31/10/25 Prof. Tom Fisher
The Generalised Fermat Equation
24/10/25 Dr Zoe Wyatt
The Wave of Translation
Abstract:
In 1834, the Scottish civil engineer John Scott Russell was conducting experiments to determine the most efficient design for canal boats. In this process, he discovered “that singular and beautiful phenomenon which I have called the Wave of Translation”. In this talk I will discuss the mathematics behind Russell’s discovery and some related ideas in other areas of mathematical analysis and modelling.
16/10/25 Dr Stephen Wolfram
Maxwell Talk – Jointly held with Cambridge University Physics Society
Computation at the Foundations of Everything — Physics and More
9/10/25 Prof. Imre Leader
Infinite Games
2024/2025
7/5/25 Prof. Nigel Hitchin
Atiyah Lecture
Gauge Theories and Algebraic Curves
Abstract:
Although Michael Atiyah was a big influence on my whole mathematical career, our collaboration in the 1970s and 80s was on instantons and monopoles, gauge theories in 4 and 3 dimensions. The talk will discuss the role of spectral curves for hyperbolic monopoles, introduced in his 1987 paper, and spectral curves for Higgs bundles, a gauge theory in 2 dimensions, where his 1955 paper, written when he was a PhD student, sheds light on some recent developments
21/3/25 Prof. Adrian Kent
Quantum Money without Quantum Memory: Unforgeable Currency in Theory and Practice
Abstract:
Quantum money was one of the first inventions in quantum information technology. The idea is to make money tokens by including some stored quantum states, whose description the bank knows but users do not. Intuitively, these tokens should be unforgeable because the quantum no-cloning theorem tells us that there is no possible way of constructing a device that will duplicate unknown quantum states. As I will explain, though, proving this intuition takes rather more work. Quantum money is not currently practical, because we have no long-term quantum memories. However, I will describe a recently proposed way of emulating its advantages without quantum memories, and an experimental demonstration. Interestingly, this involves re-examining what the advantages of quantum money actually are, which requires re-thinking the role of money in space and time.
14/3/25 Prof. Henrik Latter
Disk and rings throughout the Universe
Abstract:
Disks are ubiquitous in astrophysics and participate in some of its most important processes: star, planet, and satellite formation; the growth of supermassive black holes; the production of jets and outflows, to name but a few. Although astrophysical disks can vary by ten orders of magnitude in size and differ hugely in composition, all share many physical phenomena. This talk explores these areas of overlap, while selecting a few topics to go into further detail, such as structure formation in Saturn’s rings, gap formation in protoplanetary disks, and the huge luminosities achieved by accretion on to compact objects.
26/3/25 Prof. Ioannis Kontoyiannis
Entropy
Abstract:
After a brief historical overview of the origins of the notion of entropy in physics, we describe its explosive entry into mathematics in the 1940s and 50s, and outline some of the mathematical areas in which entropy plays a crucial role. Some examples of important recent problems where the entropy was instrumental in their solution will also be described
21/3/25 Dr. Alejandra Castro
Connections between quantum black holes and number theory
Abstract:
Modular forms play a pivotal role in the counting of black hole microstates. The underlying modular symmetry of counting formulae was key in the precise match between the Bekenstein-Hawking entropy of supersymmetric black holes and Cardy’s formula for the asymptotic growth of states. The goal of this talk is to introduce aspects of the connection between modular forms and black hole entropy, and tie it with other consistency conditions of quantum gravity (via AdS/CFT).
14/2/25 Dr. José Siqueira
Modelling for the Budding Mathematician
Abstract:
Both pure and applied mathematics are about modelling: one approximates concepts, while the other is concerned with ‘the real world’. Either form must contend with the inherent difficulties of the enterprise. This talk will discuss what ‘modelling’ entails and how the tools of category theory provide natural ways to tackle the associated challenges. This is intended as a gentle introduction to categorical thinking, and no prior background is assumed. If time permits, I will also sketch an elegant new approach to modelling complex systems based on the theory of double categories and how it is connected to assume-guarantee reasoning.
