An archive of all events, colloquia, and talks held by the Maxwell Institute for Mathematical Sciences.
The second Maxwell Institute welcome event of the 2022-23 academic year took place on Tuesday 28th March.
Where: ICMS/Bayes Centre, Main Lecture Theatre
Nine new Maxwell Institute colleagues presented talks on their work:
Boado Penas Carmen (HW)
Jiawei li (UoE)
Moritz Linkmann (UoE)
Geoff Vasil (UoE)
Matteo Capopoferri (HW)
Linhan Li (UoE)
Stefan Klus (HW)
Amanda Lenzi (UoE)
Neil Chada (HW)
The talks were followed by a reception where new staff were welcomed by colleagues from the University of Edinburgh and Heriot Watt.
The Maxwell Institute held a Mini-symposium in Analysis and PDEs on Friday 31 March from 2pm to 5:20 pm in Lecture Theatre 3, Appleton Tower.
This mini-symposium featured three speakers: Nicolas Burq (Universite Paris-Saclay and Institut Universitaire de France), Mathieu Lewin (CNRS and Universite Paris Dauphine, France), and Maciej Zworski (UC Berkeley, USA).
Title: Propagation of Smallness, Control and Stabilisation
Abstract: Control and stabilization are now well understood when controls or damping act on open sets. When they act on measurable sets only, much remains to be understood. In this talk I will present results in this direction. In most cases (but not always) the results I will present will rely on the quantitative results of Logunov/Malinnikova for harmonic functions that can be translated into control or stabilisation results for heat, waves or Schrodinger equations. Most of the results presented in the talk are in collaboration with I. Moyano (Univ. Nice).
Title: Derivation of nonlinear Gibbs measures from many-body quantum mechanics
Abstract: In this talk, I will define and discuss some probability measures in infinite dimensions, which play an important role in (S)PDE, in Quantum Field Theory and for the description of Bose-Einstein condensates. Those are Gibbs measures associated with nonlinear Schrodinger-type energies. In dimensions larger than or equal to 2, the measures concentrate on distributions, and they need to be properly renormalized. After presenting the Gibbs measures, I will explain how to derive them from many-body quantum mechanics. Joint works with Phan Thanh Nam (Munich) and Nicolas Rougerie (Lyon).
Title: Mathematics of magic angles
Abstract: Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky–Kruchkov–Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hormander’s bracket condition in a very simple setting). Recent mathematical progress also includes the proof of existence of generalized magic angles and computer assisted proofs of existence of real ones (Luskin–Watson, 2021). The results will be illustrated by colourful numerics which suggest many open problems (joint work with S Becker, M Embree, J Wittsten in 2020 and S Becker, T Humbert and M Hitrik in 2022).
The 7th David Finney Lecture was presented by Bin Yu (UC Berkeley) on 3rd May 2023.
“AI is like nuclear energy–both promising and dangerous.” Bill Gates, 2019
Data Science is a pillar of AI and has driven most of recent cutting-edge discoveries in biomedical research and beyond. Human judgment calls are ubiquitous at every step of a data science life cycle, e.g., in problem formulation, choosing data cleaning methods, predictive algorithms and data perturbations. Such judgment calls are often responsible for the “dangers” of AI.
To mitigate these dangers, we introduce in this talk a framework based on three core principles: Predictability, Computability and Stability (PCS). The PCS framework unifies and expands on the ideas and best practices of statistics and machine learning. It emphasizes reality check through predictability and takes a full account of uncertainty sources in the whole data science life cycle including those from human judgment calls such as those in data curation/cleaning. PCS consists of a workflow and documentation and is supported by our software package veridical or v-flow. Moreover, we illustrate the usefulness of PCS in the development of iterative random forests (iRF) for predictable and stable non-linear interaction discovery. Finally, in the pursuit of genetic drivers of a heart disease called hypertrophic cardiomyopathy (HCM) as a CZ Biohub collaborative project, we use iRF and UK Biobank data to recommend gene-gene interaction targets for knock-down experiments. We then analyze the experimental data to show promising findings about genetic drivers for HCM.
During the Edinburgh Science Festival in May 2023, 5 MAC-MIGS PhD students presented engaging short talks to the public at a “Mathematics Showdown” event in the Bayes Centre. The talks are available to watch here on the Maxwell Institute YouTube channel.
The third Atiyah Lecture was delivered by Prof. Edriss Titi (University of Cambridge) on 8 June 2023: Determining the Global Dynamics of Infinite-dimensional Dissipative System by a Scalar ODE – The Navier-Stokes Equations Paradigm.
