# The Annual Pascal Lecture Series

The Blaise Pascal Lecture series has been established following a generous bequest by Mrs Marjorie Roberts in 2006. Each year, a Pascal Fellow is chosen from a list of nominees, whose research interests are either in Pure Mathematics or Natural Philosophy (Applied Mathematics and Theoretical Physics).

The lectures which have taken place since 2008 are shown below.

Year |
Pascal Fellow |
Institution |
Title |

2024 | Professor Eveliina Peltola | University of Bonn (IAM) | |

2022 | Professor Sir Martin Hairer, FRS, Fields Medal 2014 | Imperial College London | A mathematical journey through scales |

2020 | Dr Nira Chamberlain | President of IMA | The black heroes of mathematics |

2018 | Professor Peter Sarnak | IAS Princeton | Integer points on affine cubic surfaces |

2017 | Professor Peter Diggle | Lancaster University | Statistical Analysis of Spatiotemporal Point Process Data |

2016 | Professor Caroline Series | University of Warwick | Mirzakhani's starting point |

2015 | Professor Dimitri Petritis | Université de Rennes, France | Gambling with classical or quantum dice: a guided tour from classical probability to quantum channels |

2014 | Professor Gary Gibbons FRS | University of Cambridge | The concept of solitons and gravity |

2013 | Professor Antti Niemi | University of Uppsala, Sweden | Strings, Integrable Models and Biology - A New Frontier? |

2012 | Professor Jose Seade | Universidad Nacional Autónoma de México | Discrete groups of automorphisms of complex projective spaces |

2010 | Professor Nick Manton FRS | University of Cambridge | Constructing Vortices |

2009 | Professor Federico Sanchez-Bringas | Universidad Nacional Autonoma de Mexico |
Second order invariants on spacelike surfaces immersed in Minkowski space |

2008 | Professor Evangelia Samiou | University of Cyprus, Nicosia | Two-radius theorems in Damek Ricci spaces |

## A mathematical journey through scales

**Abstract:** The tiny world of particles and atoms and the gigantic world of the entire universe are separated by about forty orders of magnitude. As we move from one to the other, the laws of nature can behave in drastically different ways, sometimes obeying quantum physics, general relativity, or Newton's classical mechanics, not to mention other intermediate theories. Understanding the transformations that take place from one scale to another is one of the great classical questions in mathematics and theoretical physics, one that still hasn't been fully resolved. In this lecture, we will explore how these questions still inform and motivate interesting problems in probability theory and why so-called toy models despite their superficially playful character sometimes lead to certain quantitative predictions.

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Martin Hairer studied at the University of Geneva, where he completed his PhD in Physics in 2001. He subsequently held positions at the University of Warwick (UK) and the Courant Institute (US), before moving to Imperial College London, where he currently holds a chair in probability and stochastic analysis. His work is in the general area of probability theory with a main focus on the analysis of stochastic partial differential equations.

Author of a monograph and over 100 research articles, Professor Hairer is a Fellow of the Royal Society as well as many other academies in Europe. His work has been distinguished with a number of prizes and awards, most notably the LMS Whitehead and Philip Leverhulme prizes in 2008, the Fermat prize in 2013, the Fröhlich prize and Fields Medal in 2014, a knighthood in 2016, the Breakthrough prize in Mathematics in 2020, and the King Faisal prize in 2022.

## The Black Heroes of Mathematics

**Abstract**: The 2017 film, Hidden Figures, is based on the true story of a group of black female mathematicians that served as the brains behind calculating the momentous launch of the NASA astronaut John Glenn into orbit. However, these mathematicians of colour are not the only ‘Hidden Figures’. Nira Chamberlain will discuss other inspirational men and women who overcame obstacles to prove that ‘mathematics is truly for everybody!’Black History Month, also initially known as the African American History

Month, is a month-long tradition of celebrating the achievements of the black community. It began as a way for remembering important people and events in the history of the African diaspora. The event is celebrated every year in October in the UK.

To celebrate the contributions of black role models to the field of mathematical sciences, the Department of Mathematical Sciences, Durham University is pleased to have Dr Nira Chamberlain as guest speaker, who will be delivering the annual Pascal lecture 2020.

