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 info

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 info

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 info

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

Living with AI

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 info
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 info
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 info
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 info

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 info
The dinner is at 7 o’clock on Friday, 26 January, 2024. Dress is black tie. It is held in the London Institute’s rooms, on the second floor of the Royal Institution. It starts with drinks in the Tyndall room, followed by dinner in the Bragg room, then dessert in the Faraday 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

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 info

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 info
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 info
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 info
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 info
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.

Data-driven insights into the rank of elliptic curves of prime conductors

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi-Yau four-folds.

New Calabi-Yau manifolds from genetic algorithms

Symmetries are the cornerstones of modern theoretical physics, as they imply fundamental conservation laws. Given the recent boom in AI and its successful application to high-dimensional large datasets, we approach the discovery and identification of symmetries as a machine learning task. We develop a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labelled dataset. We use fully connected neural network architectures to model the transformations and the corresponding generators. Our proposed loss functions ensure that the applied transformations are symmetries and that the corresponding set of generators is orthonormal and forms a closed algebra. One variant of our method is designed to discover symmetries in a reduced-dimensionality latent space, while another is capable of obtaining the generators in the canonical sparse representation. We give examples illustrating the discovery of the symmetries behind the orthogonal, unitary, Lorentz, and exceptional Lie groups.

Deep learning symmetries in physics from first principles

Informal panel discussion with Thomas Fink, Yang-Hui He, Alexander Kaspryzyk, Challenger Mishra, and Thomas Oliver.  

Join us for our weekly LIMS drinks in Old Post Room.

 Drinks

Danger

DANGER: Data, numbers and geometry

Mathematical theories have informed our understanding of financial markets since the stochastic modelling of Black, Scholes and Merton in the 1960s—and also helped to shape them. Generative AI, which can identify and mimic patterns in data, is already gaining traction in risk assessment, portfolio optimisation and trading strategies. Will AI challenge the belief that global financial markets are fundamentally unpredictable? Are we comfortable with the idea of it steering those markets? Can we extend the principles behind large language models to create large market models, which will predict market trends better than any human?

In this half-day event, the London Institute brings together experts in finance, mathematical physics, machine learning and large language models to discuss the potential and pitfalls of AI-driven markets. There will be six talks, then a panel discussion, followed by a drinks reception.

## Event info

This event will take place on Tuesday 25 July at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Coffee will be served from 13:30. Talks take place in Tyndall’s Parlour from 14:00. Afterwards, from 18:00, there will be drinks and further discussion in the Old Post Room. To register to attend, email akm@lims.ac.uk.

## Programme
- *13:30* Arrival
- *14:00* Nikolaos Kaplis: Geometry of language
- *14:30* Hao Ni: Generating high-fidelity synthetic time series via the path characteristic function
- *15:00* Guido Caldarelli: The physics of financial networks
- *15:30* Coffee break
- *16:00* Mikhail Burtsev: Beyond attention: breaking the limits of transformer context length with recurrent memory
- *16:30* Challenger Mishra: Mathematical conjecture generation and machine intelligence
- *17:00* Jon Hammant: Machine learning for finance in the cloud 
- *17:30* Panel discussion: Challenger Mishra, Yang-Hui He, Bharat Kasturi, Mikhail Burtsev, Nikolaos Kaplis
- *18:00* Drinks reception

The London Institute brings together experts from the worlds of finance and AI to discuss the potential and the pitfalls of AI-driven markets.

## Arabesque AI

This workshop is funded by Arabesque AI. Arabesque AI was born out of a desire to tackle key challenges faced by the asset management industry. The company operates through its cloud-based technology platform, Aether. Aether leverages AI and machine learning to deliver hyper-customisable investment strategies at scale.


We discuss initial results on characterising the geometry of sentence embeddings, a task in natural language processing. This is based on the work of Daniel Platt, Moritz Platt, Rodrigo Barbosa, Christopher Evans, et al: "The Effect of Active and Passive Voice on Sentence Embeddings”. 