Abstract: One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations, Rayleigh—Bénard convection and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters/functionals — finite number of determining modes, nodes, volume elements and other determining interpolants. Exploring this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems one can design finite-dimensional feedback control for stabilizing their solutions. Moreover, the same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete observational measurements. As a byproduct of this approach we will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in a dynamical system that is governed by an ordinary differential equations (ODE) with a globally Lipschitz vector field, named determining form. Remarkably, as a result of this machinery we will eventually show that the global dynamics of the Navier-Stokes equations is determined by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models. Notably, the existence of global determining form and of an algorithm for computing it establish a rigorous mathematical foundation for implementing Machine Learning algorithms for recovering the solutions of the Navier-Stokes solutions from their discrete observational measurements.
Short Bio: Edriss Saleh Titi is an Arab-Israeli mathematician. He is Professor of Nonlinear Mathematical Science at the University of Cambridge. He also holds the Arthur Owen Professorship of Mathematics at Texas A&M University, and serves as Professor of Computer Science and Applied Mathematics at the Weizmann Institute of Science and Professor Emeritus at the University of California, Irvine. He is a Fellow of the John Simon Guggenheim Memorial Foundation, a Fellow of the Inaugural Class of the AMS, was the recipient of a Humboldt Research Award and the SIAM Prize for the Best Paper in PDEs, and held a Senior Simons Professorship, among many other honours and awards.
The third Fitch Lecture was given by Prof. Vicky Pope at 2pm on 13th June 2023 in G.03, Bayes Centre.
About the Lecture
Prof. Pope described the mathematics and physics that go into a cutting-edge climate model, and how this is then used to provide information on how and why the climate is changing, and how it might change in the future. She also provided examples of how climate science and models can inform responses to the climate change issue which focus on developing climate resilience and sustainability. In addition, Prof. Pope discussed the important issue of unintended consequences, showing that we must look at the complete cost of solutions over their lifetimes and all aspects of their impact – not just on climate change.
About the Speaker
Prof. Vicky Pope is the Chair of the Trustee Boards of Devon Wildlife Trust, and Chair of Mathematics in Education and Industry. Her long career as a climate scientist specialising in climate modelling allowed her to provide assistance to governments and the general public, aiding in their understanding of the implications of climate change. Her work has helped to provide the information required by government bodies, and others, to reduce the negative impacts of climate change. Furthermore, Prof. Pope has helped encourage evidence-based decisions on diverse issues, including drought and the interaction between air quality and climate change.
Her current interests are diverse, encouraging wider appreciation and protection of the environment and improving access to science and mathematics education. As well as her Chair appointments, Prof. Pope is a trustee for numerous other charities and was awarded an honorary Professorship at University College London. She is the founding Editor in Chief or Climate Resilience and Sustainability: a new interdisciplinary climate change journal.
Richard Nickl (University of Cambridge) and Jana de Wiljes (University Potsdam) visited the Maxwell Institute on 27th September 2023 for an afternoon of talks on Inverse Problems.
Attendees also heard from other researchers across the Maxwell Institute, Heriot Watt and Edinburgh Universities (including Joann Altmann, Cecilia Tarpau, and Mike Davies).
James Maynard, 2022 Fields Medal winner and Professor of Number Theory at the University of Oxford, delivered the 2023 Maxwell Institute Whittaker Lecture on 7th December in the Bayes Centre.
Primes, the zeta function, and zero density estimates.
If we could prove the Riemann Hypothesis, then there would be several fantastic consequences for our understanding of prime numbers. It turns out that even if the Riemann Hypothesis is false and there are some counterexamples to it, we can still obtain many of these consequences for primes provided the counterexamples are suitably ‘rare’.
I’ll talk about this picture and recent joint work on possible ‘patterns’ of zeros. As a consequence of our new approach if the zeros of the zeta function lay on finitely many vertical lines then we obtain several results on primes which are essentially as strong as what the Riemann Hypothesis would imply.
The first Maxwell Institute social event of the 2022-23 academic year was held on Friday afternoon November 11.
Where: ICMS/Bayes Centre, Main Lecture Theatre
2:00 Welcome remarks – Laura Ciobanu (HW) and Ruth King (UoE) on behalf of MI
2:10-3:00 Short talks by new staff:
Tudor Dimofte (UoE)
Panagiota Birmpa (HW)
Moritz Linkman (UoE)
Julian Braun (HW)
Simon Wood (UoE)
3:00-3:30 Nibbles and drinks
3:30-4:30 Short talks by new staff:
James Gaunt (HW)
Serveh Sharifi Far (UoE)
Alistair Wallis (HW)
Jacob Page (UoE)
Sara Lombardo (HW)
4:30-5:00 Nibbles and drinks
A Maxwell Institute Colloquium was held on Thursday, December 1st, 2022 at 40 George Square, Lecture Theatre A.
