4-5pm

The future is full of uncertainty, but we still need to make plans and decisions based on the data we have.  Where should a hospital invest its resources to allow for changing health needs in a year's time?  Should the supermarket order extra ice cream because the summer will be warm and sunny?  Should the council road maintenance team get extra gritting salt ready for an icy winter? Making predictions is hard - and maths can help, as we’ll see in this interactive webinar.

Maths Makes a Difference is a collaboration between the Mathematics outreach teams at Oxford and Cambridge. This interactive webinar series for students in Year 12 at state schools in the UK (and Year 10 later in the year) will explore aspects of maths that make a difference to the world and society. More information and registration.

The webinars will be led by Claire Metcalfe from Cambridge and Vicky Neale from Oxford.

Wednesday, 20 January 2021

Call for applications for PROMYS Europe Connect 2021

We are delighted to announce PROMYS Europe Connect for 2021, online from 12 July to 6 August.

In view of continuing restrictions and uncertainty around Covid-19, we are designing PROMYS Europe Connect as a unique 4-week online programme that captures many of the key elements of the usual PROMYS Europe experience. PROMYS Europe is a challenging mathematics summer programme based at the University of Oxford, UK.

PROMYS Europe Connect is seeking

  • Pre-university students from across Europe (including all countries adjacent to the Mediterranean) who show unusual readiness to think deeply about mathematics;
  • Undergraduate students who would like to work with them as counsellors. 

PROMYS Europe Connect is designed to encourage mathematically ambitious students who are at least 16 to explore the creative world of mathematics. Participants will tackle fundamental mathematical questions within a richly stimulating and supportive online community of fellow first-year students, returning students, undergraduate counsellors, research mentors, faculty, and visiting mathematicians.

First-year students will focus primarily on a series of very challenging problem sets, daily lectures, and exploration projects in Number Theory.  There will also be a programme of talks, by guest mathematicians and the counsellors, on a wide range of mathematical subjects, as well as courses aimed primarily at students who are returning to PROMYS Europe for a second or third time.

PROMYS Europe is a partnership of Wadham College and the Mathematical Institute at the University of Oxford, the Clay Mathematics Institute, and PROMYS (Program in Mathematics for Young Scientists, founded in Boston in 1989).

The programme is dedicated to the principle that no one should be unable to attend for financial reasons.  Most of the cost is covered by the partnership and by generous donations from supporters. In addition, full and partial financial aid is available for those who need it.

Applications for counsellors and students are available on the PROMYS Europe websiteThe closing date for counsellor applications is 7 February.  The closing date for first-year student applications is 14 March, and students will need to allow enough time before the deadline to tackle the application problems.  PROMYS Europe Connect will run online from 12 July to 6 August.

Wednesday, 6 January 2021

Rethinking defects in patterns

Social distancing is integral to our lives these days, but distancing also underpins the ordered patterns and arrangements we see all around us in Nature. Oxford Mathematician Priya Subramanian studies the defects in such patterns and shows how they relate to the underlying pattern, i.e. to the distancing itself.

"Nature has many examples of situations where individuals in a dense group have to balance between short range repulsion (e.g., competition for resources) and long range attraction (e.g. safety in numbers). This naturally leads to an optimal length scale for separation between neighbours. If we think of each agent as a circle with this optimal length as the diameter, the most efficient packing is in a hexagonal arrangement. So, it is unsurprising that we find these group of gannets nesting on a beach in Muriwai, New Zealand forming a largely hexagonal pattern.


Figure 1: When neighbours keep their distance: gannets nesting at Muriwai Beach, New Zealand. The image on the left shows the overall hexagonal ordering of the birds while that on the right shows a penta-hepta defect (PHD). The green (pink) markers identify gannets with five (seven) neighbours instead of the usual six. Photo credit: Barbora Knobloch. Adapted from [1].

Patterns are rarely perfect and defects arise due to many factors such as boundary conditions (e.g. cliff edge), fluctuations in the background (e.g. unevenness in the ground), etc. A generic defect that arises in such hexagonal patterns is highlighted in the right panel of the picture of the gannets. Normally each gannet will have six neighbours, but here we see that the gannet marked with a green dot has only five neighbours while the gannet marked with a pink marker has seven. Such a structure consisting of a bound state with one location having five neighbours and another location seven neighbours, instead of the usual six, is called a penta-hepta defect (PHD). Hexagonal arrangements are found in many areas of physics, from patterns formed in heated fluids, to self-assembled crystals formed in both hard materials (e.g. graphene) and soft materials (e.g. star-shaped polymers [2]). Equally prevalent in all such hexagonal arrangements is the possibility for PHDs.