Geometry of language

Synthetic time series generation is gaining attention, particularly in finance where sensitive time series data is abundant. We present PCF-GAN, a state-of-the-art generative adversarial network inspired by the path characteristic function. This work is joint with Hang Lou and Siran Li.

Generating high-fidelity synthetic time series

As the total value of global financial markets outgrew that of the real economy, a web of interactions to model systemic risk was built, demanding new theoretical approaches. Through statistical physics, we develop new ways to study the network’s structure, dynamics, and stability.

The physics of financial networks

The complexity of transformer-solvable problems scales quadratically with input size. Investigating recurrent memory augmentation of pre-trained transformer models, we find scope for storage of up to 2 million tokens, exploring further with language modelling task experiments. 

Breaking the limits of transformer context length

Good mathematical conjectures epitomise milestones in discovery, and crafting them is a pattern-recogntion task, for which machine learning is tailor-made. We present a method for machine-learning a space of conjectures, and generate new ones in number theory and group theory.

Mathematical conjecture generation

How can we use machine learning for finance in the Cloud? We discuss how high performance computing provides a quicker cycle time, increasing the progression cadence of technology. We also discuss increases in heterogeneous computing that are helping shape the future of technology.

Machine learning for finance in the cloud

Challenger Mishra, Yang-Hui He, Bharat Kasturi, Mikhail Burtsev, Nikolaos Kaplis

Drinks reception

Converging futures

Speakers & panelists

Converging futures: AI and finance

Number theory is accelerating. One of the key modern branches increasing the pace of discovery is additive combinatorics, and many of its fundamental questions await answers. The structures of sets under a large number of additions remain a mystery. The links between the structure-randomness dichotomy and ergodicity in dynamical systems are only beginning to be explored. Generalisations to the recent discovery of murmurations hold the potential to unlock new results. But while we embrace modern number theory, many classical questions are yet to be addressed. Inverse problems, the tightness of inequalities, present open challenges. This millennia-old subject continues to unearth new results and refine proofs established years ago.
This event assembles number theorists to discuss modern advances, as well as classical unanswered questions. Topics include links between the structure-randomness dichotomy and ergodicity in dynamical systems, and potential breakthroughs inspired by murmurations’ recent discovery. This event is a platform for experts to exchange ideas and collaborate at the forefront of discovery.

## Event info
This event is on Monday 3 July at the London Institute for Mathematical Sciences, located on the second floor of the Royal Institution. Grab a coffee and say hi from 10:30 am, before being seated in Tyndall’s Parlour for talks to start at 11 am. There will be a drinks reception at the end of the day in the Old Post Room. To register to attend, email akm@lims.ac.uk.

## Programme

- *10:30 - 11:00* _Arrival_
- *11:00 - 11:40* Andrew Granville
- *12:00 - 12:40* Thomas Oliver
- *13:00 - 13:40* _Lunch break_
- *14:00 - 14:40* Peter Van Hintum
- *15:00 - 15:40* Marius Tiba
- *16:00 - 16:30* _Coffee break_
- *16:30 - 17:10* Ben Krause
- *17:30 - 18:10* Oleksiy Klurman
- *18:30* Drinks reception 


Number theorists gather at the London Institute to discuss cutting-edge research and present their latest work in this branch of mathematics.

Connected counting

Speakers & organisers

Connected counting: illuminating relationships in number theory

Hiroshima. Chernobyl. Three Mile Island. War, accidents and weapons of mass destruction have riddled the word “nuclear” with controversy. The echoes of the anti-nuclear movement that exploded in the 70s still linger, making it easy for people to rebut this technology. But the urgent need to address climate change demands a fresh perspective from society. There is an often-overlooked consensus among experts that nuclear energy is a viable solution. Its minimal carbon footprint, and steady advances in making the technology safe, make it a good alternative to fossil fuels, opening up the potential to secure a sustainable future.

