Dr Matthew Yu presents a unified categorical framework for gauging continuous and finite symmetries in arbitrary dimensions. It identifies electric and magnetic symmetries of G-gauge theory and captures symmetry breaking by charged matter. Using geometric categories, he extends equivariantisation to continuous groups and constructs a functorial Symmetry Topological Field Theory with boundaries, computing endomorphisms.

[Dr Matthew Yu](https://www.maths.ox.ac.uk/people/matthew.yu) is a postdoc in mathematical physics at Oxford University. He holds a PhD from the Perimeter Institute, where he studied topological phases using higher category theory and cobordism. His research interests encompass homotopical methods in quantum field theory.

Matthew Yu introduces a categorical framework for gauging symmetries and constructs Symmetry Topological Field Theories with boundaries.

Continuous gauging

Geometric categories for continuous gauging

Topological quantum field theories (TQFTs) in 2+1 dimensions describe gapped phases supporting anyons, with a successful formulation in terms of modular tensor categories. However, this framework relies on spacetime orientability. Ippo Orii surveys current efforts to extend TQFTs to non-orientable manifolds—which are closely tied to time-reversal symmetry— and highlights key physical insights guiding their construction. 

[Ippo Orii](https://sites.google.com/g.ecc.u-tokyo.ac.jp/ippoorii/about) is a graduate student in theoretical physics at the Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo. His research focuses on topological quantum field theory, with interests in symmetry, anomalies and non-orientable manifolds.

Ippo Orii surveys topological field theories and discusses the implications of recent progress in extending them to non-orientable manifolds.

Non-orientable TQFTs

 Insights from the crosscap state towards non-orientable TQFTs

In this second talk in the AI for Mathematical Sciences (AIMS) seminar series Dr Adam Zsolt Wagner explores the frontiers of automated mathematical discovery. With an emphasis on his work at Google DeepMind, he discusses tools such AlphaEvolve that aim to bridge the gap between human intuition and the vast space of potential mathematical constructions. By treating mathematical problems as search tasks within “language space” these tools can generate counterexamples, find novel configurations and significantly reduce the manual effort required to explore ideas and test complex hypotheses.

Dr Wagner demonstrates how these AI-driven approaches are being applied to real-world problems in combinatorics and beyond. He discusses the mechanics of FunSearch and AlphaEvolve, shares insights from collaborative work with Javier Gomez-Serrano and Terence Tao and provides a framework for mathematicians to determine which AI search strategies are best suited for their own research.

## Event information
This event, part of our AI for Mathematical Sciences series, takes place at 2 pm on Thursday 2 April at the London Institute for Mathematical Sciences, on the second floor of the Royal Institution. AIMS is sponsored by Nebius. This series is organised by LIMS fellows [Prof. Yang-Hui He](https://lims.ac.uk/yang-hui-he/) and [Dr Evgeny Sobko](https://lims.ac.uk/evgeny-sobko/). To register for the series please fill out the [online form](https://applications.lims.ac.uk/subscribe-to-aims). 

Adam Zsolt Wagner discusses how AI-driven search in language space accelerates mathematical discovery and progress on long-standing problems.

New maths with AI

Speaker

Searching in language space to find constructions in mathematics

A century after the Austrian theoretical physicist Erwin Schrödinger published his famous quantum wave equation, the exponential complexity of quantum many-body systems remains one of the central challenges of physics. In this lecture, Prof. Frank Verstraete presents how entanglement theory breaks this impasse through the development of tensor networks. He explains how these methods provide a powerful framework for efficiently representing and simulating strongly correlated matter, turning seemingly intractable problems into structured, computable ones.

Prof. Verstraete also demonstrates how such tensor networks reveal the organising role of quantum entanglement and make it possible to uncover emergent and generalised symmetries. These symmetries, in turn, define and classify quantum phases of matter. By combining conceptual insight with computational power, he argues that tensor networks now form an essential toolkit for understanding complex quantum systems.

## Event information
This event takes place at 4:30pm on Wednesday 4 March in the Faraday seminar room of the London Institute for Mathematical Sciences, on the second floor of the Royal Institution. Afterwards, there are drinks with the speaker in the Old Post room. LIMS colloquia are sponsored by Cognia.

Frank Verstraete shows how tensor networks harness entanglement to simulate strongly correlated quantum matter and reveal hidden symmetries.

Entanglement matters

Entanglement as an organising principle of quantum matter

Javier Gómez Serrano is a professor of mathematics at Brown University. He also collaborates with Google DeepMind to advance maths with AI.

Confirmed speaker

Ken Ono is founding mathematician of AI-startup Axiom Math. Formerly, he was a professor of mathematics at the University of Virginia.