Traditionally pattern formation techniques used to investigate defects use an amplitude-phase formulation, where a periodic pattern has a homogenous amplitude and a varying phase. The topological charge of a defect is calculated by integrating the phase around any closed curve enclosing the defect, and this quantity does not change when the defect moves. Topological defects [3] are associated with zeros of the amplitude (where the phase becomes undefined): these defects have non-zero topological charge and so they can only be eliminated or healed by interacting with another topological defect with opposite charge. On the other hand, non-topological defects, such as PHDs, have a well-defined phase everywhere (implying zero topological charge) and so were thought to be able to heal by themselves. However, if the defect has an internal structure it may persist as a result of frustration and get locked/pinned to the background periodic state: the gannets in the PHD in Figure 1 could re-arrange themselves to remove the defect, but to do so would involve all nearby gannets moving a little bit, so in practice they don't.


Figure 2: Coexisting equilibria with penta-hepta defects separating regions of hexagons with different orientations in the SH23 system [1]. All three states are dynamically metastable. 

We explore such non-topological defects in the prototypical pattern forming Swift-Hohenberg model in our recent work [1], by adopting a different point of view and thinking of defects as spatially localised structures [4]. We focus on grain boundaries separating two-dimensional hexagon crystals at different orientations (shown in Figure 2): these grain boundaries are closed curves containing a ring of PHDs. Even with the parameters all the same, the model has many different stable configurations of these grain boundaries, and solution branches connected to each of these states form isolas that span a wide range of the model parameters, opening up multiple interesting questions about such defect states. Our results will also be applicable to understanding the role of grain boundaries in two dimensional solids such as graphene [5], in which defects play a crucial role near phase transition, i.e., melting [6], and in determining bulk properties of a material."


[1] Snaking without subcriticality: grain boundaries as non-topological defects, P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, arXiv:2011.08536, 2020.

[2] Two-dimensional crystals of star polymers: a tale of tails, I. Bos, P. van der Scheer, W. G. Ellenbroek and J. Sprakel, Soft Matter, 15, 615-622, 2019

[3] The topological theory of defects in ordered media, N. D. Mermin, Rev. Mod. Phys., 51, 591-648, 1979.

[4] Spatial localisation in dissipative systems, E. Knobloch, Annu. Rev. Condens. Matter Phys., 6, 325-359, 2015. 

[5] Energetics and structure of grain boundary triple junctions in graphene, P. Hirvonen, Z. Fan, M. M. Ervasti, A. Harju, K. R. Elder and T. Ala-Nissila, Sci. Rep., 7, 1-14, 2017.

[6] Melting of graphene: from two to one dimension, K. V. Zakharchenko, A, Fasolino, J. H. Los and M. I . Katsnelson, J. Phys.: Condens. Matter, 23, 202202, 2011. 

Friday, 1 January 2021

The launch of the Oxford Online Maths Club

Happy New Year! 2021 has a lot to make up for after 2020, so we're starting with a bang with the launch of the Oxford Online Maths Club, a new weekly maths livestream from Oxford Mathematics.

The Club provides free super-curricular maths for ages 16-18. It is aimed at people about to start a maths degree at university or about to apply for one. We'll be livestreaming one hour of maths problems, puzzles, mini-lectures, and Q&A, and we'll be exploring links between A level maths and university maths with help from our Admissions Coordinator James Munro and our current Oxford Mathematics students. And you get to ask questions and share thoughts and feelings with like-minded mathematicians. 

In a nutshell, it’s free, interactive, casual, and relaxed, with an emphasis on problem-solving techniques, building fluency, and looking ahead at links to university maths. The Club follows in the footsteps of James's hugely popular weekly MAT (Mathematics Admissions Test) sessions where he went thorough entrance problems and took live questions.

Whether you're the only person you know interested in maths, or you're an entire sixth-form maths club looking for more content, we're here for you in 2021! Join us every Thursday 16:30 starting this Thursday, 7 January. 

