Symmetries play a central role in mathematics, linking algebra and geometry. Groups and algebras of symmetries provide geometric realisations of respective abstract structures, while their algebraic properties lead to unexpected geometric insights. In this session, three speakers showcase the rich interplay between these two worlds.

First, Prof. Travis Schedler of Imperial College outlines a programme to classify crepant resolutions of moduli spaces—especially Nakajima quiver varieties—revealing new structures beyond the projective case.

Next, Prof. Evgeny Feigin from Tel-Aviv University describes the geometry of the graph closure of a birational map from projective space to a Grassmannian, including its fibres and embeddings into projectivized cyclic representations of a degenerate Lie algebra.
Finally, Prof. Konstanze Rietsch of King’s College London explores new tropical versions of parametrisations for totally positive Toeplitz matrices, offering a unified perspective on their roles in representation theory and mirror symmetry, and connecting them to the quantum cohomology of flag varieties and canonical bases.

## Event information

These talks take place in the seminar room of the London Institute, on the second floor of the Royal Institution. There will be a short coffee break between talks and a longer pause to allow participants to have lunch.

## Programme

- *10:00am* Arrival & Coffee
- *10:30am* Prof. Travis Schedler: Crepant resolutions of moduli spaces and quiver varieties
- *11:40am* Prof. Evgeny Feigin: Birational maps to Grassmannians and poset polytopes
- *13:00pm* Lunch
- *13:30pm* Prof. Konstanze Rietsch: A tropical Edrei theorem

Three leading experts show how the idea of symmetry forms key interfaces between algebra and geometry through the lens of their recent work.

Symmetry day

Speakers

Symmetries in algebra and geometry

As Camille Jordan noted in 1870, problems in enumerative geometry have associated Galois groups, just as univariate polynomials do. Despite this pedigree, the subject lay dormant for a century, with systematic investigations of such Galois groups starting only in the past 15 years. Prof. Frank Sottile of Texas A&M University discusses recent advances in this area, describing the current directions and open problems.

Prof. Frank Sottile of Texas A&M outlines the history and the state of the art of a keystone topic at the interface of algebra and geometry.

Enumerative Galois

Galois groups in enumerative geometry

Prof. Ed Segal of UCL discusses derived equivalences and autoequivalences arising from wall-crossing phenomena on Calabi-Yau toric varieties. Mirror symmetry relates these to monodromy around the Gelfand–Kapranov–Zelevinsky's discriminant locus. He conjectures that intersection multiplicities in the discriminant equal the multiplicities appearing in semi-orthogonal decompositions, proving the conjecture in some cases.

Ed Segal from UCL will talk about semi-orthogonal decompositions of derived categories of toric varieties, coming from wall-crossing in GIT.

Discriminants & physics

Discriminants and semi-orthogonal decompositions

The Special Relationship between America and Britain has never been more special than in science. Yet now that the number of scientists looking to leave America is on the rise, what is Britain doing to support them? In a recent [article](https://lims.ac.uk/perspectives/the-most-valuable-cargo/) in *The Spectator*, the Director of the London Institute, Dr Thomas Fink, argued that we should create new scientific posts earmarked for Americans, and view the coming wave of talent not as a prospective win for Britain but as part of the long-term scientific alliance between our two countries.

In this event, Prof. Thomas Rosenbaum, the President of Caltech, will sit down with James Arroyo OBE, the Director of the Ditchley Foundation, for a conversation about how British science can respond to recent proposed cuts to the government funding of science in America. The discussion will be chaired by Dr Fink. There will be time for questions from guests, which will include Caltech alumni, members of the London Institute, science journalists, and others.

## Event information

The event takes place on Tuesday 24 June at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. It starts at 6:30pm.

## Programme

- *6:30pm* Arrival and drinks.
- *7:10pm* Introduction by Thomas Fink.
- *7:15pm* Presentation by Thomas F. Rosenbaum, followed by a chat between him and James Arroyo about how Britain can support America’s scientists. The conversation will be chaired by Thomas Fink.

The President of Caltech chats with the Director of the Ditchley Foundation about how Britain should respond to US science funding cuts.

Supporting US scientists

The Special Relationship between America and Britain has never been more special than in science. That relationship is now being put to the test. According to *Nature*, the number of scientists looking to leave America has recently risen by a third. The question is: what is Britain doing to support them? In a recent article in *The Spectator*, the Director of the London Institute, Thomas Fink, banged the drum for creating new scientific posts that are earmarked for Americans. 

 This event is first and foremost for Caltech alumni and postdocs. Prof. Thomas F. Rosenbaum, the President of Caltech, will begin proceedings with an update on Caltech news. There will then be a conversation between Prof. Rosenbaum and James Arroyo OBE, the Director of the Ditchley Foundation, about how British science should respond to recent proposed cuts to the American state funding of science. The discussion will be chaired by Thomas Fink and will be open to questions from guests.

## Event information

The event takes place on Tuesday 24 June at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Please come at 6:30 pm for drinks beforehand.

## Programme

- *6:30pm* Arrival and drinks.
- *7:10pm* Introduction by Thomas Fink.
- *7:15pm* Presentation by Thomas F. Rosenbaum, followed by a chat between him and James Arroyo about how Britain can support America’s scientists. The conversation will be chaired by Thomas Fink.

Caltech alumni gather at the London Institute to catch up on Caltech news and hear a discussion on how Britain can help American science. 

Caltech in London

Ageing is thought to be an inevitable consequence of breakdowns in how genetic information is processed. But mounting experimental evidence suggests aging can be slowed. To resolve this, Dr Thomas Fink, Director of the London Institute, derives a mortality equation that governs the dynamics of an evolving population with a given maximum age. His solution shows that programmed ageing cannot confer a long-term benefit.

Thomas Fink unveils a mathematical model showing programmed aging cannot be favoured by natural selection, even in a shifting environment.

The mortality equation

Martin Reeves, Senior Partner and Managing Director at the Boston Consulting Group and Trustee of the London Institute, took up the Great Highland bagpipe during the coronavirus pandemic. In this talk and performance, he delves into the noble history and mathematical structure of the instrument.. With lyrical insight and expert playing, Mr Reeves shows why the bagpipe is as rich in logic as it is in tradition and sound.

Martin Reeves explores the surprising harmony between the music and maths of the Great Highland bagpipe—complete with a live demonstration.

The mathematical piper

Prof. Michael Douglas of the Center of Mathematical Sciences and Applications at Harvard explores the frontiers of artificial general intelligence, which some experts now anticipate within 2–3 years. Drawing lessons from past milestones—such as when computers surpassed humans in chess—he discusses the development of mathematical copilots and conceptual systems that could soon enable autonomous, machine-led research.

Prof. Michael Douglas examines how advances in AI may soon lead to autonomous systems capable of making genuine mathematical discoveries.

