News

Thursday, 14 March 2019

Kristian Kiradjiev wins Gold Award at this year’s STEM for Britain

Oxford Mathematician Kristian Kiradjiev has won the Gold Award in the Mathematical Sciences category at this year’s STEM for Britain at the House of Commons on 13th March. This prestigious competition provides an opportunity for researchers to communicate their research to parliamentarians.  

Kristian’s poster covered his research into the mathematical modelling of flue-gas purification and the removal of toxic chemicals from the gas.

As reported last week, Kristian was one of three Oxford Mathematicians presenting in the Commons.

Friday, 8 March 2019

Apala Majumdar wins 2019 FDM Everywoman in Tech award

Oxford Mathematics Visiting Fellow and Reader in Applied Mathematics at the University of Bath, Apala Majumdar has been awarded the 2019 FDM Everywoman in Tech Academic Award. This is awarded to a woman in academia who has made an outstanding contribution to technology and science and whose work has made or has the potential to make a significant long-term impact in STEM.

Apala is an applied mathematician researching fundamental mathematical theories in material science. She specialises in Liquid Crystals and has published over 40 papers to date. Moreover, Apala works to inspire female researchers globally through mentorship and is deeply committed to teaching and training young people.

Apala was nominated by Oxford Mathematician and Director of the Oxford Centre for Industrial and Applied Mathematics (OCIAM), Alain Goriely, who said: “I cannot think of a more deserving candidate for an academic award for young women who are inspiring other female researchers around the world. Apala has single-handedly built an international network spanning four continents, making her one of the world leaders in her field and most internationally recognised of her generation."

The FDM Tech Awards take place in the week of International Women’s Day and celebrate 50 of the most talented individuals shaking up the tech industry.

 

Wednesday, 6 March 2019

Three Oxford Mathematicians to present their research in the House of Commons

Three Oxford Mathematicians, Kristian Kiradjiev, Liam Brown and Tom Crawford are to present their research in Parliament at this year’s STEM for Britain competition at the House of Commons on 13th March. This prestigious competition provides an opportunity for researchers to communicate their research to parliamentarians.  

Kristian’s poster covers his research into the mathematical modelling of flue-gas purification, Liam's poster researches computational models of cancer immunotherapy while Tom is researching the spread of pollution in the ocean.

Judged by leading academics, the gold medalist receives £2,000, while silver and bronze receive £1,250 and £750 respectively. 

Thursday, 28 February 2019

Heather Harrington awarded the Adams Prize

Oxford Mathematics' Heather Harrington is the joint winner of the 2019 Adams Prize. The prize is one of the University of Cambridge's oldest and most prestigious prizes. Named after the mathematician John Couch Adams and endowed by members of St John's College, it commemorates Adams's role in the discovery of the planet Neptune. Previous prize-winners include James Clerk Maxwell, Roger Penrose and Stephen Hawking.

This year's Prize has been awarded for achievements in the field of The Mathematics of Networks. Heather's work uses mathematical and statistical techniques including numerical algebraic geometry, Bayesian statistics, network science and optimisation, in order to solve interdisciplinary problems. She is the Co-Director of the recently established Centre for Topological Data Analysis.

Tuesday, 26 February 2019

Bendotaxis - when droplets are self-propelled in response to bending

We’re all familiar with liquid droplets moving under gravity (especially if you live somewhere as rainy as Oxford). However, emerging applications such as lab-on-a-chip technologies require precise control of extremely small droplets; on these scales, the forces associated with surface tension become dominant over gravity, and it is therefore not practical to rely on the weight of the drops for motion. Many active processes (requiring external energy inputs), such as those involving the use of temperature gradients, electric fields, and mechanical actuation, have been used successfully to move small droplets. Recently, however, there has been increasing interest in passive processes, which do not require external driving. One example of this is durotaxis, in which droplets spontaneously move in response to rigidity gradients (similar to the active motion of biological cells, which generally move to stiffer regions of a deformable substrate). Here, the suffix ‘taxis’ refers to the self-propulsive nature of the motion. In a recent study, Oxford Mathematicians Alex Bradley, Finn Box, Ian Hewitt and Dominic Vella introduced another such mechanism; Bendotaxis is self-propelled droplet motion in response to bending. What is particularly interesting is that the motion occurs in the same direction, regardless of whether the drop has an affinity to (referred to as ‘wetting’) the channel walls or not (‘non-wetting’), which is atypical for droplet physics.

A small drop confined to a channel exerts a force on the walls, as a result of surface tension; this force pulls the walls together when the drop wets them, and pushes them apart otherwise. By manipulating the geometry of the channel (leaving one end free, and clamping the other end), the deformation that results from this surface tension force is asymmetric—it creates a tapering in the channel. The drop subsequently moves in response to this tapering, which is towards the free end in both the wetting and non-wetting cases.

