Tuesday, 1 November 2016

North meets South - helping mathematicians to understand each other

Mathematics can look like a foreign language to those who have not studied it in depth. Even for mathematicians, it can be difficult to understand the work of colleagues in other branches of mathematics, or indeed to know what questions they are seeking to answer in their research, because the vocabularies are so specialised and technical. A huge success of modern mathematics is that it is both broad and deep: mathematicians study a wide range of topics, and our knowledge in many of these areas is now so great that in order to work at the cutting edge of research one must specialise a lot. 

There is a tension, though, because while mathematicians need to immerse themselves in their research areas in order to make progress, at the same time they also need to be aware of developments in other areas that might help their work. Above all, and perhaps contrary to some perceptions, mathematics is highly interconnected and collaborative, and very often progress happens by making connections between research areas.

Oxford Mathematics seeks to address this in a number of ways. At the physical level, the splendid new home of the department, the Andrew Wiles Building, has been designed to facilitate both deliberate collaboration and also spontaneous exchanges of mathematical ideas through communal spaces (not least the excellent Café π).

Last year, Oxford Mathematics went one step further with a new initiative aimed at helping mathematicians to get to know what their Oxford colleagues are working on, and to give early career researchers the opportunity to share their work.

The new Fridays@4 project involves a programme of weekly sessions, aimed primarily but not exclusively at graduate students and postdocs, including a mix of colloquia, skills training, and advice on personal and career development. Part of this programme is the new ‘North meets South’ colloquium, organised by early career researchers and with early career researchers as invited speakers.

The colloquia happen once or twice each term. Each features two speakers, one from the north wing of the Andrew Wiles Building (roughly corresponding to pure mathematics, although that is a rather crude way to subdivide mathematics), and one from the south wing (roughly corresponding to applied mathematics). The speakers are asked to ensure that their talks are accessible to all the mathematicians in the department, not only those in their research areas.  Last year saw talks on a selection of topics: cluster algebras, modelling data streams, topological quantum field theory and defects in liquid crystals. This week (November 4) sees the first North meets South colloquium for this academic year, featuring Emilie Dufresne and Robert Van Gorder talking about their work. Interestingly, while both are working in Applied Mathematics, much of their work has also been in Pure Maths and Emilie's talk on separating sets in Invariant Theory is indeed Pure Mathematics. She and Robert are perhaps the ultimate North meets Southers, today's modern mathematicians. 

Heather Harrington and Brent Pym were instrumental in the setting up of North meets South. “We created this internal colloquium to learn what other young mathematicians here in Oxford are working on, and to form a network of early career researchers. Last year's speakers showed great skill, delivering exciting and accessible talks about research-level mathematics - and in only 30 minutes each! As organisers, we have observed that this experience is rewarding for the speakers, but even more so for Oxford Mathematics, as it brings together mathematicians from many subfields. We hope that North meetsSouth is another example of how such events can spark interactions that cross mathematical lines.”

The success of the North meets South colloquium is in itself a reminder of why mathematicians need to talk to each other, both to ensure that they make the most of the ideas and expertise around them, and, above all, to motivate them in their work. This is not a recreational add-on, but a core component of a modern mathematician's life.

Monday, 31 October 2016

A peek round the corner - introducing undergraduates to mathematical research

What is it like to do mathematical research? Many undergraduates wonder this, as they consider whether they would like to pursue graduate studies. There is no better way for the department to answer the question than to give undergraduates the opportunity to work on their own mathematical research projects. This summer the Oxford Mathematics enabled around 50 students to carry out such projects, working with faculty and postdocs in the department.

Eliza Casapopol (Balliol College) did a project with Dr Tom Sanders. "The summer project was a great opportunity for me as a mathematician. It has given me insight into what research is about and the biggest achievement was that it taught me to enjoy asking and answering my own questions. I think the project has given me a better idea about research and its challenges compared to what I knew so far, and it allowed me to see which areas of maths I am most interested in. Even if it seems like a difficult path, these past weeks have showed me that there is so much sense of fulfilment when you understand a proof or manage to come up with a new question or idea."