This first talk in the AI for Mathematical Sciences (AIMS) seminar series features Prof. Kevin Buzzard, who presents the rapid rise of formalised mathematics in computer theorem provers such as Lean. Over the past decade, mathematicians have begun encoding proofs in precise formal languages that machines can verify line by line, eliminating ambiguity and ensuring complete logical rigour. In these systems, proofs can be checked automatically, helping to create durable, searchable libraries of trusted results and transforming how mathematicians collaborate and build on one another’s work.

Prof. Buzzard discusses the motivations of this movement, the technical bottlenecks that remain, and the growing role of AI tools that can write Lean code and help verify the output of large language models. He also reflects on leading a major project to formalise a 21st-century proof of Fermat’s Last Theorem, and what such ambitious efforts signal for the future of mathematical research.

## Event information
This event, part of our AI for Mathematical Sciences series, takes place at 2 pm on Monday 9 March at the London Institute for Mathematical Sciences, on the second floor of the Royal Institution. AIMS is sponsored by Nebius. This series is organised by LIMS fellows [Prof. Yang-Hui He](https://lims.ac.uk/yang-hui-he/) and [Dr Evgeny Sobko](https://lims.ac.uk/evgeny-sobko/). To register for the series please fill out the [online form](https://applications.lims.ac.uk/subscribe-to-aims). 

Kevin Buzzard opens AIMS with his views on what a new era of formalised maths, Lean and AI—verified proofs mean for the future of research.

Formal proofs with AI

AI and the formalisation of mathematics

Shu-Heng Shao discusses conformal twist defects in 4d Maxwell theory, where electric and magnetic fields exchange roles. He determines their operator spectrum by solving classical wave equations with twisted boundary conditions and shows that defect correlation functions split into universal and chiral current sectors. He also revisits related 2d defects and explores a ’t Hooft anomaly and its dynamical implications.

[Dr Shu-Heng Shao](https://physics.mit.edu/faculty/shu-heng-shao/) is a theoretical physicist at MIT. His research explores structural aspects of quantum field theories and lattice systems, with a focus on non-invertible symmetries, which appear in quantum systems such as the Ising model, Yang-Mills theories and the Standard Model.

Shu-Heng Shao analyses twist defects in 4d Maxwell theory, deriving their operator spectrum and a chiral sector from classical wave equations.

Cosmic string duality

Cosmic string for electromagnetic duality

Energy conditions are central assumptions in classical general relativity, where they place bounds on the stress–energy tensor and underpin key results such as the singularity theorems. Quantum theory complicates this picture: quantum fields admit states with locally negative energy density, violating classical energy conditions. These violations are not arbitrary, however, and are constrained by quantum energy inequalities that limit their size and duration.

In this course, Dr Eleni Kontou reviews the classical energy conditions, their physical motivation, and the ways in which they can fail in realistic settings. She introduces quantum field theory on curved spacetimes and explains how quantum energy inequalities arise from first principles. The course then explores the average null energy condition in detail, examining the restrictions it places on exotic spacetimes and the implications for causality, spacetime structure, and the foundations of gravitational theory.

## Event information
This is a four-part lecture series from LonTI Lectures are on Mondays at 10:30 am in the 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 website](https://lonti.weebly.com/).

Eleni Kontou explores energy conditions in classical general relativity and QFT, showing how quantum bounds constrain negative energy.

Quantum energy bounds

Classical and quantum energy conditions

Symmetry Topological Field Theory (SymTFT) enables the systematic exploration of quantum systems with generalised symmetries. Alison Warman outlines the framework of the theory and and its application to phases of matter, including gapped phases where symmetry-broken vacua carry distinct forms of topological order.  She also shows how SymTFT techniques can be used to construct Clifford-hierarchy quantum computing phase gates in 2D.

[Alison Warman](https://www.maths.ox.ac.uk/people/alison.warman) is a PhD student at Oxford University, where her doctoral research is supervised by Prof. Sakura Schäfer-Nameki. Her work focuses on generalised symmetries in fundamental quantum systems, ranging from high energy physics to condensed matter and quantum computation. 

Alison Warman explains how Symmetry Topological Field Theory classifies quantum phases and informs advances in quantum information research.

Phases and phase gates

The London Institute welcomes invited guests to an evening of piano music for four hands with the Ukrainian brothers Alexei and Sasha Grynyuk, acclaimed concert pianists whose close partnership brings this repertoire vividly to life. During the interval, Prof. Yang-Hui He offers a short mathematical interlude, reflecting on pattern and variation in music.

Johannes Brahms’ *Variations on a Theme by Robert Schumann* arise from one of the most human stories of the Romantic era. Welcomed by Robert and Clara Schumann in the early 1850s, Brahms was profoundly shaped by their circle. The variations carry something of that private world into the concert hall.

Franz Schubert’s *Variations on an Original Theme in A-flat Major* come from Vienna’s rich culture of domestic music-making. Heard together, the works trace a Romantic fascination with how a single idea can be transformed. The antique Steinway & Sons grand piano heard this evening is generously on loan from Sasha Grynyuk.