Stéphane Mallat presented Multiscale Models of Deep Neural Networks from 14:00-15:30.
Stéphane Mallat was Professor at NYU in computer science, until 1994, then at Ecole Polytechnique in Paris and Department Chair. From 2001 to 2007 he was co-founder and CEO of a semiconductor start-up company. Since 2017, he holds the “Data Sciences” chair at the Collège de France. He is a member of the French Academy of sciences, of the Academy of Technologies, and a foreign member of the US National Academy of Engineering. Stéphane Mallat’s research interests include machine learning, signal processing and harmonic analysis. He developed the multiresolution wavelet theory and algorithms at the origin of the compression standard JPEG-2000, and sparse signal representations in dictionaries through matching pursuits. He currently works on mathematical models of deep neural networks for data analysis and physics.
Deep Neural Networks
Deep neural networks have spectacular applications but remain mostly a mathematical mystery. An outstanding issue is to understand how they circumvent the curse of dimensionality to generate or classify data. Inspired by the renormalization group in physics, we show that a key element is that data distributions can be factorized and renormalized into simpler conditional probabilities across scales. They often have local dependencies and can thus can be estimated with limited databases. This is first applied to generate turbulence and cosmological fields as well as structured images. We then derive classification architectures reaching the state of the art on complex image data bases such as ImageNet.
The Maxwell Institute held its first Teaching Symposium on Monday 12th December in the Bayes Centre. Recordings of all talks are available here.
11:30 – 12:30
Michela Ottobre (HWU) – Mathematical Writing skills and Collaborative teaching
Steven O’Hagan (UoE) – Evolving course designs
12:30 – 13:30 Lunch
13:30 – 14:30
Charlotte Desvages (UoE) – Embedding collaborative practice in introductory programming courses
Alistair Wallace (HWU) – Automated marking of R assessment
Michela Ottobre (HWU) : Mathematical Writing skills and Collaborative teaching
Mathematical Writing skills are core skills for our UG students, and intertwined with (Mathematical) Reading Skills. Yet, the importance of such skills is seldom recognised by our students. This creates a gap between our expectations and student’s understanding of them.
Pamela Docherty and I have been working together on our respective first year courses to include Mathematical Writing and Reading in the syllabus and assessment schedule. In this seminar I will explain how we have coordinated and `experimented’, highlighted what I think worked well and what instead I believe needs improvement for next year.
Steven O’Hagan (UoE) – Evolving course designs
I will give a couple of case studies of course designs at Edinburgh: one from Year 1 (Calculus) and one from Year 3 (Complex Analysis). Both build on practice developed in the School over many years and on experience gained during the pandemic. The designs sought to maximise the value of whole-class activities and encourage continual student engagement. They made heavy use of online workbooks containing a mixture of written course materials, video clips and computer-based self-assessment. I will describe how they were developed, how they worked in practice and how the designs might be adapted for other courses.
Charlotte Desvages (UoE) – Embedding collaborative practice in introductory programming courses
Programming has become an essential skill for many graduates, including Mathematics students. Professional practice often relies on collaborative workflows, with demonstrable benefits to code quality, team building, and knowledge sharing. While there is a strong culture of group working in “pen-and-paper” Mathematics courses in UoE, it had been less obvious how to foster collaborative learning in computing labs. I hypothesised that mirroring elements of professional practice in the classroom would not only allow students to collaborate effectively and harvest the pedagogical benefits of peer-learning, but also provide them with valuable employability skills for today’s job market.
In this talk, I will give an overview of how I embedded different aspects of professional programming practice into two introductory Python courses for Mathematics students, respectively at pre-Honours and MSc levels. In particular, I will focus on two aspects of collaborative practice: pair programming on practical tasks in workshop sessions, and code review as a peer-assessment task. I will outline my motivations for introducing these tasks, describe practical implementation in each course, and discuss the successes and challenges found along the way.
Alistair Wallis (HWU) – Automated marking of R assessment
Over the past few years, I have been teaching two undergraduate courses, Stochastic Processes and Survival Models, which are taught at Heriot-Watt’s campuses in Edinburgh, Putrajaya and now Dubai. Recently, for both courses, I have created assignments which require students to 1) submit a report and 2) submit an R script with any code used for any simulations, data analysis, calculations, plots etc. Each student has initial information or data which is specific for their student ID and the students are required to calculate certain numerical or vector quantities, or create functions with specified inputs.