AI for discovery

AI in math and theoretical physics

Prof. Joseph Conlon from Oxford University will explore the largely unconstrained epoch between inflation and nucleosynthesis, spanning more than 30 orders of magnitude in time. String cosmology suggests this era may include moduli-dominated phases—such as kination, tracker, and matter epochs—and he introduces a novel percolation-based scenario for the formation of cosmic string networks in the very early universe.

Prof. Joseph Conlon outlines how string theory suggests new equations of state that may describe the physics of the very early universe.

Early universe strings

String theory and the early universe

Artificial Intelligence is revolutionising our understanding of life. The DeepMind programme tackling the protein folding problem, AlphaFold, ushered in a new era for protein structure prediction and earned its creators last year’s Nobel Prize in Chemistry. But that was just the beginning.

Speaking in the historic lecture theatre of the Royal Institution, Dr Mikhail Burtsev gives a tour of the leading-edge biological applications of AI, including his own, that are now making complex biological insights more accessible than ever before. AI chatbots such as ChatGPT are trained on human language, but these new AI models are instead fed billions of real DNA sequences, allowing them to perform astonishing feats like designing new biological molecules—or even whole new bacterial genomes. Their ultimate goal is to solve perhaps the most fundamental problem in all of biology—reading and understanding life’s genetic blueprint from beginning to end.

## Event information

This event, which includes research supported by the Khodorkovsky Foundation, takes place in the lecture theatre at the Royal Institution. London Institute guests are invited to join the speaker after for drinks in our private rooms.

Mikhail Burtsev explains how artificial intelligence is helping to decipher genomes in the Royal Institution’s iconic lecture theatre.

Decoding life with AI

Speaker

Dr Simon Telen leads the numerical algebraic geometry group at the Max Planck institute for Mathematics in the Sciences in Leipzig. He studies toric amplitudes—rational functions inspired by the scattering amplitudes of particle physics but associated to arbitrary simplicial polyhedral fans. He describes the algebraic properties of such amplitudes and their zero loci, exploring their Fano schemes and singular loci.

Simon Telen presents his discovery of a new connection between polynomials inspired by particle physics and classical algebraic geometry.

Geometry meets physics

Mr Pinardin is a PhD student at the University of Edinburgh. He addresses the problem of describing the linearizable subgroups of the plane Cremona group, the group of birational self-maps of the projective plane. Drawing on work with Drs Arman Sarikyan and Egor Yasinsky, Mr Pinardin gives a complete solution to this problem, equivalent to the question of G-equivariant rationality, over the field of complex numbers.

Antoine Pinardin presents a complete description of the linearisable subgroups of the plane Cremona group over the field of complex numbers.

Linearising groups

Linearisation problem for finite subgroups of the plane Cremona group

Dr Matthew Buican is a Reader in theoretical physics at Queen Mary University of London. He describes a universal relevant deformation that takes local unitary 3d N=4 superconformal field theories to topological quantum field theories, explaining, for example, how Abelian mirror symmetry is related to generalisations of level-rank duality, as well as some results that arise as a result of ’t Hooft anomaly matching. 

Dr Matthew Buican details how a universal deformation takes 3d N=4 superconformal field theories to 3d topological quantum field theories.

Conformal to topological

A universal connection between 3d N=4 SCFTs and 3d TQFTs

Symmetry plays a fundamental role in quantum systems, shaping their dynamics, enforcing selection rules and enabling the development of new algorithms for nascent quantum computers. In recent years, novel forms of symmetry, known as generalised global symmetries, have been discovered in a wide range of quantum systems. These include non-invertible symmetries and higher-group symmetries, which go beyond traditional notions of symmetry and provide new insights into the structure and behavior of quantum theories.

In this series of four lectures, Dr Hsin will explore the role of these symmetries in various quantum systems and their applications, such as their ability to constrain low-energy dynamics. The lectures will examine examples from quantum mechanics, gauge theories and lattice models, focusing on ordinary symmetries, higher-form symmetries and non-invertible symmetries, shedding light on their profound impact in modern physics.

## Event information

This is a four-part lecture series from LonTI Lectures are on Mondays at 10:30 am in the Tyndall seminar room at the London Institute, on the second floor of the Royal Institution, followed by drinks and snacks onsite. To register and attend, please visit lonti.weebly.com.

Dr Po-Shen Hsin introduces generalised symmetries in quantum systems and explores their roles in dynamics, constraints and applications.

Quantum symmetries

In a session co-chaired by Prof. Kevin Buzzard of Imperial College London and Prof. Richard Hill of University College London, researchers and PhD students discuss their projects in the area of formal proof verification in Lean. The meeting includes visitors from Imperial, UCL, the University of Cambridge and Google DeepMind as well as the London Institute, fostering collaboration across a wide swathe of mathematics.

Mathematicians across London share their work on the formalisation of mathematics using the AI proof-assistant computer language Lean.

London learns Lean

Two seminars discuss new applications of AI to quantum theory. Prof. Koji Hashimoto of Kyoto University shows how a neural network can generate arbitrary Feynman path integrals as a consequence of the universal approximation theorem. Afterwards, Dr Akio Tomiya of Tokyo Woman's Christian University introduces a transformer architecture designed for lattice QCD that outperforms existing gauge covariant neural networks.

In two consecutive talks, Prof. Koji Hashimoto and Dr Akio Tomiya discuss applying machine learning to problems in quantum field theory.

AI for QFT

Prof. Konstantinos Zoubos from the University of Pretoria will explain how Lie groupoids and their twists are the appropriate framework to discuss the true symmetries of N=2 supersymmetric orbifold gauge field theories in the planar limit. These hidden symmetries provide an explanation for certain non-trivial relations between QFT observables and point to possible integrable structures associated to these theories.

Prof. Konstantinos Zoubos will use Lie groupoids and their twists to uncover hidden symmetries of supersymmetric orbifold field theories.

Groupoids do the twist

Dr Bhavik Mehta is research fellow at Imperial College London, where he works with Prof. Kevin Buzzard, an expert on formal proof verification. Dr Mehta did his PhD at Cambridge under the supervision of Prof. Timothy Gowers and later worked on Google DeepMind’s AlphaProof, which formulates problems in Lean. In a series of talks at the London Institute, he shows how Lean can verify proofs and even help find them.

Dr Bhavik Mehta from Imperial College London gives an overview of Lean, the proof-assistant computer language used to formalize mathematics.

Co-piloting proofs

Dr Matthew Yu is a postdoc at the University of Oxford’s Mathematical Institute. He describes how topological theories that are fermionic in nature can be rigorously axiomatised in low dimensions with additional structure to accommodate their relevant physical properties. Prior to the talk, London Institute Fellow Dr Juven Wang will join Dr Yu for a discussion on the physical and mathematical context of the problem.

Dr Matthew Yu explains how fermionic topological quantum field theories can be given a categorical description in 3+1-dimensional spacetime.

