# The Annual Collingwood Lecture

A generous bequest has allowed the Department to institute an annual lecture in memory of Sir Edward Collingwood FRS. The lectures are given by mathematicians of international renown and are suitable for a general audience. We welcome visitors from other departments and from outside the University.

Edward Collingwood managed the family estate at Alnwick in Northumberland whilst simultaneously having a successful career as a research mathematician. He is probably known best for his work on the theory of Cluster Sets. He gave up a great deal of his time to medical administration and was, in addition, Chairman of the Council of Durham University from 1955 to his death in 1970. Collingwood College, Durham is named after him and the small research library in the Department began from the nucleus of his books, collected works and journals. He was knighted in 1962, elected to the Royal Society in 1965 and became President of the London Mathematical Society in 1969.

The first Collingwood Lecture was given in 1984 by Professor Christopher Zeeman FRS on "The discovery of perspective during the Renaissance". A list of subsequent lectures is given below.

Academic Year |
Date |
Speaker |
Institution |
Title |

23/24 |
23 Jan 2024 |
Professor Alison Etheridge FRS |
Oxford University | |

22/23 |
6 Jun 2023 |
Professor Christina Pagel |
University College London | |

18/19 |
21 Nov 2018 |
Professor Gwyneth Stallard |
The Open University | |

16/17 |
16 Feb 2017 |
Professor Pierre Cartier |
Institut des Hautes Etudes Scientifiques | |

15/16 |
10 May 2016 |
Professor Ray Goldstein FRS |
University of Cambridge | |

14/15 |
26 Feb 2015 |
Professor Martin Hairer FRS |
University of Warwick | |

13/14 |
30 Jan 2014 |
Professor Wendelin Werner |
ETH, Zurich | |

12/13 |
5 Nov 2013 |
Professor Peter Higgs FRS |
University of Edinburgh | |

11/12 | 13 Mar 2012 |
Professor A. O'Hagan |
University of Sheffield |
Masters of Uncertainty |

10/11 | 25 Nov 2010 | Professor Robert S. MacKay FRS | University of Warwick |
The Mathematics of Emergence |

09/10 | 6 May 2010 |
Professor Sir John Ball FRS |
University of Oxford |
Mathematics in the Public Eye: the story of Perelman and the Poincaré conjecture |

08/09 | 7 May 2009 |
Professor David Spiegelhalter FRS |
University of Cambridge |
Understanding Risk and Uncertainty |

07/08 | 23 Nov 2008 | Professor Vladimir Popov | Steklov Institute, Moscow | One and a half centuries of invariant theory |

06/07 | 2 Mar 2007 | Professor Tony Sudbery | University of York | Alice and Bob in the quantum wonderland |

05/06 | 10 Mar 2006 | Professor Frank Kelly |
University of Cambridge |
Traffic through Networks |

04/05 | 15 Feb 2005 | Professor Vladimir Turaev | University of Strasbourg | Curves and Words |

03/04 | 3 Nov 2004 | Professor Jon Keating | University of Bristol | Random Matrices and the Riemann Zeros |

02/03 | 18 Feb 2003 | Professor Don B. Zagier | College de France/MPI Bonn | The Experimental Side of Number Theory |

01/02 | 9 Nov 2001 | Professor GR Grimmett | University of Cambridge | Diffusion, Finance and Universality |

99/00 | 11 Nov 1999 | Professor NS Manton FRS | University of Cambridge | Are Particles Solitons |

98/99 | 23 Nov 1998 | Sir Michael Atiyah OM FRS | University of Edinburgh | The Icosahedron Past and Present |

96/97 | 6 Dec 1996 | Professor KW Morton | University of Oxford | Can We Trust the Numbers We Get From Our Computers |

95/96 | 5 Dec 1995 | Professor M Berry FRS | University of Bristol | Quantum Mechanics, Chaos and the Prime Numbers |

94/95 | 28 Nov 1994 | Professor PJ Green | University of Bristol | E is MC Squared: Inference by Throwing Dice |

93/94 | 3 Dec 1993 | Dr WBR Lickorish | University of Cambridge | Knots in the Nineties |