24/1/25 Dr. Maria Ivan
Strategies, infinite games and catching robbers
Abstract:
The game of cops and robbers is played on a fixed graph G. The cop picks a vertex to start at, and the robber then does the same. Then they move alternately, with the cop moving first: at each turn the player moves to an adjacent vertex or does not move. The game is won by the cop if he lands on the robber. The graph G is called cop-win if the cop has a winning strategy, and weak cop-win if the cop has a strategy that ensures that the robber is either caught or visits each vertex only finitely many times. How can we characterise the graphs on which we can, if we are smart enough, catch the robber? On a finite graph, we know the answer exactly. These are called `constructible’ graphs: obtained recursively from the one-point graph by repeatedly adding dominated vertices. What about infinite graphs? This notion totally fails to describe the cop-win graphs. In this talk we investigate the relation between constructible graphs and cop-win or weak cop-win graphs. We also investigate how these notions relate to the (weaker and more natural) notion of ‘locally constructible’ (every finite subgraph is contained in a finite constructible subgraph). It turns out that we can have exotic examples, locally constructible, on which the robber can outsmart the robber.
28/9/24 Dr. Tom Crawford
Using Maths to Clean the Ocean
Abstract:
Where does river water go when it enters the ocean? And what does this have to do with plastic pollution? Dr Tom Crawford explains his PhD thesis in Fluid Dynamics, and shares the story of how he ended up sailing around Antarctica for 6 weeks in the name of maths…
22/9/24 Prof. Anne Christine Davis
Searching for Modifications to Gravity
Abstract:
Gravity has been tested over many years in the lab and the solar system. Small deviations from Newton theory can be explained by Einstein general relativity. The modern theory of gravity has passed all tests in the solar system. However beyond the solar system there have been results over the years suggesting we may have missed something and the Universe might be undergoing a period of accelerated expansion today, possibly explained by a so-called Cosmolgical Constant. I will discuss how we can test gravity and search for deviations from the usual theory. I will show the effect the ‘cosmological constant’ has on the dynamics of the local group of galaxies and how we can search for ‘modified gravity’ using the Sun. Knowledge of general relativity is definitely not required.
15/9/24 Prof. Oscar Randal-Williams
Graph Complexes
Abstract:
I will describe a simple “combinatorial” object, Kontsevich’s graph complex GC_d, with the necessary background to understand the definition, and then explain some elementary calculations with it. I will then explain how it plays a role in a conjectural picture of the topology of the space of all homeomorphisms of R^d, as well as the space of all diffeomorphisms of D^d.
25/10/24 Prof. Richard Samworth
Stein’s Paradox
Abstract:
Stein’s paradox is one of the most striking results in Statistics. Although it appears to be a basic problem in mathematical statistics, it turns out to have profound implications for the analysis of modern, high-dimensional data. I will describe both the result and some of its consequences.
18/10/24 Prof. Imre Leader
Eating and Racing
Abstract:
We’ll consider some interesting finite games and some interesting infinite games.
10/10/24 Prof. Micheal Duff
Maxwell Talk – Jointly held with Cambridge University Physics Society
The Best of All Possible Worlds
Abstract:
According to Brandon Carter’s 1960 “Anthropic Principle” : If the laws of physics were slightly different, life could not have formed. Our existence hinges on a delicate fine tuning of the constants of Nature. The masses of the subatomic particles, the strengths of the forces between them, the expansion rate of the universe are “just right” for life. This has been dubbed the ”Goldilocks Enigma”. The traditional goal of theoretical physics is to seek a set of laws that not only describe the universe but do so uniquely. But I will describe how recent advances in Cosmology, String Theory and M-theory have led physicists to abandon this Universe in favor of the Multiverse.
2023/2024
25/4/24 Prof. Simon Donaldson FRS
Atiyah Lecture
Complex Numbers, Quaternions, Octonions and Singular Spaces
Abstract:
In the first part of the talk I will discuss the 1958 algebro-geometric paper of Atiyah “On analytic surfaces with double points”, relating smoothings and resolutions of two-dimensional double point singularities. In the second part I will review the quaternion number system, differential-geometric hyperkahler structures on four-dimensional manifolds and the ALE spaces which connect with the first part. For the last part of the talk, I will introduce the octonion number system, the exceptional Lie group G_2 and the corresponding differential-geometric structures on seven-dimensional manifolds. I will discuss some parts of the 2001 paper of Atiyah and Witten “M-Theory dynamics on a manifold of G_2 holonomy”, and questions of current research interest concerning singularities of G_2 structures
8/3/24 Prof. Anthony Challinor
Constraining the birth of our Universe
Abstract:
One hundred years ago, Hubble determined that the Milky Way does not constitute the entirety of our Universe. Two years earlier, Friedmann had derived his eponymous equations describing the evolution of homogeneous and isotropic models of the universe. These discoveries elevated cosmology to an observational and quantitative science. In the intervening century, our understanding of the Universe and our cosmic history has been transformed. Careful comparisons between measurements of fluctuations in the cosmic microwave background (CMB) and precise calculations of their expected statistical properties have been pivotal to these developments. In this talk, I will discuss how we can use the CMB to constrain the physics of the early universe and what we have learnt with recent data from space- and ground-based measurements.