In this event, Oscar-winning director Oliver Stone presents the UK premiere of his documentary film Nuclear Now. After the screening, physicist Dr Thomas Fink and historian Dr Theo Zenou will interview Mr Stone on stage, alongside nuclear scientist Dr Shannon Bragg-Sitton. They will talk about the director’s creative process and how film can shape public opinion.

## Event info

This event is at 7pm on Wednesday, 14 June at the Royal Institution. The screening and 30-minute interview take place in the Lecture Theatre. Members of the public can purchase tickets from the Royal Institution [here](https://www.eventbrite.co.uk/e/nuclear-now-uk-premiere-in-conversation-with-oliver-stone-tickets-634902950817?utm-campaign=social&utm-content=attendeeshare&utm-medium=discovery&utm-term=listing&utm-source=cp&aff=escb). London Institute guests are invited to join the speakers afterwards for private drinks and canapés in our rooms on the second floor.

The UK premiere of Oliver Stone’s new film, _Nuclear Now_, takes place in the Lecture Theatre, followed by an interview with the director.

Nuclear Now

At the dawn of the 21st century, a problem was posed in functional analysis: are the solutions to Navier–Stokes equations smooth, or do they break down after an amount of time? In the 89 years since this problem was first introduced, all attempts to solve the Navier–Stokes equations using functional analysis have failed. 

This workshop focuses on this open problem of whether solutions to Navier–Stokes equations are smooth, bringing together experts from different fields to discuss potential solutions. The highlight of this workshop will be a lecture by Prof. Eva Miranda, who was awarded the 2023 Hardy Lectureship for her outstanding contributions to mathematics and society. She will present her recent work on how hydrodynamics is capable of performing computations. Other talks will cover advances in neuromorphic computing, machine learning, and fluid computation. The workshop will also include poetry and a Bismuth sculpture related to the theme of ‘fluid computing’.

This event is funded by the London Mathematical Society Research Grant, Scheme 6. Please register your interest in attending this event [here](https://docs.google.com/forms/d/e/1FAIpQLSelRY7bYUt_nwNBC4TkFRpR0eFgMF8gWY-qpTa4S46FKvhFXg/viewform).

## Programme

- *11:30* Arrival
- *12:00* Eva Miranda, LMS Hardy lecture: From Alan Turing to contact geometry: towards a “fluid computer”
- *13:00* Lunch break
- *14:00* Giulia Marcucci: Learning at the edge of chaos: neuromorphic computing by nonlinear waves 
- *14:30* Juan Sebastian Totero Gongora: Complexity-driven neuromorphic photonics
- *15:00* Michael Negus: Machine learning methods to accelerate CFD 
- *15:30* Yang-Hui He: Machine learning mathematical structures
- *16:00* Abdullah Awad: It from bit, Song-Shi art
- *16:30* Concluding remarks and departure


The London Institute hosts a workshop on the Navier-Stokes millennium-prize problem and its connection to fluid computing and machine learning.

Towards fluid computing

Speakers

Navier-Stokes regularity, fluid computing & machine learning workshop

_tex: 
By the Euclidean algorithm, any rational number $a/q$ can be uniquely represented as a regular continued fraction. In the theory of continued fractions, in 1971 Zaremba famously made the following conjecture: there is an absolute constant $k$ such that for any positive integer $q$, there exists $a$, which is coprime to $q$, such that in the continued fraction expansion of $a/q$, all partial quotients are less than or equal to $k$. Zaremba himself conjectured that the value of this absolute constant $k$ equals 5. Two decades later, Hensley conjectured that for a large, prime denominator, a bound equal to 2 is sufficient. Ever since, further tightening this bound has been an ongoing problem in the number theory community.

In this talk and drinks, Prof. Ilya Shkredov presents a [marked improvement](https://lims.ac.uk/paper/on-korobov-bound-concerning-zarembas-conjecture/) to the well-known Korobov bound on this conjecture. In 1963, using exponential sums, Korobov proved that for any prime $q$, there exists a coprime numerator such that the partial quotients never exceed log($q$). Shkredov presents a proof that, for any sufficiently large prime $q$, the partial quotients are bounded by $O$(log $q$/ log log $q$). 