91/92 | 24 April 1992 | Professor R Penrose FRS | University of Oxford | Magic Dodecahedra and the Mystery of Quantum Entanglement |

90/91 | 26 Feb 1991 | Professor DV Lindley | University of Warwick | The Logical Analysis of Experimental Results (with Applications to Tea and Wine) |

89/90 | 13 Mar 1990 | Professor JD Barrow | University of Sussex | Why is the Universe Mathematical? |

88/89 | 25 Apr 1989 | Professor NH Kuiper | IHES, Paris | Convexity, Knots and Surfaces |

87/88 | 14 Mar 1988 | Professor D Williams FRS | University of Cambridge | Probability: Philosophy and Practice |

86/87 | 24 Feb 1987 | Dr JS Bell | CERN, Geneva | No Action at a Distance? |

85/86 | 13 Mar 1986 | Dr PM Neumann | University of Oxford | The Paris Grand Prix of 1860 |

84/85 | 18 Mar 1985 | Professor JH Conway FRS | University of Cambridge | Cantor and the Infinite |

83/84 | 3 May 1984 | Professor EC Zeeman FRS | University of Warwick | The Discovery of Perspective during the Renaissance |

## The 2023/24 Collingwood Lecture

**Professor Alison Etheridge FRS (Oxford University)**

"Some mathematical models of evolution"

23 January 2024, 1pm, CLC013

Alison Etheridge is Professor of Probability at the University of Oxford where she holds a joint appointment in the Departments of Mathematics and Statistics and a Fellowship at Magdalen College. She was an undergraduate at New College, and divided her graduate study between Oxford and McGill. She then held research fellowships in Oxford and Cambridge and positions in Berkeley, Edinburgh and Queen Mary University of London before returning to Oxford in 1997.

Abstract: How can we explain the patterns of genetic variation in the world around us? The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic, ecological and evolutionary mechanisms. How do they interact with one another, and what was their relative importance in shaping the patterns that we see today? This question lies at the heart of theoretical population genetics. Whereas the pioneers of the field could only observe genetic variation indirectly, by looking at traits of individuals in a population, researchers today have direct access to DNA sequences, but making sense of this wealth of data presents a major scientific challenge and mathematical models play a decisive role. In this lecture we'll discuss some of the ways in which we can distill our understanding into workable mathematical models.

## The 2022/23 Collingwood Lecture

**Professor Christina Pagel (University College London)**

"UK response to the Covid-19 pandemic: the vicious circles, the brilliant science and where science was not enough"

6 June 2023, 3pm, CLC013

Christina Pagel graduated with a BA in Mathematics from The Queen's College, Oxford in 1996. She also holds an MSc in Mathematical Physics from King's College London, and MAs in Classical Civilisation, Medieval History and an MSc in Applied Statistics with Medical Applications from Birkbeck College, University of London. In 2002 she was awarded a PhD in Space Physics on Turbulence in the interplanetary magnetic field from Imperial College London.

Christina is currently a professor of operational research at University College London (UCL) within UCL's Clinical Operational Research Unit (CORU), which applies operational research, data analysis and mathematical modelling to topics in healthcare. She was Director of UCL CORU from 2017 to 2022 and is currently Vice President of the UK Operational Research Society. She also co-leads UCL's CHIMERA research hub which analyses data from critically ill hospital patients. In May 2020, Christina joined the Independent SAGE committee, whose aim is to offer independent advice to the UK Government during the COVID-19 pandemic. As part of her work for Independent SAGE, she is regularly quoted in newspapers, writes for national newspapers and appeared on national and international broadcast media and various podcasts discussing the UK's response to the pandemic.

**Abstract:** Christina will discuss how the fundamental nature of COVID-19 transmission and illness led to a vicious circle of repeating waves of infection, disproportionately affecting those in more deprived communities. She will highlight where brilliant science helped to tackle the pandemic but also where it did not – especially when uncoupled from other expertise and responsive policy.