Fermionic TQFTs

Quantum Chromodynamics has been a profound source of inspiration for theoretical physicists, driving the development of key concepts such as string theory, effective field theories, instantons, anomalies, and lattice gauge theories. In this four-part lecture series, Dr Guerrieri will explore two distinct regimes of Quantum Chromodynamics—its infrared and ultraviolet limits—and the theoretical tools used to study them.

In the infrared regime, where perturbative techniques break down, Effective Field Theories provide a powerful framework. Here, Dr Guerrieri will introduce the pion EFT as a tool to study non-linearly realized symmetries and soft theorems. In the ultraviolet regime, where Quantum Chromodynamics becomes amenable to perturbative analysis, he will discuss the Operator Product Expansion and renormalization group equations, focusing on their application to deep inelastic scattering, a cornerstone in the discovery of quarks and gluons.

## Event information

This is a four-part lecture series from LonTI Lectures are on Mondays at 10:30 am in the Tyndall seminar room at the London Institute, on the second floor of the Royal Institution, followed by drinks and snacks onsite. To register and attend, please visit lonti.weebly.com.

Dr Andrea Guerrieri explores QCD in two distinct regimes to illustrate its pivotal role in shaping our understanding of fundamental physics.

QCD’s IR & UV limits

Dr Alex Davies leads Google DeepMind’s initiative for AI and mathematics. In a discussion with our scientists and staff, he gives his take on AlphaGeometry, which solves hard but known problems, and the possibilities for AI-assisted discovery of new conjectures. He also considers the scope and limits of automated theorem proving, and how the human mind makes shortcuts that machines must spell out pedantically. 

Dr Alex Davies of Google DeepMind leads an interactive discussion about AI-assisted mathematical discovery and automated theorem proving.

Automating maths

Prof. Alexander Migdal, a member of the Institute for Advanced Study and formerly at Princeton and the Landau Institute, presents an exact solution to the Navier-Stokes equations for incompressible fluid flow in the context of decaying turbulence. He introduces the framework of the Euler ensemble, which maps turbulent dynamics onto discrete states represented by regular star polygons with rational vertex angles.

Prof. Alexander Migdal presents an infinite dimensional manifold of exact solutions to the Navier-Stokes equations for decaying turbulence.

Solving Navier-Stokes

In 1974, Stephen Hawking calculated that black holes emit particles, eventually evaporating away completely. This was the origin of the black hole information paradox, in essence a conflict between quantum theory, which demands that information on particle interactions cannot be lost, and general relativity, which predicts that once a particle crosses the event horizon, no information can ever escape.

In this four-part series, Dr Tarek Anous will give a technical introduction to the paradox. Beginning with a review of quantum path integrals, he will outline the laws of black hole thermodynamics before deriving the Unruh effect, which predicts that a detector with a uniformly accelerated observer measures a temperature related to the observer’s proper acceleration. This will be followed by a session on classical and quantum information theory. The series concludes with a toy model of the paradox and a review of what is left to understand.

### Event information

This is a four-part lecture series from LonTI. Lectures are on Mondays at 10:30 am in the Tyndall seminar room at the London Institute, on the second floor of the Royal Institution, followed by drinks and snacks onsite. To register and attend, please visit [lonti.weebly.com](https://lonti.weebly.com/).


Dr Tarek Anous introduces the 50-year-old black hole information paradox and discusses some of the questions that have yet to be answered.

Black hole paradox

In this seminar, Dr Wang will explore key problems in high energy theoretical physics. First, he will discuss the puzzle of why there are three families of quarks and leptons in the Standard Model. He will argue that a dimensional reduction of the Standard Model to a 2-dimensional conformal field theory with chiral central charge c-=0 mod 24 is both modular invariant and framing anomaly free. This result, he will show, is constrained by the Hirzebruch signature theorem, gravitational Chern-Simons theory, and 3 E8 quantum Hall states.

Secondly, Dr Wang will propose a new model of unification constructed from a mod 16 nonperturbative global anomaly cancellation that adds a symmetry-extended anomalous topological quantum field theory to the Standard Model. With this fresh approach to physics beyond the Standard Model, Dr Wang will address long-standing questions relating to the properties of neutrinos, the composition of Dark Matter and the cause of leptogenesis.

Dr Juven Wang proposes novel solutions to open problems in high-energy phenomenology via new analogies from modern ultra quantum matter.

Ultra Unification

Simon Norton, who died in February 2019, is best remembered by mathematicians for his remarkable contributions to group theory. A child prodigy, Norton earned a First in mathematics from the University of London while he was still studying at Eton College.

Later, at the University of Cambridge, he would work with John Conway, co-authoring the ATLAS of Finite Groups, the definitive catalogue of different types of symmetry. The pair’s paper on monstrous moonshine, published in 1979, proposed an unexpected deep connection between the monster group and the j-function. Seemingly far-fetched, the conjecture was proved in 1992 by Richard Borcherds and is recognised as one of the great unifying conjectures in modern mathematics.

In the second Simon Norton Lecture, Prof. Leonard Soicher will explain what the monster is and its importance before describing Norton’s research on the internal structure of the group and its subgroups. He will also discuss his work with Norton and Conway on the projective plane presentation for the bimonster. We are grateful to Simon’s family for their generous support of this lecture.

## Event information

The event takes place on Wednesday 12 February at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. To book a place, [register here](https://tally.so/r/wdQKDK). 

## Programme

- *5:30pm* Arrival and drinks, Old Post Room
- *6:00pm* Prof. Yang-Hui He, Introduction, Tyndall seminar room
- *6:10pm* Prof. Leonard Soicher, “Simon and the Monster”
- *7:10pm* Drinks and canapes, Old Post room

In the second Simon Norton Lecture, Prof. Leonard Soicher will convey the mystery and the magic of the monster group to a general audience.

A friendly monster

The London Institute was founded in 2011 and incorporated on 10 February. This marked the birth of Britain’s only independent research organisation in the physical sciences. The date is also the feast day of St Scholastica, who lived in Italy in the 6th century and is the patron saint of learning and Benedictine nuns.

One of the few surviving stories about St Scholastica is that one night, after her brother St Benedict had dined with her, he rose to leave. She prayed to God to make it rain, so he would be compelled to stay and they might continue talking. So it came to pass. St Scholastica is therefore indelibly associated with evenings filled with good conversations, which continue into the night.

To celebrate our founding and our belief in the value of community, we hold an annual St Scholastica’s Feast in our rooms. This formal dinner is for our scientists and staff and a handful of distinguished guests. It takes place on a Friday near the day of our incorporation.

## Event information

The dinner is at 7 o’clock on Friday, 14 February in our rooms on the second floor of the Royal Institution. Dress is black tie. We meet in the Old Post Room, then head to dinner in the Faraday Room and dessert in the Bragg Room, followed by drinks in the Old Post Room.

We hold an annual formal dinner in our rooms, to mark the anniversary of our founding and affirm our belief in the importance of community.