Rosemary Walmsley (Worcester College) worked with Professor Alison Etheridge, and said "my summer project has been a fantastic opportunity to experience mathematical research, and has given me a really valuable taste of what it would be like to do postgraduate study. It's been great to be able to get engrossed in an area I knew relatively little about beforehand - probabilistic models used in genetics - and to be able to explore it in a less prescribed way than I am used to with lecture courses. The project was more varied than I expected: reading papers, posing new questions, working on these questions, discussing ideas with my supervisor and DPhil students and writing up what I'd done. I would very much recommend doing a summer project!"

Shati Patel (Lady Margaret Hall) was part of a group of students who worked with Dr Jennifer Balakrishnan. "I really enjoyed doing a summer project as it's a completely different way of working compared to term time lectures and problem sheets. Not only was I introduced to some interesting theory about elliptic curves, I also got some general experience writing code and working with the command line. Our project involved about 10 people so it was incredibly useful to help each other and share ideas."

There is no doubt that this window on to a possible future is vital. Students not only learned about the process of research and its mix of personal focus and collaboration, they also were given a taste of new areas of mathematics beyond their current experience, the areas where Oxford mathematicians are leading the world in research and where these students might one day join them.

Three students received financial support from the London Mathematical Society. Eliza Casapopol (Balliol College), Daniel Fletcher (Oriel College) and Lorenzo Sarnataro (Worcester College) were among the 36 students selected nationally to receive funding, which was matched using funds from the Mathematical Institute and the Engineering and Physical Sciences Research Council (EPSRC). Further students were supported by EPSRC funding distributed through the Oxford Mathematical, Physical and Life Sciences Division. Oxford Mathematics' own funding enabled a significantly larger number of high-achieving students to experience what it is like to do mathematical research.

Thursday, 20 October 2016

So just who were the Merton Scholars?

In its first five hundred years Oxford University had many fine mathematicians, astronomers and philosophers – from the Merton scholars of the early 14th century to the newly appointed Savilian Professors of Geometry and Astronomy in the 17th century. Indeed some of the most sophisticated mathematical discussions of the Middle Ages took place at Oxford in the 14th century.

Find out more in the latest in our Oxford Mathematics History series.

PDF icon Early Mathematicians.pdf




Thursday, 20 October 2016

Savilian Professor Nigel Hitchin reflects

As he retires from the the Savilian Chair of Geometry, Oxford Mathematician Nigel Hitchin reflects in this interview with Martin Bridson. From early mathematical inspiration at school in Duffield, Derbyshire, Nigel recalls his often unplanned progress via Jesus College, Oxford, Princeton, Cambridge and Warwick, before his final return to Oxford.

Along the way such luminaries as Michael Atiyah and Simon Donaldson play their part as Nigel talks about time spent with physicists in Cambridge, the Eureka moments when the answers take shape, to his final reflections on a career where the name Hitchin is attached to so many of the tools of modern geometry and which culminated in the award of the 2016 Shaw Prize.





Wednesday, 19 October 2016

Fashion, Faith and Fantasy - Roger Penrose Public Lecture now online

What can fashionable ideas, blind faith, or pure fantasy have to do with the scientific quest to understand the universe? Surely, scientists are immune to trends, dogmatic beliefs, or flights of fancy? In fact, Roger Penrose argues that researchers working at the extreme frontiers of mathematics and physics are just as susceptible to these forces as anyone else.

In this lecture, based on his new book, Roger argues that fashion, faith, and fantasy, while sometimes productive and even essential, may be leading today's researchers astray, most notably in three of science's most important areas - string theory, quantum mechanics, and cosmology.





Tuesday, 18 October 2016

Hummingbirds, umbrellas and hopper poppers do it. But why not as quickly as expected?