## The 2018/19 Collingwood Lecture

**Professor Gwyneth Stallard (The Open University)**

"The beauty of fractals"

21 November 2018, 4pm, CLC013

Professor Stallard's research is in the area of complex dynamics and concerns the iteration of transcendental meromorphic functions. She has made fundamental contributions to the theory of the dynamics of transcendental complex functions and has made important discoveries concerning the dimension of Julia sets. Her insight and originality have established major results in the subject. Her work is characterised by the successful application of hard analytic techniques and, as she readily admits, by stubborness.

Gwyneth has a long standing interest in the issues surrounding women's careers in mathematics and chaired the London Mathematical Society's Women in Mathematics Committee from 2006 to 2015. This work was recognized by the award of an OBE in 2015. In 2016, she was honoured as part of the Suffrage Science Scheme, run by the Medical Research Council's Clinical Sciences Centre. She was among 12 women receiving awards in 2016 to celebrate their scientific achievements in maths and computing, and their ability to inspire others.

**Abstract:** In this talk we discuss the fascinating structure of geometrical objects known as fractals, beginning with classic fractal sets such as Cantor sets and the von Koch snowflake. We will then explore fractals which arise as Julia sets in the subject of complex dynamics. These are sets on which the iterates of a function behave chaotically and they have structures such as a Cantor bouquet and an infinite spider’s web. Major advances in complex dynamics have often come from applications of powerful techniques in topology and complex analysis – many of which are described in the classic text 'Collingwood and Lohwater'.

## The 2016/17 Collingwood Lecture

**Professor Pierre Cartier (Institut des Hautes Etudes Scientifiques, Paris)**

"Is there a future for the cosmic Galois group?"

16 February 2017, 4pm, CLC013

Professor Cartier studied at the École Normale Supérieure in Paris from 1952 to 1954 and obtained his doctorate from the Université de Paris in 1958, with a thesis entitled `Derivations and Divisors in Algebraic Geometry'. By 1955, he had become a full member of the Bourbaki group (Bourbaki is the pseudonym of a group of (mainly) French mathematicians who publish an authoritative account of contemporary mathematics). He was appointed Professor in the Faculty of Science at Strasbourg in 1961, remaining there until he moved to the Institut des Hautes Études Scientifiques at Bures-sur-Yvette ten years later. In addition to this post he was director of research at the Centre National de la Recherche Scientifique from 1974. In 1982 he left the Institut des Hautes Études Scientifiques becoming a professor at the École Polytechnique (1982-88) and at the École Normale Supérieure from 1988. Professor Cartier has written papers on a broad range of mathematical topics including algebraic geometry, number theory, group theory, probability, and mathematical physics. He was awarded the Ampère Prize of the French Academy of Sciences in 1979.

**Abstract:** About twenty years ago, Pierre Cartier observed a similitude between the renormalization group in physics - in the construction given by Alain Connes and Dirk Kreimer - and the Grothendieck Teichmuller group in arithmetic geometry. The initial guess was a little too optimistic, based on numerical calculations by Broadhurst and Kreimer. A revised version suggests a connection between the Galois theory of transcendental numbers (as envisionned by Grothendieck) and a possible new kind of symmetry group in high energy physics: the `cosmic Galois group'. This direction has been developed by Francis Brown, giving explicit `superselection rules'. Much more is expected.

## The 2015/16 Collingwood Lecture

**Professor Ray Goldstein FRS (University of Cambridge)**

"Evolution of Biological Complexity"

10 May 2016, 4pm, CLC013 (Calman Learning Centre)

Professor Ray Goldstein received his PhD in 1988 from Cornell University and has held academic appointments at the University of Chicago, Princeton University and the University of Arizona. He was appointed Schlumberger Professor at the University of Cambridge in 2006. He is an internationally recognised leader in the fields of biological physics and nonlinear dynamics. He is distinguished for having made important mathematical contributions to those subjects as well as pioneering experimental discoveries. His broad-ranging contributions include classic work on the dynamics of pattern formation driven by long-range forces, the differential geometry of interfacial pattern formation, and the explanation for the shapes of stalactites. He has made seminal experimental contributions to the study of active matter, including developing a class of green algae as model organisms for the study of biological fluid dynamics, the physics of multicellularity, and the synchronization of eukaryotic flagella. Professor Goldstein was elected a Fellow of the Royal Society in 2013.