St Scholastica’s Feast

Our Christmas party takes place in our rooms in the Royal Institution on the first Friday of December. It’s a chance to celebrate the contributions of our supporters and collaborators—which is why most attendees are from outside the Institute.
Because our financial year ends in December, our Christmas party is also a chance to take stock of the past year and set out our vision for the next.

Attending our 2024 Christmas party are one High Commissioner, two British knights, three of our Trustees, four science writers, five major donors, six Alexanders, seven RI members, eight tech upstarts, nine Erdös 3-ers, ten LIMS Fellows, eleven rock climbers, and twelve Russian and Ukrainian theorists.

Among the 78 attendees are the Director of the International Centre for Mathematical Sciences, the Director of the Royal Institution, and members of the Khodorkovsky and Sakharov foundations, who have supported our programme of fellowships for Russian and Ukrainian researchers.

## Event information

Our Christmas party is on Friday, 6 December, from 6 o’clock until late. It is held in the London Institute’s Faraday Room and Old Post Room, on the second floor of the Royal Institution. The party is for members of the Institute and their guests and those who have supported us.

Our Christmas party is an opportunity to celebrate our supporters and collaborators, review the past year, and set our plans for the future.

Christmas party

Combinatorial geometry serves as a bridge between various areas of mathematics, such as discrete and algebraic geometry, combinatorics, and algebra, facilitating the transfer and abstraction of ideas across these fields and their applications. In particular, over the last decade, an explosive exchange of ideas between algebraic geometry and combinatorial convexity has led to breakthroughs in both disciplines, the resolution of many landmark conjectures, Fields Medals, and even new entries in the mathematics subject classification.

The two talks at our meeting survey these advances and present an up-to-date overview of combinatorial geometry to the broader mathematical community. It is a pleasure for the London Institute to welcome the speakers, whose contributions to the field have largely shaped its landscape. The presence of combinatorial geometry in the research of many mathematicians and physicists at the Institute—spanning topics from algebraic geometry and representation theory to mirror symmetry and beyond—illustrates its central, connecting role across mathematics.

## Event information

The collquium takes place on Monday 18 November at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution in Mayfair. Please enter the door under "Ri" flags and take the stairs.

The London Institute will host two talks by leading experts, surveying recent advances at the intersection of geometry and combinatorics.

Matroids are combinatorial structures that track “independence” relations on a set. A key example is the linear independence of vectors in a vector space. While not all matroids arise from a vector space, those that don't often exhibit surprising algebraic behavior as if they do. Breakthroughs over the last decade have introduced a suite of tools from, and inspired by, algebraic geometry to prove inequalities for matroids, including the 'matroid Hodge theory' developed by Adiprasito, Huh, Katz, and others. I'll begin by motivating matroids and aim to conclude with enough about my work in progress with Andy Berget to illustrate how its central tool differs from matroid Hodge theory.

Matroid inequalities from algebraic geometry

The factorization problem asks under what conditions two birationally isomorphic varieties, X and Y, share a common iterated smooth blowup, Z. Oda specifically conjectured that this holds for toric varieties. We provide an affirmative answer if the blowups are allowed to be at rationally smooth centers, solving a weighted version of Oda's conjecture and addressing a separate conjecture of Alexander in PL topology.

The Oda conjecture

Combinatorial geometry

Combinatorial geometry colloqioum

Conjectures can inspire new branches of pure mathematics and theoretical physics. They usually come from spotting patterns and applying instinct. Recently, there has been a surge of interest in using automated pattern detection to help humans form conjectures. Because in mathematics there are no coincidences, mathematical data is immune from the false positives and false negatives that plague physical measurement.

In this two-day workshop, the London Institute brings together physicists and mathematics to explore how AI can speed up theoretical research. The topics addressed range from geometry to string theory to representation theory.

This is the fourth in the series of annual DANGER workshops (Data, Numbers, Geometry and Representation theory). The series was created by Alexander Kasprzyk from Nottingham University, Thomas Oliver from Westminster University, and Yang-Hui He from the London Institute. This is a hybrid workshop. Those unable to attend in person can join online via Zoom.

## Event information

This workshop takes place on Thursday 24 and Friday 25 August at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution in Mayfair. To register to attend this workshop, please visit the conference [website](https://sites.google.com/view/danger4workshop/home).

## Programme

The London Institute hosts a two-day workshop for theorists to discuss and explore the links between data science, AI and pure mathematics.

Thursday 8 August

Arrival

Proposing good conjectures is at least as valuable as proving theorems. Good conjectures capture our attention and orient our efforts, acting like guide posts on the 'mazy paths to hidden truths' (Hilbert). This talk will touch on three aspects of computer-aided conjecture generation: matching numerical values, symbolic regression and generative models. I will present a Zipf-type law for Physics equations, discuss how genetic algorithms can be used to identify new identities of Rogers-Ramanujan type and present a number of conjectural generating functions for holomorphic line bundle cohomology on certain complex projective varieties.

Computer-aided conjecture generation in maths and physics

Lunch

In this talk, we explore the asymptotic behaviour a certain sequence, which contains the coefficients of a series called the regularized quantum period. This is conjectured to be a complete invariant of Fano varieties and its can be considered as a numerical fingerprint of each Fano variety. We discuss how these results originate from the study of large datasets using data analysis and machine learning techniques and show how they can be used to visualise the landscape of these objects. This is joint work with Tom Coates and Alexander Kasprzyk.

Asymptotic formulae for the regularized quantum period

How can we make sure that a machine learning system gives results as it should, is not biased or is safe to use–as we would like for self-driving cars, for example? We cannot. Not fully. But we can try! This talk will be an incursion in theoretical computer science and formal verification. We will explain ways one can model machine learning systems to try to obtain partial guarantees on them, and how to enforce some safety property.

Modelling machine learning systems

Topological data analysis (TDA) offers a plethora of methods and tools to study the “shape of data”. This talk is about work towards implementing a TDA tool inside a theorem prover, leading to software that is formally proved to implement the intricate mathematical theory underlying TDA. The theorem prover of choice for this project is Cubical Agda, which implements ideas from Homotopy Type Theory and offers a logic to directly reason about (homotopy types of) topological spaces. We give a taste of what it’s like to reason in this theorem prover, presenting a formalisation of discrete Morse theory for graphs, and discuss challenges for turning this formalisation into a fully fledged tool for TDA.

Formalising topological data analysis in Cubical Agda

Break

We provide a theoretical argument explaining the experimental observations of Yang-Hui He about the learnability of primes, and posit that the Erdős-Kac law would very unlikely be discovered by current machine learning techniques. Numerical experiments that we perform corroborate our theoretical findings. This is joint work with A. Alistair Rocke.