## Event info

This event is at 5pm on Wednesday, 29 March, on the second floor of the Royal Institution. Grab a drink and say hi before the talk starts at 5:15 in Tyndall’s Parlour. After the talk, everyone meets for discussion over drinks in the Old Post Room. To attend, email at@lims.ac.uk.


Prof. Ilya Shkredov discusses Zaremba’s elegant 1971 conjecture in the theory of continued fractions, and explores the bounds relating to it.

 Bounding Zaremba

A new bound concerning Zaremba’s conjecture

Special geometries play a key role in string theory and gravitational theories in general. The 6 compact dimensions of the 10-dimensional (super) string are often taken to be a Calabi-Yau manifold. Such compactification is important as it allows us to realise our 4-dimensional spacetime within the theory, and leaves some of the original supersymmetry unbroken. When fluxes are included to describe the analogies of EMFs within the compact space, the supersymmetry condition is modified and defines a new class of manifolds that generalise the notion of a Calabi-Yau space.

In this LonTI series, Prof. Daniel Waldram introduces some formalism and tools useful for characterising such geometries. An outline of purely geometrical backgrounds is given, using general notions of a G-structure and special holonomy. Backgrounds that have non-trivial fluxes are then described. The narrative of the series is guided by applications to string phenomenology and the AdS/CFT correspondence.

## Event info
---
This is a three-part lecture series from LonTI. Lectures are on Mondays at 10:30 am in Tyndall’s Parlour 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/).

Prof. Daniel Waldram introduces the formalism and tools for characterising geometries in gravitational theories, such as Calabi-Yau manifolds.

Geometry and fluxes

Geometry is so much more than measurements of shapes. It is the heart of physics and mathematics. Inscribed above the door of Plato’s Academy in Athens was “Let no one ignorant of geometry enter!”

In this renowned Friday Evening Discourse, Prof. Yang-Hui He takes us through time and space, from geometry’s classical beginnings to the Renaissance and the Enlightenment, to our current understanding of spacetime itself. In the 19th century Gauss and Riemann considered the relaxation of Euclid’s axioms, resulting in the explosion of post-Euclidean geometry, paving the way for Einstein’s theory of relativity and the birth of modern physics. Singularity solutions of spacetime geometry gives us the discovery of black holes. String theory, attempting to unify quantum mechanics with relativity, delivers a brave new world of quantum geometry. Anyone at risk of ignorance can rest assured, they will soon be fit to enter through Plato’s doors.

This event takes place in the lecture theatre at the Royal Institution. London Institute guests are invited to join the speaker after for drinks and canapés in our private rooms.

<a href="https://youtu.be/z8jdndd-x7w">Watch the event</a>

At the Royal Institution's Friday Evening Discourse, Prof. Yang-Hui He recounts the creation of modern physics at the hands of geometry.

A revolution in geometry

How geometry created modern physics

Imagination is crucial for creativity and success. In business, where it underpins competitive differentiation and profits, the timescales for reinvention are contracting, while many resources, such as capital, are increasingly scarce. This trend is also reflected in scientific discovery, where the stakes are higher than ever.

In this event, our Trustee Martin Reeves debunks some of the romantic myths surrounding imagination—the idea that it is a solitary enterprise, relying on a eureka moment—and analyses what imagination is and how we may take a systematic approach to harness it, to become powerful “imagination machines”. 

Illustrating his argument, he highlights key examples of businesses that have cultivated imagination, explaining how they secure a competitive advantage. He also delves into the story of the Royal Institution, where transformative discoveries have been made, in this historic imagination powerhouse.

This event takes place in the lecture theatre at the Royal Institution. 

<a href="https://www.youtube.com/embed/Z-ChwkxFMg4">Watch the event</a>

Our Trustee Martin Reeves explores imagination at its core, rethinking previous romantic notions, asking if we can harness it systematically.