To facilitate marking, I have developed a relatively simple system with R and Python for automatically marking their code. The reports are then manually assessed using SpeedGrader / Turnitin.
In this talk, I want to briefly explain the rationale for using this set-up, challenges that were faced and advice for anyone considering something similar, along with a brief demonstration of the set-up. Most of this talk will not be particularly Python or R specific so could be of interest to anyone interested in having a system for automatically marking code used in mathematical calculations.
On Thursday 15th December 2022, the Maxwell Institute held its annual PhD Student Conference at ICMS.
A total of 26 PhD students presented on a range of topics, including Statistics, Probability, Financial Mathematics, Pure Mathematics, Operational Research, Applied Mathematics, and Analysis. Additional discussions were held about job prospects after PhD completion.
The second annual Davey Fitch Lecture (part of the Maxwell Institute’s Distinguished Lectures series) took place on Friday 4th November 2022 in 50 George Square.
Professor Arnaud Doucet (University of Oxford and Research Scientist at DeepMind) delivered a talk titled “An Unlikely Journey”.
I will discuss how mathematical sciences, industry (and many random events) have shaped my academic career over the past three decades. Starting from the French countryside and a PhD in Electronic Engineering on “Predictive monitoring of neutron sensors”, this long and highly unlikely journey will take us to countries on four continents. I will illustrate how industry problems have been a constant source of inspiration for my research. Very early on, I developed a long-standing interest in stochastic filtering while working in a nuclear research centre. This subsequently motivated me not only to work in many applied domains such as robotics, computer vision and epidemiology but also led me to far away destinations. The journey is not yet over.
After having moved from engineering to statistics, my association with DeepMind has reshaped my research agenda and I will discuss some of my recent machine learning work on generative modelling and uncertainty quantification.
The second Atiyah Lecture was given by Professor Thaleia Zariphopoulou on 13 May 2022 at 14:00 in Bayes Centre at Room G.03.
Professor Thaleia Zariphopoulou is a Greek-American mathematician specializing in mathematical finance. She is the Presidential Chair in Mathematics and the Neuhaus Centennial Professor of Finance at the University of Texas at Austin. Zariphopoulou earned a B.S. in electrical engineering from the National Technical University of Athens in 1984. She then went to Brown University for graduate studies in applied mathematics and earned her master’s degree in 1985 and her Ph.D. degree in 1989 under the supervision of Wendell Fleming. She was an assistant professor at Worcester Polytechnic Institute and an associate professor at the University of Wisconsin-Madison, before she moved to the University of Texas at Austin in 1999. Thaleia was the first holder of the statutory Oxford-Man Chair in Quantitative Finance, Mathematical Institute, University of Oxford from 2009-2012. She became a fellow of the Society for Industrial and Applied Mathematics in 2012. Zariphopoulou was an invited speaker at the 2014 International Congress of Mathematicians in Seoul.
Human-machine interaction models and stochastic optimization
This talk will offer an introduction to human-machine interaction (HMI) models in asset allocation (e.g. robo-advising) and a discussion on the related modeling and mathematical challenges. Modeling difficulties stem from the limited ability to quantify the human’s risk preferences and describe their evolution, but also from the fact that the stochastic environment, in which the machine optimizes, adapts to real time incoming information that is exogeneous to the human. Furthermore, the human’s risk preferences/goals and the machine’s actions may evolve at different scales. This dynamic interaction creates an adaptive cooperative game with both asymmetric and incomplete information exchange between the two parties. As a result, challenging questions arise on, among others, how frequently the human and the machine should communicate, how much information can the machine accurately detect, infer and predict, how should the human’s (over)reaction to exogeneous events and realized performance be processed and tamed by the machine, and how the performance of the machine could be compared with the one of a human advisor. Such HMI models give rise to new, non-standard optimization problems that combine adaptive stochastic control, time-inconsistency, stochastic differential games, optimal stopping, multi-scale analysis, and learning.
The 2022 David Finney Lecture was held on 13th October in Appleton Tower.
Professor Mark Girolami (University of Cambridge) presented on “Data Centric Engineering: A New Concept?”.
Lord Kelvin stated that ‘To Measure is to Know’. The role of experiment, empirical observation and data have always been core to the development of the empirical laws of the engineering sciences and associated professions. So, what does Data Centric Engineering actually mean? Is this yet another bandwagon of hype or is there substance behind the term? This talk will highlight the role of recent advances in the mathematical and statistical sciences and how they are transforming the study and practice of engineering.