23/2/24 Prof. Urlich Sperhake
Exploring the Universe with Gravitational Waves
Abstract:
In September 2015, the LIGO gravitational-wave observatory made the first direct detection of a gravitational-wave signal. This Nobel Prize winning achievement marks the dawn of a new era in observational physics and astronomy offering a vast range of unprecedented opportunities to look at (or listen to) our Universe through a new medium. In this talk we present an overview of the theoretical background of this new type of radiation and the methodology how we can observe them. We also discuss the most exciting discoveries made with the about 100 gravitational-wave events that have been detected over the following years and present an outlook on future endeavours of this new rising field of research.
17/11/23 Prof. Nikku Madhusudhan
The Hycean Paradigm in the Search for Life Elsewhere
Abstract:
The search for life elsewhere is the holy grail of exoplanetary science. The detection of atmospheric signatures of habitable Earth-like exoplanets is challenging due to their small planet-star size contrast and thin atmospheres with high mean molecular weight. A new class of habitable exoplanets, called Hycean worlds, promises to expand and accelerate the search for planetary habitability and life elsewhere. Hycean planets are expected to be temperate ocean-covered worlds with H2-rich atmospheres. Their large sizes and extended atmospheres, compared to rocky planets of the same mass, make Hycean worlds significantly more accessible to atmospheric spectroscopy. Several temperate Sub-Neptunes have been identified in recent studies as candidate Hycean worlds orbiting nearby M dwarfs that make them highly conducive for transmission spectroscopy with the James Webb Space Telescope (JWST). Recently, we reported the first JWST spectrum of a possible Hycean world, K2-18 b, with detections of multiple carbon-bearing molecules in its atmosphere. In this talk, we will present observational constraints on the atmospheric composition of K2-18 b, its atmospheric temperature structure, clouds/hazes, chemical disequilibrium and the possibility of a habitable ocean underneath the atmosphere. We will discuss new observational and theoretical developments in the characterisation of candidate Hycean worlds, and their potential for habitability.
3/11/23 Dr. Scott Melville
How Unique is Quantum Gravity?
Abstract:
Our most fundamental physical theories turn out to be the most tightly constrained. There are more mathematical restrictions on quantum mechanics than on classical mechanics, and many more restrictions on relativity than on non-relativistic dynamics. In this talk, I will describe this steady march towards a maximally constrained (hopefully unique) theory of everything, and give an example of how to put this perspective into practice: namely the calculation of the classical gravitational waves emitted by two black holes merging (which turns out to be much easier if you first quantise everything, make use of how constrained quantum mechanics is, and take the classical limit only at the very end).
20/10/23 Prof. Richard Nickl
Bayesian Inferences and the Laplace-Bernstein-von Mises Theorem
Abstract:
Bayesian inference has been widely used in the statistical sciences and applied mathematics – systematically so since Laplace’s ‘Theorie analytique’ from 1812. It has seen a ‘dark age’ of subjectivist thinking in the 20th century due to computational infeasibility, but then emerged, with the advent of modern Markov chain Monte Carlo methods in the 1990s, as a popular paradigm for data driven inference under uncertainty. Nowadays Bayesian algorithms are among the most commonly used methods in statistics and machine learning. We will discuss some mathematical theorems that explain when one can trust such algorithms in the high-dimensional context of modern data science.
13/10/23 Prof. Bela Bollobas
Long Life Problems
Abstract:
My talk will be about some problems that were raised many decades ago, and in spite of numerous partial solutions are still unsolved. I do not mean the greatest problems in mathematics like P vs NP, the Riemann Hypothesis, and other Millennium Problems, but problems and their partial solutions that can be enjoyed by all, from freshmen to Part III students. With a bit of luck, a member of the audience will achieve instant fame by producing a breakthrough.