The imagination machine

The imagination machine: revisiting creativity

Design is at the interface of function, beauty and order. Throughout history, it has had many points of contact with mathematics, which is the language for describing patterns. These have ranged from the theory of proportion, to symmetry and architecture, to [grid systems and modularity](https://lims.ac.uk/perspectives/designing-web-design/), to simplicity and industrial design, to self-similar fractals and aperiodic tilings, to the dependence of behaviour on the layout of space, to the design of intelligent organisations.

At this meetup, a mix of designers, artists, physicists and mathematicians talk about the present and future of design and mathematics. Four five-minute talks are interspersed throughout the evening to stimulate informal conversation. The purpose of the event is to explore ideas and build connections.

The event takes place at the London Institute for Mathematical Sciences, on the second floor of the Royal Institution in Mayfair. Drinks are provided. If you would like to attend, please email at@lims.ac.uk.

Designers and theorists talk about the intersection of design and mathematics in visualisation, architecture, digital design and industry.

Design meets mathematics

Design meets maths

In particle physics, models of interactions between an endless variety of possible particles in any number of spatial dimensions can be constructed, with each situation described by its own quantum field theory. Conformal field theories are a special type of quantum field theory, in that they are scale invariant. They have key applications and provide insights in condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory.

In this LonTI series, Dr Andreas Stergiou motivates the study of conformal field theory, contextualising the “old” way of studying it as endpoints of renormalisation group flows. The conformal bootstrap method is introduced, which allows for computing formulas that encode relationships between particles described by the conformal field theory. The method exploits conformal symmetries to solve for critical exponents, constraining the correlation functions. Both analytical and numerical implementations are discussed.

## Event info
---
This is a four-part lecture series from LonTI. Lectures are on Mondays at 10:30 am in Tyndall’s Parlour 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 Andreas Stergiou delivers an introduction to the conformal bootstrap method which is used to constrain and solve conformal field theories.

Conformal bootstrap

Conformal field theories and conformal bootstrap

We are delighted to host the London Algebra Colloquium for their final seminar of 2022. For over seventy years, algebraists from across London and beyond have met regularly at this colloquium to discuss research and ideas in this sphere of pure mathematics. 

In this talk, Frank Neumann presents the Hochschild cohomology of differential graded categories. This is a homology theory that admits the characteristic homomorphism, a natural map to the graded centre of its derived category. When interpreted as an edge homomorphism in a spectral sequence, this map can be studied systematically, enabling its application to examples in algebra, geometry, topology and physics. The framework can be extended beyond DG-categories to A-infinity-categories.

As a festive treat and at the invitation of London Institute Fellow Prof. Yang-Hui He, a regular attendee of LAC, this meeting will be held in Tyndall’s Parlour at the London Institute, on the second floor of the Royal Institution.

Previous talks given by the London Algebra Colloquium and further details about the society can be found on their [website](https://londonalgebra.wordpress.com). 



The London Institute hosts guest speaker Dr Frank Neumann and the London Algebra Colloquium for their final seminar of 2022.

Cohomology and sequences 

Hochschild cohomology and spectral sequences

In February, Prof. Maryna Viazovska won the Fields Medal for her work on sphere-packing formulas. She is only the second woman to receive the award, which is the most acclaimed in mathematics, in its 86-year history.

In this event, Prof. Viazovska will speak with London Institute Fellow and BMUCO advisor Prof. Yang-Hui He about her proof that the E8 lattice provides the densest packing of identical spheres in eight dimensions, her contributions to related external problems, and her efforts with interpolation problems in Fourier analysis. She will also touch on her efforts with geometric optimisation problems and her championing of academic freedom in light of the war in Ukraine, where she was born and many of her relatives remain. 