Professor Claire Voisin is famous for her work in algebraic geometry. She won the European Mathematical Society Prize in 1992, the Sophie Germain Prize in 2003, the Ruth Lyttle Satter Prize in 2007, and the Clay Research Award in 2008. Voisin was an invited speaker at the 1994 International Congress of Mathematicians in Zurich, and a plenary speaker at the 2010 International Congress of Mathematicians in Hyderabad. In 2009 she became a member of the German Academy of Sciences Leopoldina. In 2016, she received the Gold medal of the French National Centre for Scientific Research. In 2017, she received the Shaw Prize in Mathematical Sciences. Claire Voisin was elected Fellow of the Royal Society in 2021.
Hyper-Kahler manifolds form a special class of compact Kahler manifolds with trivial canonical bundle. They are higher-dimensional generalizations of K3 surfaces, and a number of deformation classes of hyper-Kahler manifolds can be constructed starting from either a K3 or abelian surface. In the first lecture, I will introduce them and describe some of their general properties, from the viewpoints of Riemannian geometry, topology, and algebraic geometry. The second lecture will present further classical results that will be needed in the last lecture, devoted to the proof of a simple topological characterization of hyper-Kahler manifolds of Hilb^2(K3) deformation type (joint work with Debarre, Huybrechts, and Macri).
Evgeny Shinder will talk about Jacobians and derived equivalence of elliptic K3 surfaces. He explained a question of Hassett and Tschinkel on whether every Fourier-Mukai partner of an elliptic K3 surface is isomorphic to one of its Jacobians. This question has both geometric and arithmetic significance, in particular it’s relevant for the D-equivalence => L-equivalence conjecture and behaviour of rational points under derived equivalence. The answer to the Hassett-Tschinkel question is positive in Picard rank two under a coprimality assumption, and is negative in general. The proofs rely on Mukai’s techniques of moduli spaces, Derived Torelli Theorem, Hodge lattices and counting formula for Fourier-Mukai partners. This was joint work with Reinder Meinsma.
Egor Yasinsky spoke on Birational automorphisms of algebraic surfaces over non-closed fields. Birational automorphisms of the projective plane (or, equivalently, automorphisms of the field of rational functions in two variables of order 2) were studied already by the Italian school of algebraic geometry – Bertini, Castelnuovo, and Enriques. However, their (more or less) complete understanding became possible only due to modern tools of birational geometry, e.g. the minimal model program, the Sarkisov program, etc. Birational automorphisms of the complex projective plane are pretty well understood today, but for planes over algebraically non-closed fields the situation is much more complicated. In the first part of the talk, Egor reviewed what is known about birational involutions of projective planes over various fields. In the second part, he covered the joint work with I. Cheltsov, F. Mangolte and S. Zimmerman, in which they classified birational involutions of the real projective plane.
The first Fitch lecture was given by Professor John Aston on 5th May 2021: Statistics, Science and Government: A Statistician as a CSA.
John Aston, Harding Professor of Statistics in Public Life, is based in the Statistical Laboratory, Dept of Pure Maths and Mathematical Statistics at the University of Cambridge. From 2017-2020 he was Chief Scientific Adviser to the Home Office. He is an applied statistician who works in areas including medical imaging and official statistics, and was a founding director of the Alan Turing Institute.
Statistics and data have always had a role in government, and censuses, for example, have been carried out for thousands of years to allow governments to make decisions. As a departmental chief scientific adviser, it was always important to me to make sure that the evidence was well considered in the policy-making process. In this talk, I’ll give some examples of how statistics was combined with other science in the policy process, and how hopefully the current appetite for science in government will continue to lead to better and better policy decisions. I very much hope this will encourage other mathematicians and statisticians to consider getting involved in science advice for themselves.
The first Atiyah Lecture was given by Professor Jean-Pierre Bourguignon on 11th January 2021. You can watch a recording of the lecture online here.
What is a Spinor?
This was the title of the lecture Sir Michael gave in September 2013 at IHES on the occasion of the farewell conference for my retirement as Director.
This was most appropriate as I learned a lot from him about this subject. It is true that mathematicians struggled for a long time to get acquainted with spinors. It is in sharp contrast with the fact that physicists adopted them without hesitation as soon as Paul-Adrien Maurice Dirac showed they were essential to formulate a quantum equation invariant under the Poincaré group.
Indeed spinors have a number of features that make them both subtle and powerful to deal with mathematical problems. Of great importance are of course the natural differential operators universally defined on spinor fields, namely the Dirac and the Penrose operators.
The purpose of the lecture is to revisit historical steps taken to master these objects, explore their remarkable geometric content and present some mathematical problems on which they shed light.