Using a combination of scaling arguments and numerical solutions to a mathematical model of the problem, the team were able to verify that it is indeed the capillary induced elastic deformation of the channel that drives the experimentally observed motion. This model allowed them to understand the dynamic nature of bendotaxis, and predict the motion of drops in these deformable channels. In particular, they identified several interesting features of the motion; counter-intuitively, it is predicted (and observed) that the time taken for a drop to move along the channel decreases as it increases in length. However, relatively long channels are susceptible to ‘trapping’, whereby the force exerted by the drop is sufficient to bring the channel walls into contact. It is hoped that understanding the motion will pave the way for its application on a variety of scales - for example, drug delivery on a laboratory-scale, and self-cleaning surfaces on a micro-scale.

Thursday, 21 February 2019

Oxford Mathematics Student Tutorial now online

The Oxford Mathematics educational experience is a journey, a journey like any other educational experience. It builds on what you learn at school. It is not unfamiliar and we don't want it to invisible. But it has aspects that are different. One of these is the tutorial system. Students have lectures. But they also have tutorials based on those lectures where they sit, usually in pairs, with a tutor, go through their work and, critically, get to ask questions. It is their tutorial.

Having streamed the First Year Students' Dynamics lecture last week and interviewed the students as they left the lecture, we now present the tutorial as it happened. Even if you are not a mathematician we hope the lectures and tutorial give you an insight in to how things work in Oxford. And maybe even encourage you, or someone you know, to think about giving Oxford a go. Or just giving maths a go.

 

 

 

 

 

 


 

Wednesday, 20 February 2019

When does one of the central ideas in economic theory work?

The concept of equilibrium is one of the most central ideas in economics. It is one of the core assumptions in the vast majority of economic models, including models used by policymakers on issues ranging from monetary policy to climate change, trade policy and the minimum wage. But is it a good assumption? In a paper just published in Science Advances, Oxford Mathematicians Marco Pangallo, Torsten Heinrich and Doyne Farmer investigate this question in the simple framework of games, and show that when the game gets complicated this assumption is problematic. If these results carry over from games to economics, this raises deep questions about economics models, and when they are useful for understanding the real world.

To understand what equilibrium is, it helps to think about a simple example. Kids love to play tic-tac-toe (also known as noughts and crosses), but at around eight years old they learn that there is a strategy for the second player that always results in a draw.  This strategy is what is called an equilibrium in economics. If all the players in the game are rational they will play an equilibrium strategy. In economics, the word rational means that the player can evaluate every possible move and explore its consequences to their endpoint and choose the best move. Once kids are old enough to discover the equilibrium of tic-tac-toe they quit playing because the same thing always happens and the game is really boring. One way to view this is that, for the purposes of understanding how children play tic-tac-toe, rationality is a good behavioural model for eight year olds but not for six year olds.

In a more complicated game like chess, rationality is never a good behavioural model. The problem is that chess is a much harder game, hard enough that no one can analyse all the possibilities, and the usefulness of the concept of equilibrium breaks down. In chess no one is smart enough to discover the equilibrium, and so the game never gets boring. This illustrates that whether or not rationality is a sensible model of the behaviour of real people depends on the problem they have to solve. If the problem is simple, it is a good behavioural model, but if the problem is hard, it may break down.

Theories in economics nearly universally assume equilibrium from the outset. But is this always a reasonable thing to do? To get insight into this question, Pangallo and collaborators study when equilibrium is a good assumption in games. They don’t just study games like noughts and crosses or chess, but rather they study all possible games of a certain type (called normal form games). They literally make up games at random and have two simulated players play them to see what happens. The simulated players use strategies that do a good job of describing what real people do in psychology experiments. These strategies are simple rules of thumb, like doing what has worked well in the past or picking the move that is most likely to beat the opponent’s recent moves.

The authors find that the prevalence of cycles in the structure of games is a very good indicator of the likelihood that strategies do not converge to equilibrium. This point is illustrated in the figure below. Cycles are indicated with red arrows in the payoff matrices – namely, the tables filled with numbers in the first row. When cycles are present, many learning dynamics are likely to perpetually fluctuate instead of converging to equilibrium. Fluctuating dynamics are colored red or orange in the panels of the second and third rows.

The theory then suggests that equilibrium is likely to be a wrong behavioral model when the game has a cyclical structure. When are cycles prevalent, and when are they rare?

When the game is simple enough, in the sense that the number of actions available to each player is small, cycles are rare. When the game is more complicated, whether or not cycles are common depends on whether or not the game is competitive.  If the incentives of the players are lined up, cycles are rare, even if the game is complicated.  But when the incentives of the players are not lined up and the game gets complicated, cycles are common. 

These results match the intuition about noughts and crosses vs. chess: complicated games are harder to learn and it is harder for players to coordinate on an equilibrium when one player’s gain is the other player’s loss. The main novelty of the paper is that the authors develop a formalism to make all this quantitative. This is confirmed in the figure below, which shows the share of cycles (dashed lines) and the non-convergence frequency of six learning algorithms (markers) as the complicatedness and competitiveness of a game vary. (The payoff correlation Γ is negative when the game is competitive.)