For the conversation and Q & A session, which Prof. Viazovska will attend by live-stream, participants are invited to join digitally or in person in the London Institute’s Tyndall’s Parlour on the second floor of the Royal Institution. In either case they must complete this [form](https://docs.google.com/forms/d/e/1FAIpQLSeA2KpKOo1pRwwtDE3iQsPMV6sZGgFgttd7BPqT6bx90atymA/viewform). This event is organised by the science NGO, [BMUCO](https://www.bmuco.org), in partnership with the London Institute.

The Ukrainian mathematician Prof. Maryna Viazovska, who won this year’s Fields Medal, joins us for a virtual interview and discussion.

Spheres of influence

In conversation with Prof. Maryna Viazovska

In terms of quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. He will present new exact solutions of Navier-Stokes and Euler equations and investigate in detail the loop equation suggested in the early 1990s. The loop equation plays the same role in the theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation.

In this limit, he finds a fixed point of the loop equation he calls Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalising Kelvin's theorem. The Clebsch field of this solution has a nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.

## Programme

Mon 5 Dec 2022

- *10:00* Lecture 1
- *12:00* Lecture 2 

- Wed 7 Dec 2022

- *10:00* Lecture 3
- *12:00* Lecture 4
- *1:00* Q&A

Over the course of four lectures, Prof. Alexander Migdal will present 35 years of research concerning his new approach to strong turbulence.

Strong turbulence

Speaker

Quantum theory of classical turbulence

### Theoretical life

The London Institute’s “Theoretical life” series tackles basic science questions about life itself, of which biology is just one instance. It is done in conjunction with bit.bio, with whom we work to decode the operating system of life. Topics range from the structure of genetic computation, to the thermodynamics of emergent self-replication, to cell programming and out-of-equilibrium switching, to a mathematical theory of programmed ageing, to learning and high-dimensional inference.

After each talk, there is a high-level panel discussion, followed by drinks and discussion on site.

### Evolution and Occam

Why does evolution favour symmetric structures when they only represent a minute subset of all possible forms? Just as monkeys randomly typing into a computer language will preferentially produce outputs that can be generated by shorter algorithms, so the coding theorem from algorithmic information theory predicts that random mutations, when decoded by the process of development, preferentially produce phenotypes with shorter algorithmic descriptions. Since symmetric structures need less information to encode, they are much more likely to appear as potential variation. Combined with an arrival-of-the-frequent population genetics mechanism, this evolutionary Occam’s razor predicts a much higher prevalence of low-complexity (high-symmetry) phenotypes than follows from natural selection alone. It explains striking patterns observed in protein complexes, RNA secondary structures, a gene regulatory network and Richard Dawkins’ biomorphs model of development.

### Finding transcription factors for cellular reprogramming

Cellular reprogramming is erasing a specific cellular identity from a population of cells and establishing another, such as turning skin cells to muscle cells. Transcription factor overexpression is the most efficient and robust method to exchange cellular identity.

Transcription factors are proteins that bind to specific gene regulatory elements in the genome and control gene expression. I will talk about experimental and predictive methods to identify transcription factors that can install specific cellular identities, for generation of human cell types for biomedical research and cell-based therapies.



The algorithmic nature of evolution implies an exponential bias towards simpler phenotypes, explaining an observed preference for symmetry.

Evolution and Occam

Does evolution have an inbuilt Occam’s Razor?

The pioneering work of Bekenstein and Hawking in the 1970s showed that black holes have thermodynamic properties like temperature and entropy, just like the air in a room. This leads to the question: can we account for the thermodynamic entropy of a black hole as a statistical entropy of an ensemble of microscopic states? One of the big successes of string theory is to answer this question in the affirmative for a large class of black holes. The aim of these lectures is to introduce these ideas to early researchers in high energy physics and mathematical physics.

This four-part series by ​​Prof. Sameer Murthy provides an orientation and the basic ideas and techniques used in black hole microstate counting. The lectures will cover the following topics in succession: the thermodynamics of black holes; the entropy of black holes in string theory; a special class of black holes called BPS black holes; and an illustration of some actual calculations of entropy. They take place over four weeks in Tyndall’s Parlour at the London Institute for Mathematical Sciences, on the second floor of the Royal Institution.