**Abstract:** Long after the first unicellular organisms arose in the primordial soup, they evolved to become multicellular and to divide up life’s processes into distinct cell types. One of the most fundamental issues in evolutionary biology is the nature of this transition: What is the advantage of being larger? What are the driving forces behind the appearance of distinct cell types? In this talk I will describe an approach to these questions based on the study of a particular class of organisms, using experimental techniques from physics and fluid dynamics, and the interpretation of those experiments with mathematical techniques from the area of stochastic nonlinear dynamics.

## The 2014/15 Collingwood Lecture

**Professor Martin Hairer (University of Warwick)**

Fields Medallist 2014

"Taming Infinities"

26 February 2015, 4pm, CLC013 (Calman Learning Centre)

Professor Martin Hairer is one of the 2014 recipients of the Fields Medal, `for his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations'. The Fields Medal, officially known as the International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The Fields Medal is often viewed as the greatest honour a mathematician can receive. The Fields Medal and the Abel Prize have often been described as the `mathematician's Nobel Prize'.

**Abstract:** Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.

## The 2013/14 Collingwood Lecture

**Professor Wendelin Werner (ETH Zurich)**

Fields Medallist 2006

"Randomness and the continuum"

30 January 2014, 4pm, W103 (the Appleby Lecture Theatre)

Professor Wendelin Werner is one of the 2006 recipients of the Fields Medal, `for his contribution to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory'.

The Fields Medal, officially known as the International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The Fields Medal is often viewed as the greatest honour a mathematician can receive. The Fields Medal and the Abel Prize have often been described as the `mathematician's Nobel Prize'.

**Abstract:** One can have a rather intuitive perception of the fact that space and time can be continuous, which is very directly related to the mathematical notion of continuity of functions. On the other hand, when one thinks of random phenomena, the natural examples that first come to mind are of discrete nature, such as coin tossing. The conceptual question of how "randomness" can be split up into and reassembled from infinitesimal little pieces turns out to be quite tricky. It is related to contemporary research in mathematics that we shall illustrate via some concrete examples.

## The 2012/13 Collingwood Lecture

**Professor Peter Higgs (Emeritus, Edinburgh University)**

Nobel Prize in Physics 2013

with an introduction by

Professor Steve Abel (Department of Mathematical Sciences, Durham)

"Electroweak Symmetry Breaking and the Higgs boson"

5 November 2013, 4.15pm, CLC013 (Calman Learning Centre)

**Abstract:** The impressive developments in physics during the first half of the 20th Century made it seem likely that ALL phenomena, from the atomic scale to the edge of the visible universe, were governed solely by two known laws, that of classical general relativity, Einstein’s generalisation of Newtonian gravity, and quantum electrodynamics, the quantum version of Maxwell’s electromagnetic theory. But gravitational and electromagnetic interactions are long range forces, and the discovery of radioactive beta decay revealed the existence of a new, short range, force - the weak interaction. At the beginning of the 1960s, the theoretical interpretation of such weak interactions seemed to pose insuperable obstacles.

In 1964, Professor Peter Higgs, and independently, Professor Robert Brout together with Professor Francois Englert, postulated the existence of scalar fields pervading the universe (Brout-Englert-Higgs fields) that dynamically generate elementary particle masses and naturally give rise to weak short range forces. This insight opened up new perspectives on the unity of the laws of nature. It also predicted a new scalar particle.

On 4th July 2012 the international particle accelerator facility, CERN in Geneva, announced that it had discovered this particle, by then commonly referred to as `the Higgs', some 50 years after it was first predicted. This discovery was widely hailed as one of the most important advances of all time in our understanding of the Universe.

In this Collingwood lecture, Professor Abel explains why the Higgs became such a crucial guide to the mathematical underpinnings of Nature, beginning with the connection between symmetry and conservation laws, spontaneous symmetry breaking, the connection between mass and spin, and encompassing the Standard Model of particle physics. He also discusses why the discovery presents a difficult conundrum, and describes our current attempts to solve it.

Professor Higgs will provide an overview of the scientific circumstances that led to his insightful theoretical discovery in 1964.