Friday, 18 December 2020

Peter Michael Neumann OBE (28 December 1940 - 18 December 2020)

We are very sad to hear the news of the death of Peter Neumann earlier today. Peter was the son of the mathematicians Bernhard Neumann and Hanna Neumann and, after gaining a B.A. from The Queen's College, Oxford in 1963, obtained his D.Phil from Oxford University in 1966.

Peter was a Tutorial Fellow at The Queen's College, Oxford and a lecturer in the Mathematical Institute in Oxford, retiring in 2008. His work was in the field of group theory. He is also known for solving Alhazen's problem in 1997. In 2011 he published a book on the short-lived French mathematician Évariste Galois.

In 1987 Peter won the Lester R. Ford Award of the Mathematical Association of America for his review of Harold Edwards' book Galois Theory. In 2003, the London Mathematical Society awarded him the Senior Whitehead Prize. He was the first Chairman of the United Kingdom Mathematics Trust, from October 1996 to April 2004 and was appointed Officer of the Order of the British Empire (OBE) in the 2008 New Year Honours. Peter was President of the Mathematical Association from 2015-2016.

Tuesday, 15 December 2020
Sunday, 13 December 2020

The Oxford Mathematics E-Newsletter - our quarterly round-up of our greatest hits

The Oxford Mathematics e-newsletter for December is out. Produced each quarter, it's a sort of 'Now That's What I Call Maths,' pulling together our greatest hits of the last few months in one place.

It's for anyone who wants a flavour of what we do - research, online teaching, public lectures, having a laugh.

And it's COVID-lite. Click here.

Saturday, 12 December 2020

Full 2nd Year Oxford Mathematics Undergraduate course publicly available for the first time

Over the past few weeks we have made 7 undergraduate lectures publicly available, sampling a range of topics from Geometry to Differential Equations. Today & over the next 2 weeks for the first time we're showing a full course on our YouTube Channel. Ben Green's 2nd Year 'Metric Spaces' (the first half of the Metric Spaces and Complex Analysis course)' gets to grips with the concept of distance. 

We are making these lectures available to give an insight in to life in Oxford Mathematics. All lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated worksheet. Course materials can be found here




Friday, 4 December 2020

Roger Penrose's Nobel Lecture and presentation of Prize

This Tuesday, 8th December, from 8am GMT onwards (repeated) you can watch 2020 Physics Laureate and Oxford Mathematician Roger Penrose's specially recorded Nobel Lecture in which he talks about the background to and genesis of his work on Black Holes which won him the prize; and also where our understanding of Black Holes is taking us. 

On the same day Roger will be presented with the Nobel diploma and medal at the Swedish Ambassador’s Residence in London and you can watch this as part of the Nobel Prize Awards Ceremony from 3.30pm GMT on Thursday 10 December. Watch both here

As Roger said on receiving the news of the award: "In 1964 the existence of Black Holes was not properly appreciated. Since then they have become of increased importance in our understanding of the Universe and I believe this could increase in unexpected ways in the future."

Roger Penrose is one of our greatest living scientists. His work on Black Holes provided the mathematical tools needed by experimentalists to go and find Black Holes. His fellow prize winners, Andrea Ghez and Reinhard Genzel went and did just that.

However, Roger's work has ranged much further than just the Universe, from twistor theory to quasi-periodic tiling, spin networks to impossible triangles, a range that perhaps might not be so encouraged in academia today.

Now in his 90th year Roger is still researching and writing. He will give an Oxford Mathematics Public Lecture in January 2021 to celebrate the Nobel Prize.

Photography below and above by Professor Alain Goriely.  Updated photographs further below of Roger receiving the Nobel Medal and Diploma from the Swedish Ambassador in London on 8 December.

Sunday, 22 November 2020

Our latest Online Student Lecture - 2nd Year Linear Algebra

The latest in our Autumn 2020 series of lectures is the first lecture in Alan Lauder's Second Year Linear Algebra Course. In this lecture Alan (with help from Cosi) explains to students how the course will unfold before going on to talk specifically about Vector Spaces and Linear Maps.

All lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated worksheet. The course materials and worksheets can be found here.

That's Cosi on the left.