## Event information
The event takes place on Friday 13 February at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Attendance is by invitation only. Arrival is from 5 pm in the Old Post Room. The concert, which will be held in Faraday’s Study, begins at 6 pm and will be followed by drinks with the musicians into the evening.

The Grynyuk brothers perform piano music for four hands by Brahms and Schubert, with a mathematical interlude by LIMS Fellow Yang-Hui He.

Piano and patterns

Performers

In *The Road to Reality* and *Fashion, Faith and Fantasy*, theoretical physicist Roger Penrose criticises string theory from conceptual, technical and methodological perspectives, alongside sociological reflections on the string theory community. In this talk, Dr James Read assesses Penrose’s conceptual and technical criticisms, focusing on objections that invoke the idea of functional freedom and their implications.

[Dr James Read](https://www.philosophy.ox.ac.uk/people/james-read) is a philosopher of physics at Oxford University. His research focuses on the foundations of spacetime theories such as general relativity, as well as scrutinising what some of physicists’ best theories, including quantum mechanics, say about the nature of reality.

Philosopher James Read explores how ideas of functional freedom shape the renowned mathematician Roger Penrose’s critique of string theory.

Functional freedom

On functional freedom and Penrose's critiques of string theory

This 14-week course examines how artificial intelligence can be integrated into contemporary mathematical research. More than a general overview of AI, it focuses on concrete applications in mathematical discovery, drawing on recent research papers and hands-on coding exercises.

The course covers training and deploying models, as well as using algorithmic optimisation and reinforcement learning to search for proofs, examples and counterexamples. Participants also learn to use AI-assisted coding to design computational experiments and build reasoning tools that function as research companions.

The course is developed by Nebius, a global AI infrastructure company that assists Microsoft, Meta and Cursor in training and deploying frontier models at scale. One key focus of Nebius’s research programme is AI for mathematics, including collaborations with the Stevens Institute of Technology and Caltech, and support for the London Institute’s new AI in Mathematical Sciences seminars.

## Event information
This is a 14-week course. Seminars are on Wednesdays at 10am in the seminar room at the London Institute, on the second floor of the Royal Institution, and conclude at 1 pm, with a break for coffee and tea. To register and attend, please visit the [Nebius Academy website](https://academy.nebius.com/ai-for-math).

The London Institute hosts a 14-week course by Nebius Academy on using AI to explore ideas and generate and test conjectures in mathematics.

Learn AI for maths

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 Cambridge University, he collaborated with John Conway on several influential projects, co-authoring a landmark paper on monstrous moonshine. Published in 1979, the paper proposed a deep connection between the monster group and the *j*-function. The seemingly far-fetched conjecture was proved in 1992 by Richard Borcherds and is now recognised as one of the great unifying conjectures in modern mathematics.

Simon was also an important contributor to the five-author ATLAS of Finite Groups—a definitive catalogue of symmetry types, containing essential information on the sporadic simple groups and the smaller members of the infinite families. In the third Simon Norton Lecture, Prof. Robert Curtis discusses Simon’s involvement in the ATLAS from its inception to its publication, explaining the crucial role finite groups play across mathematics and science.

## Event information 
The event takes place on Thursday 5 March 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
- *5:30pm* Arrival and drinks, Old Post Room.
- *6:00pm* Prof. Yang-Hui He, Introduction, Tyndall seminar room.
- *6:10pm* Prof. Robert Curtis, “Simon Norton and the ATLAS of Finite Groups”
- *7:10pm* Drinks and canapes, Old Post room

In the third Simon Norton Lecture, Robert Curtis will describe Simon’s pivotal contributions to the classic ATLAS of Finite Groups.

Simon and the ATLAS

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 pm on Friday, 27 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

The product replacement algorithm generates random elements of a finite group, yet performs far better than theory predicts. A key parameter is the largest size of a minimal generating set. In this talk, building on Julius Whiston’s result that the symmetric group on n points has size n–1, Prof. Colva Roney-Dougal presents new tight bounds for all its subgroups and explores their wider impact on product replacement.

[Colva Roney-Dougal](https://research-portal.st-andrews.ac.uk/en/persons/colva-roney-dougal/) is a professor of pure mathematics at St Andrews. She holds a PhD from Queen Mary University of London and is also known for her work on popularising mathematics, particularly on the radio. Her research focuses on group theory and computational algebra.

Colva Roney-Dougal gives new bounds on minimal generating sets in permutation groups and shows what they reveal about product replacement.

Permutation group limits

Minimal generating sets of permutation groups

Gauge theories play an essential role both in condensed matter and high-energy physics. Topological gauge theories also have mathematical applications, as they provide invariants of manifolds. Dr Pavel Putrov explores the relationships between gauge theories with finite and continuous gauge groups that have the same underlying algebraic group, considered over a finite field and over complex numbers, respectively.