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Nira Chamberlain is the current president of the Institute of Mathematics and its Applications (IMA). He has more than 25 years of experience of writing mathematical models/simulation algorithms that solve complex industrial problems. He has developed mathematical solutions within many industrial sectors, including spells in France, the Netherlands, Germany and Israel. In 2015 Dr Chamberlain joined the exclusive list of 30 UK mathematicians who are featured in the autobiographical reference book Who’s Who. He has held visiting positions in many prestigious UK universities.

## Integer points on affine cubic surfaces

**Abstract**: A cubic polynomial equation in four or more variables tends to have many integer solutions, while one in two variables has a limited number of such solutions. There is a body of work establishing results along these lines. On the other hand very little is known in the critical case of three variables. For special such cubics, which we call Markoff surfaces, a theory can be developed. We will review some of the tools used to deal with these and related problems. Joint works with Bourgain/Gamburd and with Ghosh

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Peter Sarnak is a South African-born mathematician who has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is also on the permanent faculty at the School of Mathematics of the Institute for Advanced Study. He is known for his work in analytic number theory. Peter is the recipient of many prestigious prizes, such as the George Pólya Prize (1998), the Ostrowski Prize (2001), the Levi L. Conant Prize (2003), the Cole Prize (2005) and the Wolf Prize (2014).

## Statistical Analysis of Spatiotemporal Point Process Data

**Abstract:** A spatiotemporal point process, P, is a stochastic model for generating a countable set of points (x(i), t(i)) ∈ IR2 × IR+, where each x(i) denotes the location, and t(i) the time, of an event of interest. A typical data-set is a partial realisation of P restricted to a specified spatial region A and time-interval [0,T], possibly supplemented by covariate information on location, time or the events themselves. In this talk, I will first give examples of different interpretations of this scenario according to whether only one or both of the sets of locations and times are stochastically generated. I will then discuss in more detail methods for analysing spatiotemporal point process data based on two very different modelling approaches, log-Gaussian Cox process models; and conditional intensity models, and describe applications of each in the context of human and veterinary epidemiology.

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Peter Diggle is a Fellow of the Royal Statistical Society and Distinguished University Professor in the Lancaster Medical School, and

holds a part-time post at the University of Liverpool, Department of Epidemiology and Population Health. He also has adjunct

appointments at the Johns Hopkins University School of Public Health, Columbia University International Research Institute for Climate and Society, and Yale University School of Public Health. He is a trustee for the Biometrika Trust, a member of the Advisory Board for the journal Biostatistics, chair of the Medical Research Council’s Strategic Skills Fellowship Panel and President-Elect of the Royal Statistical Society.

His research concerns the development and application of statistical methods relevant to the biomedical and health sciences.

## Mirzakhani's starting point

**Abstract:** In 2014, Maryam Mirzakhani of Stanford University became the first women to be awarded the Fields medal. The starting point of her work was a remarkable relationship called McShane’s identity, about the lengths of simple closed curves on certain hyperbolic surfaces. The proof of this identity, including the Birman-Series theorem about simple curves on surfaces, uses only quite basic ideas in hyperbolic geometry which I will try to explain. We will then look briefly at Mirzakhani’s ingenious way of exploiting the identity and where it led.

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Caroline Series is Emeritus Professor of Mathematics at the University of Warwick, where she has been since 1978. Following her first degree at Somerville College, Oxford, she won a Kennedy Scholarship to Harvard where she did a Ph.D. under George Mackey. Her early research was in dynamical systems and chaos, for which she won an LMS Junior Whitehead prize in 1987. She held an EPSRC Senior Research Fellowship in 1999 -- 2004. A flavour of her recent work, about the geometry of three dimensional hyperbolic manifolds and the fractal sets associated to their symmetry groups, can be gained from her widely praised book Indra's Pearls, coauthored by D. Mumford and D. Wright, recently republished in paperback (CUP 2015).

Professor Series has served the mathematical community on many committees both national and international, and was a member of both the RAE and REF maths panels. She has given many distinguished public lectures. Throughout her career she has taken a leading role in encouraging women mathematicians. In 2014, she was awarded the first Senior Anne Bennett prize of the London Mathematical Society and she is vice-chair of the newly formed IMU Committee for Women in Mathematics.