## Black hole mechanics

Lecture 1 reviews the basics of black hole thermodynamics, including the four laws of black hole mechanics, analogous to the usual four laws.

## Black hole entropy

Lecture 2 describes the ideas underlying the counting of the statistical entropy of black holes in string theory, and implications for it.

## BPS black holes

Lecture 3 covers a special class of black holes for which the entropy can be calculated and the corresponding microscopic ensemble of states.

## Don’t drink and derive

Lecture 4 shows how to actually calculate the statistical entropy in practice, in particular for some simple examples of BPS black holes.



Prof. Sameer Murthy talks about the thermodynamic entropy of a black hole and how to formulate it within a statistical physics foundation.

Black hole microstates

Introduction to black hole microstate counting

Quantum field theory is typically not solvable in closed form and represents a complex and accurate description of elementary particle physics. Integrable quantum field theories, while sufficiently simplified, still exhibit a rich range of effects. They normally reside in a 1+1 D world, and are characterised by exact S-matrices which are calculable in closed form from fundamental physical requirements such as unitarity, crossing symmetry and the location of bound states. Thanks to their vast symmetry, such models are an ideal playground for understanding non-perturbative quantisation. The theory of integrable systems also plays a role in recent advances in string theory.

In this LonTI series, Prof. Alessandro Torrielli describes the duality between two integrable systems: the 2D Sine-Gordon model and the 2D Thirring model. He outlines the classical and quantum Sine-Gordon models, in particular the spectrum, S-matrices and underlying quantum symmetry. He then presents the duality with the Thirring model. Prerequisite knowledge is kept to a minimum, and lecture notes with exercises are available online.

## Classical Sine Gordon

Lecture 1 gives an introduction to classical and quantum integrability, and describes the Sine-Gordon model and its soliton solutions.

## Quantum Sine-Gordon

Lecture 2 sets out quantisation of the classical picture, and algebraic aspects of quantum Sine-Gordon, such as relations with spin-chains.

## Sine-Gordon and Thirring duality

Lecture 3 describes the duality between the Sine-Gordon model and the Thirring model and provides a particular test using form factors.

Prof. Alessandro Torrielli talks about integrable quantum field theories and the duality between the 2D Sine-Gordon and 2D Thirring models.

Sine-Gordon/Thirring duality

Sine-Gordon and Thirring model duality

A swirling quasi-particle, a knot of twisting field lines, or a subatomic hurricane? Skyrmions aren’t easy to describe in words, yet that’s the challenge Prof. Nicholas Manton tackles in his new book, Skyrmions: A Theory of Nuclei.

In this event, Prof. Manton takes us through the  60-year history of Skyrme theory, which was first proposed in 1962 by British physicist Tony Skyrme as a way of modelling subatomic entities like protons and neutrons using convoluted twists in the quantum pion field. It was later superseded by quantum chromo-dynamics and its quarks, but, more recently, Skyrmions have been revived by researchers working on the structure of atomic nuclei, because quantum chromo-dynamics is extremely hard to solve. ​In his book, Prof. Manton touches on some of the variants of Skyrme theory—which incorporate mesons heavier than pions, and extend the theory to include particles such as kaons—before introducing the Sakai-Sugimoto model, which relates Skyrmions to gauge theory. A different type of Skyrmion describes magnetic fields that can be coaxed into forming vortex-like swirls in some special materials.

Afterwards there will be a discussion. Then we will celebrate the launch of Prof. Manton’s book, which is replete with beautiful colour illustrations.

## Programme

- *14:00* Seminar on skyrmions by Prof. Nicholas Manton
- *15:00* Discussion and book launch

Prof. Nicholas Manton will talk about the 60-year history of Skyrme theory, as he launches his new book on the subject.