[Dr Pavel Putrov](https://www.ictp.it/member/pavel-putrov#biography) is a research scientist at ICTP. He did his PhD in theoretical physics at the University of Geneva. He was the Sherman Fairchild Prize Postdoctoral Fellow at Caltech and a postdoc at the IAS. His work focuses on mathematical aspects of quantum field theories.

Pavel Putrov explores surprising connections between 3D topological gauge theories based on finite groups and their continuous counterparts.

Finite vs continuous

A relationship between gauge theories with finite and continuous gauge groups

In its ninth iteration, String Data 2025 brings together leading theorists, mathematicians and computer scientists for three days to discuss how open problems in theoretical physics can be addressed through the use of techniques from computer science. Held jointly this year at the Infosys Centre and the London Institute for Mathematical Sciences, the programme features keynote talks, presentations, poster sessions and panel discussions.

Participants examine how machine learning and data science can shed light on deep problems such as the structure of the string landscape, the dynamics of quantum fields and the geometry underpinning fundamental theory. With speakers drawn from top institutions worldwide, the event aims to build new bridges between disciplines and spark breakthroughs in foundational physics.

This year’s programme also includes, on Tuesday, a public-facing talk and panel on the parallels between the neural networks underlying AI and quantum field theory.

## Event information
This conference takes place from Monday 8 to Wednesday 10 December.  On Tuesday 9 December the event will be hosted by the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution in Mayfair. On other days, the event will take place at the Infosys Centre. For further details, please visit the conference [website](https://indico.global/event/14913/overview).

The London Institute co-hosts the leading international conference at the intersection of string theory, data science and machine learning.

String Data 2025

Far more than other nations such as Germany and America, Britain suffers from a dearth of independent institutes. That means that the lion’s share of our research takes place at universities, where all too often scientists must divide their time between research, teaching and administrative duties. 

In this event, the Tony Blair Institute will unveil its vision for a new kind of research organisation. The Lovelace disruptive invention laboratories will offer an alternative model, which is designed to pursue ambitious, high-risk ideas with long-term vision.  Characteristics that will set them apart from universities include the offer of early-career independence for scientists, a reliance on internal review as opposed to chasing grants, and directors with genuine autonomy. Partly inspired by the example set by the London Institute, the labs aim to accelerate discovery, attract world-class talent and bridge the gap between fundamental discovery and marketable innovation. 

## Event information
The event takes place on Thursday 20 November at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Attendance is by invitation only. There will be speeches from 7 pm in our seminar room, followed by drinks in the Old Post Room.

The Tony Blair Institute unveils its vision for a new kind of research institution to policymakers and the press at the London Institute.

Launching Lovelace labs

_tex: In 1936, Bartel van der Waerden predicted that among all monic integer polynomials of degree $n$ with absolute values of coefficients at most $H$, only about $O(H^{n-1})$ fail to have the full symmetric Galois group. This had been proved only for degrees up to four, through work by van der Waerden and later Sam Chow and Rainer Dietmann. In this seminar, Prof. Manjul Bhargava presents a proof of the conjectured bound in all degrees, showing that polynomials of large degree with smaller-than-expected Galois groups form only a thin exceptional set.

## Event information
The seminar, which will be held in the Faraday room, starts at 3 pm and is followed by a Q&A.

In this seminar, Fields Medalist Manjul Bhargava proves the van der Waerden conjecture on counting polynomials with small Galois groups.

Van der Waerden proven

Galois groups of random polynomials

Algebraic geometry has always been an intrinsically interdisciplinary subject, shaped by ideas from algebra, geometry, physics and number theory. Yet genuinely collaborative work requires more than shared interests: it demands that researchers gain a clear view of one another’s methods, motivations and results.

This workshop brings together graduate students and early-career mathematicians to foster exactly that exchange. Participants prepare and deliver research talks, present papers, and learn to navigate unfamiliar areas through the talks of their peers. The meeting blends research and education in a focused, supportive environment, allowing young algebraic geometers to broaden their technical range, sharpen their communication skills, and engage deeply with neighbouring subfields. By the end, they not only strengthen their understanding of algebraic geometry’s many facets but also build the relationships and shared vocabulary needed to begin new collaborative projects.

## Event information
This workshop takes place on Wednesday 26 November 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/agworkshopforphd).

The London Institute hosts a workshop on algebraic geometry for PhD students to share their research results and initiate collaborations.

Arrival

In this talk we observe that the categorical structure of a flop occurs for some well-known non-commutative resolutions of a nodal curve. We describe the flop-flop spherical twists, and give a geometric interpretation in terms of Landau–Ginzburg models. The resolutions are all weakly crepant but not strongly crepant, and we formulate an intermediate condition that distinguishes the smaller ones.

A categorical flop in dimension one

In this talk, we prove Homological Mirror Symmetry for the family of hypersurfaces in P(1, 1, 1, k). We construct an explicit mirror map, as well as a new construction exhibiting the special McKay correspondence on the A-side.