Machine Learning of the Prime Distribution

Calabi-Yau compactifications of string theory lead to quantum field theories in four dimensions with chiral matter. Calculating parameters of the low-energy effective theory in general compactifications requires the Ricci-flat metric on the Calabi-Yau manifold. Such metrics are not known analytically. In this talk, we discuss how to approximate the Ricci-flat metric using neural networks. The accuracy of the numerical metrics is assessed for K3 and the quintic threefold. In the standard embedding, we calculate Yukawa couplings for compactifications on various Calabi-Yau geometries. This is an initial step toward a first principles calculation of particle masses from string theory.

Precision string phenomenology

Friday 9 August

In this talk, we explore the transformative potential of off-the-shelf reinforcement learning (RL) algorithms in accelerating solutions to complex, research-level mathematical challenges. We begin by illustrating how these algorithms have achieved a 10× improvement in areas where previous advances of the same magnitude required many decades. A comparative analysis of different network architectures is presented to highlight their performance in this context. We then delve into the application of RL algorithms to exceptionally demanding tasks, such as those posed by the Millennium Prize problems and the smooth Poincaré conjecture in four dimensions. Drawing on our experiences, we discuss the prerequisites for developing new RL algorithms and architectures that are tailored to these high-level challenges.

Math+AI=AGI

Reproducing kernel Hilbert C*-module (RKHM) is a generalisation of Reproducing kernel Hilbert space (RKHS) and is characterised by a C*-algebra-valued positive definite kernel and the inner product induced by this kernel. The advantages of applying RKHMs to data analysis instead of RKHSs are that we can enlarge representation spaces, construct positive definite kernels using the product structure in the C*-algebra, and use the operator norm for theoretical analyses. We show fundamental properties in RKHMs for data analysis, such as a minimization property of the orthogonal projection and representer theorems. Then, we propose a deep RKHM, which is constructed as the composition of multiple RKHMs. This framework is valid, for example, for analysing image data.

Reproducing kernel Hilbert C*-module for data analysis

We explore the idea of using machine learning to determine rational points on weighted projective varieties. Our work leads to many interesting questions on the geometry of weighted projective spaces and how machine learning models can be used to study weighted  spaces. While mathematically it seems that such models would be superior to more traditional models,  it is still an open question to provide computational evidence to support such claims.

Machine models for weighted spaces

We use machine learning to study the moduli space of genus two curves. During the talk, we will focus on the locus Lₙ of genus two curves with (n,n)-split Jacobian. More precisely, we design a transformer model which, given moduli points (i.e., values for the Igusa invariants), determines if the corresponding genus two curve is in the locus Lₙ, for n = 2, 3, 5, and 7. During this study we discover some interesting arithmetic properties which seem difficult to guess otherwise. For example we show  that there are no rational points p∈ Lₙ with weighted moduli height ≤ 2 in any of L₂, L₃, and L₅.

Machine learning in the moduli space of genus two curves

Rather than relying on existing data sets to data mine, the combination of computational algebra and data science techniques allows one to simultaneously create algebraic data sets and to then apply standard data science techniques in order to try to find patterns in this generated data through exploratory data analysis. Ultimately, this could help complement mathematical intuition in guiding hypothesis formulation and theorem proving. We will show examples from recent work on cluster algebras and Clifford algebras, such as machine learning interesting new Clifford invariants of linear functions, using the example of Coxeter elements as transformations of interest with links to other, well-known, geometric invariants.

Applying a computational algebra and data science pipeline to Clifford algebra and cluster algebra data

DANGER 2024

DANGER: Data, Numbers and Geometry

Birational geometry studies when two algebraic varieties are isomorphic outside lower-dimensional subsets. Quot schemes were constructed by Grothendieck to parameterise families of sheaves with fixed numerical invariants. Stability is a key property of a sheaf over an algebraic variety, characterising, for examples, the existence of flat Hermitian metrics.

In this seminar, Prof. Birkar will explore the moduli of sheaf stable pairs, examining their relationships with Quot schemes and birational geometry. The focus will be on Fano varieties and their singularities. Drawing from the recent paper "Sheaf Stable Pairs, Quot-Schemes, and Birational Geometry" by Birkar, Jia, and Sheshmani, this presentation connects moduli theory of sheaf stable pairs with birational geometry. It offers in-depth analysis of moduli on smooth projective curves and explores calculations in low degrees. This seminar aims to bridge these concepts and give their implications for algebraic geometry.

## Event information

The event is on Wednesday 31 July at the London Institute, which is on the second floor of the Royal Institution. The seminar starts at 5pm and is followed by a Q&A. There will be drinks afterwards.

In this seminar, Fields Medalist Prof. Caucher Birkar talks about sheaf stable pairs in algebraic geometry, especially for Fano varieties.

Stable pairs

In our private and professional lives, from the way we communicate with one another to how we form mathematical conjectures, the impact of artificial intelligence will be transformative. Writers and editors from MIT Technology Review have been following and commenting on the rise of AI for years. In this panel discussion, which is organised by the MIT Club of Great Britain, members of the magazine’s editorial team talk about what is exciting them in artificial intelligence, how they think about the technology, and what the future holds. The senior AI editor, Will Douglas Heaven, and the senior AI reporter, Melissa Heikkilä, will be moderated by the magazine’s executive editor, Niall Firth. The event will conclude with a Q&A.

Founded in 1899, MIT Technology Review is an independent award-winning bimonthly magazine, which reports on the newest technologies and their commercial, social and political impacts. The MIT Club of Great Britain is the association for MIT’s UK-based alumni. The event is open to members and non-members of MIT Club, but tickets must be booked before the event using the link below. 

## Event information

The event is on Wednesday 5 June at the London Institute, which is on the second floor of the Royal Institution. Come for a glass of wine from 18:30. The panel discussion starts at 19:00 and is followed by a Q&A. There will be drinks afterwards. Please book your tickets here.

## Programme

- *6:30pm* Arrival and drinks, Old Post Room
- *7:00pm* Panel discussion and Q&A, Tyndall Parlour, with panellists Will Douglas Heaven and Melissa Heikkilä, moderated by Niall Firth
- *8:00pm* Drinks

Journalists from _MIT Technology Review_ discuss some of the risks and opportunities around AI and how the magazine will be covering them.

The future of AI

Speakers and panellists

The Langlands program of number theory, or what is also known as Langlands duality, was proposed by Robert Langlands, in the late 1960s. It is a unified scheme for many results in number theory ranging from quadratic reciprocity, which is hundreds of years old, to modern results such as the Shimura-Taniyama-Weil conjecture (as proved by Andrew Wiles) which implies Fermat’s last theorem. The duality has a geometric version which makes it amenable to tools in complex algebraic geometry, where it can also be realized in quantum field theory.

In this Bragg Lecture Series, Prof. Meng-Chwan Tan will first furnish a pedestrian introduction to Langlands duality, and its geometric version which manifests itself in physical gauge theory. He will then proceed to expound on its generalizations which involve complex surfaces instead of curves, and also elucidate the relationship between the gauge-theoretic and original algebraic CFT approaches to the geometric duality by Kapustin-Witten and Beilinson-Drinfeld, respectively, via string/M-theory.