Skyrme theory at 60

Good conjectures can inspire new branches of mathematics and theoretical physics. They usually come from spotting patterns and applying instinct. Because in mathematics there are no coincidences, automated pattern detection is immune to the false positives that plague experimental science. Can machines help us make mathematical discoveries and speed up theoretical research? 

There has been a recent surge of interest in using machine intelligence to study various aspects of geometry, topology and representation theory. Theoretical physics has helped drive these developments, through the percolation of ideas and analogies and by providing an effective testbed for learning methods. One pertinent example is machine-driven studies of the algebraic geometry of Calabi—Yau spaces, which appear as extra-dimensions in string theory.

In this one-day symposium, we discuss how to assist and automate new mathematical discovery by exploiting novel computation and deep learning, as well as insights into the theory of learning itself. At 4pm there is a high-level panel discussion for a broader audience on the future of AI-assisted discovery, and the roles of academia versus industry. This is followed by drinks and dinner on site.

## A two-way street

Several of the success stories in AI-assisted mathematics have relied on the deep learning architectures of neural networks and their remarkable ability to act as universal function approximators. Well-known feature attribution techniques recently led to exciting new results in knot theory and representation theory. Even number theory, normally resistant to such approximations, is now proving susceptible to certain forms of learning.

But the reverse is also true: mathematics is shedding light on deep learning. As we develop better theories of learning, profound connections to well-studied areas of mathematics are becoming apparent. The Kolmogorov-Arnold representation theorem seems structurally similar to neural networks, with deep connections to Hilbert’s 13th problem. More recently, researchers established that training in neural networks, when viewed geometrically, is a variant of Ricci flow. Using such links to leverage modern machine learning algorithms will lead to surprising breakthroughs in mathematics and physics, as well as a better understanding of machine learning itself.

## Programme

Old Post Room
- *10:00* Coffee, meet and greet 

- Tyndall Parlour
- *10:20* LIMS, AI and mathematics: Thomas Fink 
- *10:30* Opening remarks: Challenger Mishra

- *10:40* Knot theory and machine learning: Andraz Juhasz
- *11:00* Machine learning with mathematicians: Alex Davies
- *11:20* Teaching theorems to computers: Kevin Buzzard

- Old Post Room
- *11:40* Coffee and discussion

- Tyndall Parlour
- *12:00* Machine-Learning Mathematical Structures: Yang-Hui He
- *12:20* Towards a human-like reasoning system: Mateja Jamnik
- *12:40* Draft, sketch and prove: Wenda Li

- *13:00* Local pub lunch and sandwiches on site in Bragg’s Dining Room

- Tyndall Parlour
- *14:00* Peter Braam 
- *14:20* Syllogistic reasoning and beyond: Tiansi Dong

- Old Post Room
- *14:40* Coffee and discussion

- Bragg’s Dining Room 
- *15:00* Speed dating: AI vs mathematics
 
- *16:00* Break

- Tyndall Parlour
- *16:15* High-level panel discussion with business leaders and journalists

- *17:45* Drinks in the Old Post Room
- *19:00* Dinner in Faraday’s Study


## Talks

To encourage understanding and discussion, talks are 20 minutes long and can have at most three slides. A blackboard is available to explain ideas and results.

## Venue

We will meet in the Tyndall Parlour of the London Institute, which is on the second floor of the Royal Institution.

## Financial support

This symposium is funded by the Henderson Institute at the Boston Consulting Group. For those outside of London, we can cover travel to London and overnight accommodation. Additional sponsorship comes from the Accelerate Program for Scientific Discovery at The Computer Laboratory, Cambridge. 

<a href="https://docs.google.com/forms/d/e/1FAIpQLSc6CFfd723x6gRK2Ec-Q4lMlKjX7h76OisthWhNOEdgB6ZMPg/viewform?usp=sf_link">Register</a>

The London Institute hosts a day symposium on using AI to speed up mathematical discovery, followed by a panel discussion, drinks and dinner.

Stable pairs

5 pm, 31 Jul 2024

Event info

Speakers