A Homological mirror symmetry for some orbifold Del Pezzo surfaces 

We demonstrate some unexpected properties that simple gluings of the simplest derived categories may have. We consider two special cases: the first is an augmented curve; the second is an ideal point gluing of curves. We construct unexpected exceptional objects contained in these categories and discuss their orthogonal complements.

Compact type generations of curves

Lunch

In this talk we discuss how generalised Kummer varieties degenerate to natural symplectic subvarieties of the Hitchin system for curves with genus larger than 2. Given a closed immersion between arbitrary smooth complex projective varieties one can take the deformation to the normal cone family. For curves in symplectic surfaces, the corresponding deformation to the normal cone space is an open subset of the relative moduli space of sheaves. 

Moduli of Sheaves and Deformation to the Normal Cone

In this talk we will describe Gopakumar–Vafa (GV) invariants associated to cAn singularities. This involves generalising GV invariants to crepant partial resolutions of cAn singularities, and showing that they satisfy Toda’s formula. We then study filtration structures on the parameter space of contraction algebras associated to cAn crepant resolutions and, finally, numerically constrain the possible tuples of GV invariants that can arise. 

GV invariants associated to cAn singularities

The notion of categorical absorption of singularities will be introduced. This is an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. For a projective variety X with isolated ordinary double points categorical absorption will be described.

Categorical absorptions of singularities and degenerations

Algebraic exchange

Workshop on algebraic geometry

Edward Frenkel is a professor of mathematics at UC Berkeley, a member of the American Academy of Arts and Sciences and winner of the Hermann Weyl Prize. He is also the author of *Love and Math*—part memoir, part masterclass in mathematical storytelling—which became a *New York Times* bestseller and has been published in 20 languages.

In this event, Prof. Frenkel joins London Institute Fellow Yang-Hui He and our chief science writer Ananyo Bhattacharya to reflect on his intellectual journey and tackle some of the most pressing questions in mathematics today. How can researchers argue for the value of basic science in a world of competing priorities? Will AI one day make human mathematicians obsolete?

The evening also offers a rare chance to hear an insider’s take on the Langlands programme—sometimes called the “grand unified theory of mathematics”—famed for its depth, difficulty and breathtaking beauty.

There will be time for audience questions and discussion over drinks.

## Event information
The event is on Friday 12 December at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. Entry is by invitation only. The interview starts at 5:30pm in our Faraday seminar room, followed by Q&A at 6:30pm. Afterwards there are drinks with the speaker into the evening, in the Faraday and Old Post rooms.

Edward Frenkel charts his path from Moscow to Berkeley and the quest for a unified theory of maths with the London Institute and its guests.

Meet Edward Frenkel

Prof. Manjul Bhargava is one of the foremost mathematicians of our time. His pioneering work in number theory has reshaped the geometry of numbers and earned him the Fields Medal, mathematics’ highest honour. A passionate advocate of India’s mathematical past and future, he also holds positions at the Tata Institute of Fundamental Research and other leading Indian institutions. Beyond mathematics, he is a scholar of Sanskrit and an accomplished tabla player.

At this reception in his honour, Prof. Bhargava discusses the promise of Indian mathematics and the urgency of supporting it. Dr Thomas Fink, Director of the London Institute, will describe our Ramanujan Junior Fellowships scheme, which will bring exceptional young Indian theoretical physicists and mathematicians to the London Institute for three transformative years.

The London Institute is  seeking visionary supporters to help realise this programme and strengthen our mission to advance fundamental research.

## Event information
The event takes place on Tuesday 25 November at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. The event is limited to 80 people and is by invitation only. There will be speeches at 6pm in our Faraday seminar room, followed by drinks with the speaker into the evening.

We hail India’s elite mathematical talent—and our Ramanujan Junior Fellowships scheme to nurture it—with Fields Medalist Manjul Bhargava.

Meet Manjul Bhargava

[Prof. Daniel Jafferis](https://www.physics.harvard.edu/people/facpages/jafferis) reviews a tensor–matrix model for 2d CFT data constrained by crossing symmetry, and topological expansion conjectured to match pure 3d gravity. He discusses sources of divergences and what remains to establish the duality, then turns to a BCFT extension linked to an open bootstrap. The resulting expansion corresponds to gluing tetrahedra, with the model reproducing certain 3d gravity amplitudes.

Prof. Daniel Jafferis is a physicist at Harvard University. He co-discovered key AdS/CFT dualities and proposed the F-theorem for 3D supersymmetric flows. Prof. Jafferis’s works on tensor–matrix models, holography and the unexpected links between mathematics, geometry and gravity.

Daniel Jafferis outlines a tensor–matrix model constrained by crossing, its proposed gravity dual and a BCFT version encoding amplitude data.