## Gambling with classical or quantum dice: a guided tour from classical probability to quantum channels

**Abstract**: Starting from a simple example (sharp gambles with a classical dice) we shall describe progressively more involved systems (like unsharp gambles with classical dice and hidden Markov chains). Then a physical experiment showing the insufficiency of classical probability to describe Nature will be explained and the notion of quantum probability and of repeated unsharp quantum measurements will be introduced. We shall conclude with a limit theorem concerning repeated quantum measurements.

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Dimitri Petritis is based at the Université de Rennes. He joined the Institut de Recherche Mathématique de Rennes there in 1989, and since 2008 has held the position of `Professor of exceptional class'. Dimitri's research is broadly concerned with uncertainty and randomness in physical systems, in the context of probability theory, ergodic theory, mathematical physics, and information theory. Dimitri received his Ph.D. from École Polytechnique on `Zero Mass Effects in Quantum Field Theory' (1984). His subsequent research has focused on statistical mechanics and stochastic processes, with recent work concerning queueing systems and processes in disordered media. Other interests include classical and quantum information, and mathematical biology.

Dimitri has held a number of visiting professor positions around the world, and has supervised a number of doctoral students

## The concept of solitons and gravity

**Abstract: ** I will begin by providing a short history of the soliton concept, as intended to eliminate the distinction between particles on the one hand and fields on the other. I will then discuss how the idea has fared in General Relativity and bring the story up to date with some remarks about extreme black holes, p-branes and fuzz-balls. If time permits, I will also mention some recent work on how solitons in the absence of gravity can give rise to interesting "emergent" spacetimes.

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Gary Gibbons is Professor of Theoretical Physics in DAMTP, Cambridge University, where he has spent most of his career. He was elected a Fellow of the Royal Society in 1999, and to a Professorial Fellowship at Trinity College Cambridge in 2002.

## Strings, Integrable Models and Biology - A New Frontier?

**Abstract:** The biological function of a protein depends critically on its three dimensional geometry. But at the moment we do not know how the shape of a protein could be deduced from the DNA sequence alone. As a consequence the protein folding problem endures as one of the most important unresolved problems in science, it addresses the origin of life itself. In this talk we shall argue, that the shape of a protein can actually be determined from very general principles, that are also utilized in the context of string theory and integrable models. We shall argue, that there is a universal energy function which relates to the discrete nonlinear Schrodinger equation, the paradigm integrable lattice model, that describes all known folded protein structures. We show how to derive this energy function from fundamental geometrical concepts. We show that it supports soliton solutions, that describe

folded proteins with a precision where the root-mean-square distance between an experimental crystallographic structure in Protein Data Bank and its theoretical description is less than the radius of a carbon atom. We present a number of examples of numerical simulations that show how a protein folds. The simulations are performed with laptop computer, and the simulation proceeds practically as fast as the folding does in vivo.

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The Pascal Fellow 2013 is Antti Niemi, CNRS Professor of Theoretical Physics at the University of Tours in France and also at Uppsala

University in Sweden. He has worked on a variety of areas in theoretical physics from quantum field theory to classical solitons, and his current research interests involve various aspects of biophysics and mathematical biology. Professor Niemi is a member of the Royal Swedish Academy of Sciences. In 1994, he was awarded the Goran Gustafsson Prize by the Royal Society of Sciences of Sweden.

## Discrete groups of automorphisms of complex projective spaces

**Abstract:** Classical Kleinian groups are discrete groups of automorphisms of the Riemann sphere, which can be regarded as

being the complex projective line CP1. The study of these type of groups has played for decades a major a role in various areas of

mathematics. In this talk we shall discuss how this theory generalizes to discrete groups of automorphisms of the complex projective space CP2, and more generally of CPn. This includes the particularly interesting class of discrete groups of holomorphic isometries of complex hyperbolic spaces.

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The Pascal Fellow 2012 is Jose Seade, Professor in the Mathematics Institute of the National Autonomous University of Mexico (UNAM). Professor Seade has been a member of the Mexican Academy of Sciences since 1983 and President of the Mexican Mathematical Society in 1986-87. He has also been President of the Third World Academy of Sciences since 2003.

In 2012, Professor Seade was awarded the Ferran Sunyer i Balaguer Prize together with Dr Angel Cano and Dr Juan Pablo Navarrete for a monograph entitled `Complex Kleinian Groups'.

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