Many of the problems encountered by economic actors are too complicated to model easily using a normal form game. Nonetheless, this work suggests a potentially serious problem. Many situations in economics are complicated and competitive. This raises the possibility that many important theories in economics may be wrong: If the key behavioural assumption of equilibrium is wrong, then the predictions of the model are likely wrong too. In this case new approaches are required that explicitly simulate the behaviour of the players and take into account the fact that real people are not good at solving complicated problems.

 

 

Tuesday, 19 February 2019

Fano Manifolds Old and New

Oxford Mathematician Thomas Prince talks about his work on the construction of Fano manifolds in dimension four and their connection with Calabi-Yau geometry.

"Classical algebraic geometry studies the vanishing loci of finite collections of polynomial equations; usually under some conditions that ensure this locus has some desirable properties. The first objects studied in this subject (in its modern history) were Riemann surfaces, one dimensional objects over the complex numbers, topologically equivalent to $n$-holed tori. The attempt to replicate the classification of curves in the context of algebraic surfaces made by the 'Italian school' led by Castelnuovo and Enriques in the early 20th century led to a fundamental insight: the classification divides naturally into two distinct problems. First one studies a courser 'birational' classification of surfaces, before analysing the surfaces within each birational class. In two dimensions the second problem has a simple solution: these surfaces are related by 'blowing up' and 'blowing down', explicit operations first described by Noether. This became the model and prototype for the modern subject of birational geometry, which developed rapidly in the later 20th century, with fundamental contributions made by Hironaka, Mori, Shafarevich, and others.

In the contemporary treatment of the subject, a particularly privileged role is given to two classes of algebraic (or complex analytic) objects. The Calabi-Yau manifolds, generalising elliptic curves; as well as a particular 'minimal' class of surfaces called K$3$ surfaces, and Fano manifolds. Fano manifolds play a key role in Mori's minimal model program, itself a sweeping higher-dimensional generalisation of key methods used in the classification of surfaces. In particular this program led to the spectacular Mori-Mukai classification of Fano manifolds in dimension three, building on work on Iskovskikh.

The construction of Fano manifolds in dimension four is thus a central open problem, and the focus of ongoing research. My own interest relates to their connection with Calabi-Yau geometry: a very rough analogy would say that a Fano manifold is to a Calabi-Yau manifold what a manifold with boundary is to a manifold. Recent ideas from string theory - in particular from the field of mirror symmetry - have introduced a number of new tools to the study of Calabi-Yau manifolds, particularly following Kontsevich, Strominger-Yau-Zaslow, and Gross-Siebert. Following Givental and Kontsevich the subject of mirror symmetry has also been extended to incorporate Fano manifolds, and suggests an approach to their construction via toric degeneration. The focus of my own research is to develop these insights to produce systematic constructions of Fano manifolds along quite a different line from that taken in birational geometry. Recent progress includes a new construction of surfaces with certain classes of singularities and the classification of 527 'new' Fano fourfolds - obtained in joint work with Coates and Kasprzyk - as complete intersections in 8-dimensional toric manifolds."

Tuesday, 19 February 2019

Oxford Mathematics Student Lecture live streamed for the first time

One of our aims in Oxford Mathematics is to show what it is like to be an Oxford Mathematics student. With that in mind we have started to make student course materials available and last Autumn we filmed and made available a first year lecture on Complex Numbers. And last week, as we promised, we went a step further and livestreamed a first year lecture. James Sparks was our lecturer and Dynamics his subject. In addition, we interviewed students as they left the lecture in preparation for filming a tutorial which will also be made available later this week. 

It has taken over 800 years to get here, but we are delighted to be able to share what we do and show that it is both familiar and challenging. The lecture is below together with the interviews. We welcome your thoughts. The tutorial will follow.


 

 


 

 

 

 

 

Tuesday, 12 February 2019

Love and Maths - first ever live streaming of student lecture this Thursday, 14th Feb,10am

Lecture theatre 1

It's Valentine's Day this Thursday (14th February in case you've forgotten) and Love AND Maths are in the air. For the first time, at 10am Oxford Mathematics will be LIVE STREAMING a 1st Year undergraduate lecture. In addition we will film (not live) a real tutorial based on that lecture.

The details:
LIVE Oxford Mathematics Student Lecture - James Sparks: 1st Year Undergraduate lecture on 'Dynamics', the mathematics of how things change with time
14th February, 10am-11am UK time

Watch live and ask questions of our mathematicians as you watch

https://www.facebook.com/OxfordMathematics
https://livestream.com/oxuni/undergraduate-lecture

For more information about the 'Dynamics' course: https://courses.maths.ox.ac.uk/node/37555

The lecture will remain available if you can't watch live.

Interviews with students:
We shall also be filming short interviews with the students as they leave the lecture, asking them to explain what happens next. These will be posted on our social media pages.

Watch a Tutorial:
The real tutorial based on the lecture (with a tutor and two students) will be filmed the following week and made available shortly afterwards
https://www.youtube.com/channel/UCLnGGRG__uGSPLBLzyhg8dQ

For more information and updates:
https://twitter.com/OxUniMaths
https://facebook.com/OxfordMathematics

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