Tensor–matrix gravity

Tensor models, 3d gravity and simplicial decomposition

Prof. Dmitry Kaledin gives two lectures on the frontiers of modern algebraic geometry. In the first, he explores how classical Hodge theory extends—or fails to extend—to the non-commutative realm. In non-commutative geometry, de Rham cohomology is replaced by periodic cyclic homology, yet many foundational questions remain unresolved and there is no general theory over the complex numbers. Prof. Kaledin outlines a potential new framework for constructing a genuine non-commutative Hodge theory, building on insights from p-adic and crystalline settings and revealing unexpected connections between geometry and arithmetic.

In his second lecture, he presents a general approach to building a homotopically
enhanced category theory, based on Alexander Grothendieck’s idea of a “derivator”. This work aims to make abstract structures more flexible and computationally powerful, offering practical tools for researchers working across geometry, topology and mathematical physics.

## Event information

The first lecture, at 1:00 pm on Monday, 24 November, is part of the MAGIC algebraic geometry seminar series and will take place at Imperial College. The second is at 3:00 pm on Friday, 28 November and will take place in the seminar room of the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. 

In two lectures, Dmitry Kaledin explores progress towards a non-commutative Hodge theory and a theory of homotopically enhanced categories.

Homotopy & categories

Non-commutative Hodge theory and enhancing categories

Prof. Ran Tessler explores how the amplituhedron offers a geometric framework to understand the surprising connections between scattering amplitudes in N=4 Super Yang–Mills theory and cluster algebras. These links are not only conceptually rich but also essential for practical amplitude calculations. He presents a key cluster relation, its proof, and a new conjectural generalisation expressed in the language of operads.

[Prof. Ran Tessler](https://www.weizmann.ac.il/math/tessler/home) is a mathematician at the Weizmann Institute whose work bridges geometry, combinatorics and physics. He studies scattering amplitudes, amplituhedra and their links to cluster algebras, exploring the deep structures that connect mathematics and quantum theory.

Ran Tessler reveals how the amplituhedron connects quantum field scattering amplitudes to cluster algebras through geometry and symmetry.

Geometry of amplitudes

Amplituhedra and cluster structures on scattering amplitudes

Dr Brandon Rayhaun shows how the irreversibility of renormalisation group flow, combined with anomalies—including those in the space of coupling constants—can be used to constrain the infrared phases of defects in well-known quantum field theories. As an illustration, he presents new evidence for a conjectural “spin–flux duality”, which describes how line operators map across particle–vortex duality in 2+1 dimensions.

[Dr Brandon Rayhaun](https://www.ias.edu/scholars/brandon-rayhaun) is a Leinweber Physics Member at the Institute for Advanced Study. He did his PhD at Stanford. He works on problems at the interface of high energy theory, condensed matter physics and pure mathematics, using quantum field theory and string theory as frameworks.

Brandon Rayhaun examines the long-distance physics of defects in QFTs through anomalies and the irreversibility of renormalisation flow.

Defects and duality

Anomalies of defect parameter spaces and a spin-flux duality

When Richard Feynman died in 1988, his last blackboard bore a curious note to himself: “To learn: Bethe Ansatz.” The Bethe Ansatz, introduced by Hans Bethe in 1931, is a method for finding the spectra of Hamiltonians in quantum integrable systems such as the Heisenberg magnet. For decades it helped produce astonishing results and conjectures, though no one quite knew why it worked.

In this talk, Prof. Edward Frenkel revisits this enduring mystery. In collaboration with Boris Feigin and Nicolai Reshetikhin, he reinterpreted the Bethe Ansatz through the lens of the geometric Langlands correspondence, expressing its spectra in terms of mathematical objects known as opers. This framework led to powerful generalisations involving q-characters of quantum affine algebras and q-opers, bridging quantum physics and modern geometry.

Prof. Frenkel explores these ideas and reflects on why the Bethe Ansatz—an old key to new symmetries—still stood at the top of Feynman’s list.

## Event information
The event takes place on Thursday 11 December at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution.

Edward Frenkel illuminates Feynman’s final quest—to learn the Bethe Ansatz—with fresh insights from the geometric Langlands correspondence.

Feynman’s last wish

Quantum integrable systems: from Bethe Ansatz to Langlands duality

By analytically studying the repeated composition of simple Boolean functions, Dr Thomas Fink uncovers the spontaneous emergence of a law of parsimony. It is the first exact solution of a nontrivial input-output map, and suggests an explanation for why neural networks are biased towards simplicity in the models that they generate. The general result applies whenever composition of more than one input occurs.

[Dr Thomas Fink](https://lims.ac.uk/thomas-fink/) is the founding Director of the London Institute and Charge de Recherche in the French CNRS. He studied physics at Caltech, Cambridge and École Normale Supérieure. His work includes statistical physics, combinatorics and the mathematics of evolvable systems.

Our director Thomas Fink explains why the repeated application of simple logics induces a bias towards simplicity, with applications to AI.