## Event information
This is a four-part Bragg lecture series. Lectures are from June 3-6, Monday to Thursday, at 11:00 am, in Bragg’s Parlour at the London Institute in the Royal Institution.

Prof. Meng-Chwan Tan gives a pedagogical introduction to geometric Langlands duality and its generalizations via gauge and string theory.

Geometric Langlands

Geometric Langlands and its generalizations via gauge and string theory

Scientific discovery is the success story of mankind. But its future is not a foregone conclusion. Now as always, discovery relies on the framework within which it operates: the incentives and organisational structures that we put in place in order to encourage and catalyse genius.

In Britain, we have an extraordinary record of discovery, which is epitomised by the remarkable run of breakthroughs Michael Faraday made at the Royal Institution. Yet if we’re to enhance our standing as a scientific nation, we need to innovate the way we innovate. That means improving how discovery is funded and rewarded, as well as increasing the variety of organisations that do discovery, from the ancient universities to younger institutes such as the London Institute for Mathematical Sciences, which is based in the Royal Institution.

At the global level, science has a unique role to play. As the language of patterns, it is universal, transcending political differences. Science diplomacy keeps channels of communication open between nations otherwise at odds. Yet when troubled times lead to a migration of talent, we must make scientists welcome through initiatives such as the London Institute’s programme of sanctuary fellowships for Russians and Ukrainians, which is the largest in the world. Meanwhile, we must compete and cooperate with countries such as China and India, which are recruiting and producing some of the best researchers. 

This symposium, which takes place in the rooms where Faraday worked, brings together scientists, innovators, policymakers and new organisations to debate the conditions within which genius can flourish. Tomorrow’s discoveries will be built according to the incentives and organisational structures that we enact today.

## Event info

This half-day symposium is on Tuesday 21 May, from 2:30 p.m. until late. To attend, please email Justine Crean, jc@lims.ac.uk. The talks and discussion take place in the Lecture Theatre of the Royal Institution. There are drinks from 6:00 p.m. in the Library and Georgian Room.

## Programme

<h3 class="pr-h3">Speeding up discovery</h3>

- Lecture Theatre
- *2:30* Introduction, by Dr Thomas Fink
- *2:35* Lord Willetts
- *2:50* Chi Onwurah, MP
- 3:05 Panel discussion, moderated by Tom Whipple
- Dr Thomas Fink
- Dr Ilan Gur
- Chi Onwurah, MP
- Arkady Volozh
- Lord Willetts
------
- Library and Georgian Room
<li class="pr-hl"></li>
- *4:00* Coffee

<h3 class="pr-h3">Global scientific cooperation</h3>

- Lecture Theatre
- *4:30* Sir Martyn Poliakoff
- *4:45* George Freeman, MP
- 5:00 Panel discussion, moderated by Ehsan Masood
- Sarah Myers Cornaby
- Prof. Carlos Henrique de Brito Cruz
- HE Vikram Doraiswami
- George Freeman, MP
- Sir Martyn Poliakoff
- *5:50* Closing, by Sir Richard Catlow
------
- Library and Georgian Room
<li class="pr-hl"></li>
- *6:00* Drinks

We are hosting a half-day symposium for scientists, innovators and policymakers to debate the framework within which genius flourishes.

Organising genius

Speakers and panelists

Organising genius: scientific progress and global cooperation

In recent years, advances in cosmology, quantum gravity, and gravitational wave detection have captured the public imagination. We have seen scientific and cinematic attempts to present black holes visually both in the images taken by the Event Horizon Telescope and in the special effects of the Hollywood film Interstellar. Meanwhile new and intriguing questions are being raised. Are we any closer to understanding the nature of dark matter, for example? Are the gravitational waves that have been detected by LIGO and Virgo correctly described by general relativity? How is information preserved in the evaporation of a black hole? Is there a holographic theory of gravity? 

This event is the latest in the series of London Gravity Meetings, which were launched five years ago and take place once every academic term. Their goal is to contribute to Britain’s grand tradition of discovery in the field of gravitation, which stretches from Penrose to Newton, by bringing together researchers who are working on all aspects of gravitation, so they can discuss recent advances, exchange ideas and foster collaborations. 

## Event information
The event takes place on Wednesday 28 February at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Please come at 14:00 for a chat over coffee. We’ll then move to Tyndall’s Parlour for the talks, which will start at 14:30. There will be a coffee break between talks and drinks afterwards in the Old Post Room.

## Programme
- *14:00* Arrival and coffee
- *14:30* Dr. Christoph Kehle: Retiring the third law of black hole thermodynamics
- *15:30* Coffee break
- *16:00* Dr Aron Kovacs: On the Cauchy problem in effective field theories of gravity
- *17:00* Drinks

Researchers working on all aspects of gravity, from gravitational waves to black holes, discuss the latest developments in their field.

London Gravity Meeting

The holographic principle is an approach to describing quantum gravity in the bulk in terms of the corresponding quantum field theory on the boundary. Originally this principle was formulated as AdS/CFT correspondence between string theory in anti-de Sitter spaces (AdS) and 4-dimensional conformal field theories (CFT). Nowadays, it is applied in a much wider context as a powerful toolkit for studying strongly coupled quantum field theories. Recently, quantum information measures such as entanglement entropy, for example, have begun to play an important role in understanding fundamental features of the holographic principle, such as the emergence of spacetime and quantum properties of black holes.

This event is the second meeting of the HoloUK network, which is kindly supported by Fundamental Physics UK. It brings together a UK-based community of experts in holography, gravity, and quantum systems to discuss recent advances in our understanding of conformal field theories and black hole physics. The aim is to provide an opportunity to exchange ideas and collaborate at the forefront of discovery.

## Event information
The event takes place on Monday 26 February at the London Institute for Mathematical Sciences, which is located on the second floor of the Royal Institution. Enjoy a coffee and chat from 13:00, before being seated in Tyndall’s Parlour for talks to start at 14:00. There will be a drinks reception at the end of the day in the Old Post Room.

## Programme

- *13:00* Arrival & Coffee
- *14:00* Dr Gabriel Wong: 3d gravity from an ensemble of approximate CFTs
- *15:00* Coffee & discussion
- *16:00* Dr Matthew Dodelson: Black hole bulk-cone singularities
- *17:00* Drinks reception

Experts in holography, gravity and quantum systems discuss advances in our knowledge of quantum field theory and black hole physics.

HoloUK 2

Simon Norton, who died in February 2019, is best remembered by mathematicians for his remarkable contributions to group theory. A child prodigy, Norton earned a First in mathematics from the University of London while he was still studying at Eton College.

Later, at the University of Cambridge, he would work with John Conway, co-authoring the ATLAS of Finite Groups, which is the definitive catalogue of different types of symmetry. The pair’s paper on monstrous moonshine, which was published in 1979, proposed an unexpected deep connection between the monster group and the j-function. Seemingly far-fetched, the conjecture was proved in 1992 by Richard Borcherds and is recognised as one of the great unifying conjectures in modern mathematics.