Many elastic structures have two possible equilibrium states. For example umbrellas that become inverted in a sudden gust of wind, nanoelectromechanical switches, origami patterns and even the hopper popper, which jumps after being turned inside-out. These systems typically move from one state to the other via a rapid ‘snap-through’. Snap-through allows plants to gradually store elastic energy, before releasing it suddenly to generate rapid motions, as in the Venus flytrap . Similarly, the beak of the hummingbird snaps through to catch insects mid-flight, while technological applications are increasingly exploiting snap-through instabilities.

In all of these scenarios, it is the ability to repeatedly generate fast motions that gives snap-through its utility. However, estimates of the speed of snap-through suggest that it should occur more quickly than is usually observed. In their research published n Nature Physics, Oxford Mathematicians Michael GomezDominic Vella and Derek Moulton study the dynamics of snap-through in detail, showing that, even without dissipation, the dynamics slow down close to the snap-through transition. This is reminiscent of the slowing down observed in critical phenomena (for example the time taken for oscillations in the climate to die down is thought to grow larger as a ’tipping point’ is reached). As well as providing a handheld demonstration of such phenomena, the work provides a new tool for tuning dynamic responses in applications of elastic bistability: for example it shows that to obtain faster snap-through in applications such as robotics, the system needs to be pushed well beyond the snap-through transition. 


Tuesday, 4 October 2016

Using geometry to choose the best mathematical model

Across the physical and biological sciences, mathematical models are formulated to capture experimental observations. Often, multiple models are developed to explore alternate hypotheses.  It then becomes necessary to choose between different models.

Oxford Mathematician Heather Harrington and colleagues from the United States have explored the problem of model selection by regarding mathematical models as geometric objects in space. In general, model selection is a hard problem, but recasting it in geometric terms allows the authors to give a new methodology for selecting the best explanation of observed phenomena, thereby bringing recent groundbreaking developments in nonlinear algebra to the study of biological and other complex systems.

Specifically their paper, published today in the Royal Society Journal Interface, considers polynomial models (e.g., mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimisation using numerical algebraic geometry. The authors use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. This approach exploits the geometric structures relating models and data, and demonstrates its utility on examples from cell signalling, synthetic biology, and epidemiology.

Tuesday, 4 October 2016

Frances Kirwan wins Suffrage Science award

The Clinical Sciences Centre based at Imperial College in London has launched a new initiative to celebrate women in maths and computing. As a new branch of the existing Suffrage Science scheme, it will encourage women into science, and to reach senior leadership roles.

Women make up no more than four in ten undergraduates studying maths. Suffrage Science aims to make a difference. There are currently two sections, one for women in the Life Sciences, and one for those in Engineering and the Physical Sciences. Now there is a third specialism, for women in Maths and Computing. Twelve women will receive awards to celebrate their scientific achievements and ability to inspire others. Oxford Mathematician Frances Kirwan FRS is one of the first recipients of the award for Mathematics.

The awards themselves are pieces of jewellery, designed by students at the arts college Central Saint Martins-UAL, and inspired by science. One, a silver bangle, holds a secret. Engraved on the inside, and hidden beneath a layer of silver, is what many mathematicians consider the most beautiful equation in mathematics, Euler’s equation.


Wednesday, 21 September 2016

H is for Homology - The Oxford Mathematics Alphabet

A life belt, a coffee cup, a jumping ball, a beach ball. What do these objects have in common? What sets them apart? Questions like these come under the mathematical umbrella of topology. And the theory of homology enables us to explore and understand them. Find out more in the latest in our Oxford Mathematics Alphabet.

Saturday, 17 September 2016

Roger Heath-Brown in conversation with Ben Green

Roger Heath-Brown is one of Oxford's foremost mathematicians. His work in analytic number theory has been critical to the advances in the subject over the past thirty years and garnered Roger many prizes.

As he approached retirement, Roger gave this interview to Ben Green, Waynflete Professor of Mathematics in Oxford and himself a leading figure in the field of number theory. In the interview Roger reflects on his influences, his achievements and the pleasures that mathematics has given him.