Why AI works

Why AI works: The spontaneous emergence of a law of parsimony

To many, science appears as a kind of modern magic—the only magic that works. In the Soviet Union, an officially atheist state, science was elevated even further, becoming a substitute for religion.  The DAU project began as a biopic of Lev Landau, the Nobel Prize–winning physicist and icon of Soviet science. But it evolved into something far more ambitious: a vast social and artistic experiment exploring what it means to be Soviet, to be modern and to believe in science. 

As part of the project, hundreds of participants lived and worked inside a reconstructed Soviet research institute, their lives filmed in an unprecedented exploration of the scientific worldview. Among the works born of the ongoing project was *DAU. Natasha*, which won the Silver Bear at the Berlin International Film Festival in 2020.

In this talk, Prof. Dmitry Kaledin offers a rare insider’s view of DAU—from its portrayal of Soviet scientific culture to its deeper inquiry into the meaning of science itself.

## Event information
The event takes place on Thursday 27 November 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/3lokj6).


Dmitry Kaledin reflects on the DAU project—an audacious theatrical experiment probing science and belief in the Soviet Union and beyond.

The magic of DAU

DAU: the magic that works

Matter and fundamental forces are governed by non-abelian gauge theories and general relativity. These theories look very different, but recent research has uncovered the “double copy”—an exact correspondence between their mathematical structures. Prof. Chris White reviews its theoretical origins and practical uses, highlighting links to string theory, exotic field theories, and open questions in pure mathematics.

[Prof. Chris White](https://www.seresearch.qmul.ac.uk/cfp/people/cwhite/) is a theoretical physicist at Queen Mary University of London. He studied at the University of Cambridge, before holding positions in Amsterdam, Durham and Glasgow. His research focuses on theories of particle physics, and their relationship to gravity and beyond. 

Chris White traces the origins and impact of a remarkable new correspondence linking non-abelian gauge theories with general relativity.

The double copy

The double copy: new connections in physics and mathematics

Plato was fascinated by the intricate relationship between science and music. We have explored it, too, in events focusing on the mathematics of the violin and the bagpipes. Now we are delighted to welcome to our rooms a Steinway & Sons grand piano, which is generously being lent to us by the Ukrainian concert pianist Sasha Grynyuk.  

This Steinway Model B grand piano has a distinguished lineage. Before Grunyuk, it belonged to the concert pianist and teacher Noretta Conci-Leech, who studied under Arturo Benedetti Michelangeli. It has also been played by such virtuosi as Evgeny Kissin, Claudio Abbado, Maurizio Pollini, Antonio Pappano and Alfred Brendel. Now it occupies our seminar room, which was once the study of Michael Faraday.

Since it arrived in July, the Faraday grand, as we’re calling it, has been played by, among others, our trustee Ben Delo and, frequently, our fellow Juven Wang. We have also enjoyed recitals from Grynyuk himself and look forward to more in the future. 

The Ukrainian concert pianist Sasha Grynyuk has generously lent us a Steinway grand piano, which was once played by Kissin and Brendel. 

Music to our ears

Matrix models—systems in which randomness and algebraic structure meet—play a central role across theoretical physics and mathematics. Often seen as zero-dimensional analogues of field theories, they also lie at the heart of deep connections linking string theory, integrability, quantum gravity, Yang–Mills theory, combinatorics, geometry, and representation theory.

In this four-part lecture series, Dr Levkovich-Maslyuk provides a guided, pedagogical journey through the subject: starting with motivation and definitions, through reduction to eigenvalue form, to continuum limits and spectral curves, and finally to orthogonal polynomials, phase transitions, and links to two-dimensional gravity. Along the way he sketches the broader vista of loop equations, topological recursion, and integrability. The series offers both newcomers and seasoned theorists a unified and illuminating view of how matrix models continue to shape modern research.

## Event information

This is a four-part lecture series from LonTI. Lectures are on Mondays at 10:30 am in the 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.

Fedor Levkovich-Maslyuk traces how matrix models reveal striking connections between quantum gravity, string theory and integrability.

Random matrices

Introduction to matrix models

Our universe is chiral—it knows its left from its right. Yet theories with chiral fermions are notoriously difficult to simulate. Dr Rishi Mouland applies powerful symmetry-based techniques to derive the phase diagrams of strongly coupled two-dimensional gauge theories, revealing new, readily simulable models that realise chiral infrared phases and illuminate fresh pathways to understanding the physics of our universe. 

[Dr Rishi Mouland](https://mou.land/) is a postdoc at Imperial College London. He did his PhD at King’s College London followed by postdoctoral research at Cambridge. His interests range from generalised symmetries in strongly coupled gauge theory to quantum gravity and the holographic bootstrap.

Rishi Mouland demonstrates how generalised notions of symmetry can be used in novel ways to probe strongly coupled quantum field theories.