In the inaugural Simon Norton Lecture, which marks the fifth anniversary of his death, Prof. Peter Cameron will celebrate some of Norton's achievements, as well as discuss the properties of “Norton algebras” which Robert Griess used to prove the existence of the monster group in 1980. We are grateful to Simon’s family for their generous support of this lecture.

## Event information

The event takes place on Monday 12 February at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. To book a place, email smc@lims.ac.uk.

## Programme

- *17:30* Arrival and drinks, Old Post Room
- *18:00* Prof. Yang-Hui He, Introduction, Tyndall seminar room
- *18:10* Prof. Peter Cameron, “Simon Norton, Norton algebras, and spherical 3-designs”
- *19:10* Drinks and canapes, Old Post room

In the inaugural Simon Norton Lecture, Prof. Peter Cameron celebrates the mathematician's achievements and talks about Norton algebras.

A monstrous talent

The London Institute for Mathematical Sciences was founded in 2011 and incorporated on 10 February. This marked the birth of Britain’s only non-university research centre for physics and mathematics. The date is also the feast day of St Scholastica, who lived in Italy in the 6th century AD and is the patron saint of learning and Benedictine nuns.

One of the few surviving stories about St Scholastica is that one night, after her brother St Benedict had dined with her, he rose to leave. She prayed to God to make it rain, so he would feel forced to stay longer and they might continue their conversation. So it came to pass. St Scholastica is therefore indelibly associated with evenings filled with good conversations, which continue into the night.

To celebrate our founding, and our belief in the importance of community, we hold an annual St Scholastica’s Feast in our rooms at the Royal Institution. This formal dinner is for our scientists and staff, as well as a handful of distinguished guests from the worlds of science, business and the arts. It takes place on a Friday near the day of our incorporation.

## Event information
The dinner is at 7 o’clock on Friday, 26 January, 2024. Dress is black tie. It is held in our rooms on the second floor of the Royal Institution. We meet in the Old Post Room, then head to dinner in the Faraday Room, dessert in the Bragg Room, and drinks in the Old Post Room.

The application of data-greedy machine-learning algorithms relies on access to large datasets. In the financial sector, data is often limited owing to privacy constraints and the complexity of financial time series. Models for synthetic time series generation have arisen as a promising approach to mitigate the issue of data availability while respecting privacy considerations. Moreover, these models may provide novel solutions for risk management and deal with imbalance problems in credit default prediction.

This workshop, which focuses on the latest advances in high-fidelity generative models for financial time series, will bring together academics and industry practitioners. It will feature presentations on state-of-the-art generative models, which are designed to tackle practical challenges in financial time series. There will also be a hands-on tutorial on how to use the python pipeline to empirical financial data. Finally, a panel discussion will focus on applications where synthetic time series is already starting to transform the finance sector and the future implications of generative AI. 

## Registration

<button type="button">[Register](https://forms.gle/5XYAL1aAwfUgTHbq8)</button>

## Event information

The event takes place on Wednesday 24 January at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Please come at 09:00 for coffee beforehand. We will move to Tyndall’s Parlour for the talks, tutorial and panel discussion, which will start at 09:30 and last until 13.30 or thereabouts.

## Funding acknowledgement

The workshop is supported by the EPSRC under the program grant EP/S026347/1 (DataSig) and the Ecosystem Leadership Award under the EPSRC Grant OobfJ22100020 and The Alan Turing Institute.

## Programme

This workshop will focus on the latest advances in synthetic time series generation and the promise they hold for financial modelling.

Arrivals

Opening remark

Simulation of scenarios is needed to estimate the loss distributions for dynamic portfolios; we use GANs and exploit the joint elicitability property of VaR and ES. We prove a universal approximation theorem and show that the training of GANs may be formulated as a max-min game.

Tail-GAN: Learning to Simulate Tail Risk Scenarios

Choosing and training (generative) models are only parts of an end-to-end machine learning pipeline. Other relevant considerations include deployment at scale, cost and modelling improvements through ensembling techniques. We explore tentative answers to some of these points.

Practical Considerations on Synthetic Data Generation in Quantitative Finance

Coffee break

In the ICAIF Regime-Switching Financial Time Series Generation competition, we use a parameter estimation framework for SDE-based synthetic data generation, leveraging the Signature Wasserstein-1 metric. Implementation and model sensitivity to parameter tuning are also discussed.

Synthetic Data Generation from Stochastic Differential Equation Models Calibrated with Signature Wasserstein 1 Metric

We present a pipeline tailored for financial time series generation encompassing preprocessing and training generative models. Model are evaluated with metrics that ensure accuracy & reliability in financial contexts. Finally, we showcase its application in real-world scenarios.

Advanced Framework for Generating Financial Time Series Data

Panel chair: Lingyi Yang (Oxford); Panellists: Terry Lyons (Oxford), Rama Cont (Oxford), Jian Chen (Point 72), Francois Buet-Golfouse (JPMorgan Case), Mikhail Burtsev (London Institute)

Panel discussion

Ending remarks

Light lunch

Synthetic data in finance

Speakers and panel members

Advancing finance with synthetic time series generation

In recent years, advances in cosmology, quantum gravity, and gravitational wave detection have captured the public imagination. We have seen scientific and cinematic attempts to present black holes visually both in the images taken by the Event Horizon Telescope and in the special effects of the Hollywood film Interstellar. Meanwhile new and intriguing questions are being raised. Are we any closer to understanding the nature of dark matter, for example? Are the gravitational waves that have been detected by LIGO and Virgo correctly described by general relativity? How is information preserved in the evaporation of a black hole? Is there a holographic theory of gravity? 

This event is the latest in the series of London Gravity Meetings, which were launched five years ago and take place once every academic term. Their goal is to contribute to Britain’s grand tradition of discovery in the field of gravitation, which stretches from Penrose to Newton, by bringing together researchers who are working on all aspects of gravitation, so they can discuss recent advances, exchange ideas and foster collaborations. 

## Event information
The event takes place on Thursday 16 November at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Please come at 14:00 for a chat over coffee. We’ll then move to Tyndall’s Parlour for the talks, which will start at 14:30. There will be a coffee break between talks and drinks afterwards in the Old Post Room.

## Programme
- *14:00* Arrival and coffee
- *14:30* Prof. Ruth Gregory: Thermodynamics of acceleration
- *15:30* Coffee break
- *16:00* Dr Stephen Green: Orthogonality of Kerr quasinormal modes
- *17:00* Drinks

Researchers working on all aspects of gravity, from gravitational waves to black holes, discuss recent developments in the field.

Euclid’s Elements has been translated more often than any book except the Bible. Within it, the author discusses harmonic relations with reference to the golden ratio, ideas that have influenced everything from Renaissance architecture to music and the making of the Stradivari violin. Dividing a string, as happens whenever a violin is played, creates an expression of harmonic ratios which are found in the design of the violin and throughout nature, just as they are in music. When you listen to a violin, you are listening to mathematics. 

In this event, these Euclidean ratios will be illustrated in various ways. Robert Brewer Young, a master luthier who has studied the philosophy of logic and mathematics, will explain their origins and sketch them with chalk and compass on a blackboard. The same proportions will be simultaneously demonstrated in musical accompaniment by the violinist Elena Urioste, and London Institute Fellow Prof Yang-Hui He, who is an amateur violinist. Both will perform on Stradivari violins. 

## Event information
This event takes place on Wednesday 1 November at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Please come for a glass of wine at 18:30, before being seated in Tyndall’s Parlour for the start of the event at 19:00. There will be drinks afterwards in the Old Post Room.

The luthier Robert Brewer Young explains the geometry of the violin, with musical accompaniment on two violins made by Stradivari himself.

Listening to maths

Speaker and Performers

The holographic principle is an approach to describing quantum gravity in the bulk in terms of the corresponding quantum field theory on the boundary. Originally this principle was formulated as AdS/CFT correspondence between string theory in anti-de Sitter spaces (AdS) and 4-dimensional conformal field theories (CFT). Nowadays, it is applied in a much wider context as a powerful toolkit for studying strongly coupled quantum field theories. Recently, many important characteristics of quantum systems were added to the holographic dictionary. In particular, the entanglement and computational complexity were successfully understood via the properties of geometric objects in the bulk space.

This event is the first meeting of the HoloUK network, which is kindly supported by Fundamental Physics UK. It brings together a UK-based community of experts in holography, gravity, and quantum systems to discuss recent advances in our understanding of conformal field theories and holographic complexity. The aim is to provide an opportunity to exchange ideas and collaborate at the forefront of discovery.

## Event information
The event takes place on Wednesday 27 Sep at the London Institute for Mathematical Sciences, which is located on the second floor of the Royal Institution. Enjoy a coffee and chat from 13:00, before being seated in Tyndall’s Parlour for talks to start at 14:00. There will be a drinks reception at the end of the day in the Old Post Room.

## Programme

- *13:00* Arrival & Coffee
- *14:00* Dr Kuo-Wei Huang: From d=2 Virasoro blocks to something analogous in d=4
- *15:00* Coffee & discussion
- *16:00* Prof. Michal Heller: Holographic complexity beyond proposals and beyond anti-de Sitter 
- *17:00* Drinks reception

Experts in holography, gravity and quantum systems discuss advances in our knowledge of conformal field theories and holographic complexity.

Launching HoloUK

Conjectures can inspire new branches of pure mathematics and theoretical physics. They usually come from spotting patterns and applying instinct. Recently, there has been a surge of interest in using automated pattern detection to help humans form conjectures. Because in mathematics there are no coincidences, mathematical data is immune from the false positives and false negatives that plague physical measurement.

In this two-day workshop, the London Institute brings together physicists and mathematics to explore how AI can speed up theoretical research. The topics addressed range from geometry to string theory to representation theory.

This is the third in the series of annual DANGER workshops (Data, Numbers, Geometry and Representation theory). The series was created by Alexander Kasprzyk from Nottingham University, Thomas Oliver from Westminster University, and Yang-Hui He from the London Institute. This workshop is the first one in the series that is in-person.

## Event information
This workshop takes place on Thursday 24 and Friday 25 August at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution in Mayfair. To register to attend this workshop, please visit the [conference website](https://sites.google.com/view/danger3workshop/home) or contact at@lims.ac.uk.

Thursday 24 August

We consider the problem of determining whether a toric variety is a Q-Fano variety—Fano varieties that have mild singularities called terminal singularities. For 8-dimensional Fano toric varieties X of Picard rank 2, a feedforward neural network predicts whether or not X has terminal singularities. We use the network to give the first sketch of the landscape of Q-Fano varieties in 8 dimensions. We formulate and prove a new global, combinatorial criterion for a toric variety of Picard rank 2 to have terminal singularities. This shows that machine learning is a powerful tool in developing mathematical conjectures and accelerating discovery.

Machine learning detects terminal singularities

We present a novel rank classification method based on deep convolutional neural networks. We compare our method with eight simple neural network models of the Mestre-Nagao sums, which are widely used heuristics for estimating the rank of elliptic curves.
We evaluate our method on two datasets: the LMFDB and a custom dataset consisting of elliptic curves with trivial torsion, conductor up to 10^30, and rank up to 10. We find that the CNNs outperform the Mestre-Nagao sums on the LMFDB dataset. On the custom dataset, the performance of the CNNs and the Mestre-Nagao sums is comparable. This is joint work with Domagoj Vlah. 


Ranks of elliptic curves and deep neural networks

Calabi-Yau links are constructed for all 7555 weighted projective spaces with Calabi-Yau 3-fold hypersurfaces. Topological properties such as the Crowley-Nordström invariants and Sasakian Hodge numbers are computed, leading to new invariant values and some conjectures on their construction. Machine learning methods are implemented to predict these invariants, as well as to optimise their computation via Gröbner bases.

Machine learning Sasakian and G2 topology on contact Calabi-Yau 7-manifolds

Convolutional neural networks without activation parametrise semialgebraic sets of real homogeneous polynomials that admit a certain sparse factorization. We investigate how the geometry of these semialgebraic sets (e.g., their singularities and relative boundary) changes with the network architecture. We explore how these properties affect the optimisation of a loss function for given training data. This talk is based on joint work with Guido Montúfar, Vahid Shahverdi, and Matthew Trager. 

Understanding linear convolutional neural networks via sparse factorisations of real polynomials

Drinks

We'll head out to The Market Tavern  together.

Informal dinner

Friday 25 August

Conjectures hold a special status in mathematics. Good conjectures epitomise milestones in mathematical discovery. Crafting conjectures can be understood as a problem in pattern recognition, for which machine learning is tailor-made. I propose a framework that allows a principled study of a space of mathematical conjectures. Using this and exploiting domain knowledge and machine learning, we generate a number of conjectures in number theory and group theory. We give evidence in support of some resulting conjectures and present a new theorem. This talk concludes with posing some general questions about the pipeline.

Mathematical conjecture generation and machine intelligence

We explore the intersection of data science and elliptic curves of prime conductors. After a quick introduction to elliptic curves, the celebrated Birch and Swinnerton-Dyer conjecture is introduced. The original insight is discussed, concerning the traces of Frobenius and what they know about data attached to elliptic curves. Along the way, we present experiments done on the largest known dataset of such curves: the Bennett-Gherga-Retchnizer dataset. Posing questions based on observations, we discuss the tension between the data and the conjecture stipulating that the average rank should be 1/2, and the bias between the distribution of 2-torsion coefficients and the rank distribution. To conclude, we discuss how machine learning models can predict the rank based on the traces of Frobenius.

Galois groups in enumerative geometry

2 PM, 23 Jul 2025