Gapping chiral fermions

Phases of 2d Gauge Theories and Symmetric Mass Generation

In later life, Alexander Grothendieck was captivated by simple graphs encoding holomorphic maps to the sphere. He called them dessins d’enfants—“children’s drawings”. Decades earlier, Gerard ’t Hooft noted that the 1/N expansion in gauge theory mirrors the genus expansion in string theory. Dr Edward Mazenc shows how recent work on AdS/CFT reveals Grothendieck’s insight as the simplest form of gauge–string duality.

[Dr Edward Mazenc](https://www.edwardmazenc.com/) is a research fellow at ETH Zurich. He did his PhD at Stanford and was a Kadanoff Fellow at the University of Chicago. He studied at MIT and Cambridge. His research explores the insights that gauge/string duality offers into the fundamental structure of spacetime.

Edward Mazenc shows how Grothendieck’s dessins d’enfants reveal a deep link between gauge theory, string theory and the AdS/CFT duality.

Strings from graphs

Grothendieck & ’t Hooft: Dessins d’Enfant from AdS/CFT

Integrable models in 1+1 dimensions allow exact solutions of scattering problems and often the complete derivation of spectra. But these models raise a deeper question: what really are the particles being scattering? In low dimensions, unexpected phenomena emerge. Prof. Alessandro Torrielli explores spin and statistics in 1+1 dimensions, delving into one manifestation of this strange behaviour in AdS_3 string theory.
[Alessandro Torrielli](https://www.surrey.ac.uk/people/alessandro-torrielli) is a professor at the University of Surrey. He has held postdoc positions at Padova, HU Berlin, MIT, Utrecht and York. His research interests focus on quantum integrable systems, particularly those appearing in the context of the AdS/CFT correspondence.

Alessandro Torrielli examines spin, statistics and strange particle behaviour in 1+1-dimensional integrable models and AdS_3 string theory.

Integrable statistics

Integrable models, spin and statistics

In 1739, Euler devised the Tonnetz, a lattice to visualise harmonic relations in music. In this talk, Prof. Konstanze Rietsch revisits Euler’s Tonnetz from a modern point of view, exploring Tonnetze on tiled surfaces. She presents examples on triangulated tori related to crystallographic reflection groups, as well as a diatonic case tied to finite geometry, yielding elegant new visualisations of chord progressions.

[Konstanze Rietsch](https://www.kcl.ac.uk/people/konstanze-rietsch) is a professor at King’s College London. She works on the geometry of flag varieties and is also interested in total positivity, a theory that allows many geometric objects occurring in Lie theory to be defined, in some sense, over the positive real numbers.

Konstanze Rietsch explores Euler’s Tonnetz through the lens of modern geometry, revealing new links between music, symmetry and harmony.

Tiling and Tonnetze

Generalising Euler’s Tonnetz

Dr Stephen Wolfram is one of the most restless and original talents in science. Since publishing his first paper in particle physics at just 15, he has created the pioneering software Mathematica, launched the knowledge engine WolframAlpha and developed a radical new framework for fundamental physics.

In this event, Dr Wolfram sits down with our director Thomas Fink and science writer Ananyo Bhattacharya to talk about the big questions. What are the deepest challenges facing theorists today? How should science be organised and communicated? And how might AI-assisted discovery transform maths and physics?

The evening offers a rare opportunity to learn about Dr Wolfram’s intellectual journey and ambitions for the future, as he reflects on his decades-long pursuit of new kinds of science, from cellular automata to his Physics Project to computational insights into life and learning. There will also be time for audience questions and discussion over drinks. 

## Event information

The event takes place on Monday 13 October at the London Institute for Mathematical Sciences, which is on the second floor of the Royal Institution. The event is limited to 80 people and is by invitation only. The interview will start at 4:00pm in our Faraday seminar room, followed by Q&A at 5:00 pm. At 5:45 pm we will have drinks and canapes with the speaker into the evening.

<a href="https://www.youtube.com/embed/8SD9WgPCZ28?si=Xf5TIxWk7SZw-hX8">Watch the event</a>

Theoretical physicist and innovator Stephen Wolfram discusses the big questions facing science with the London Institute and its guests.

Meet Stephen Wolfram

The latest advances in quantum computing are pushing the limits of our ability to describe and simulate out-of-equilibrium dynamics of quantum systems. Dr Alexey Milekhin introduces powerful theoretical tools to address these problems, focusing on the Sachdev–Ye–Kitaev model and the emerging framework of “entanglement in time”, an information-theoretic approach designed to probe dynamical properties of quantum systems.

[Dr Alexey Milekhin](https://pa.as.uky.edu/users/ami426) is an assistant professor at the University of Kentucky. He has a PhD from Princeton University and did postdocs at UC Santa Barbara and Caltech. His work focuses on statistical mechanics and the dynamics of many-body quantum systems, including quantum gravity.

Alexey Milekhin explores how the Sachdev–Ye–Kitaev model and “entanglement in time” can reveal new ways of understanding quantum dynamics.

Entanglement in time

Quantum dynamics meets quantum information

Since the discovery of the thermodynamics of black holes, a key challenge has been uniting statistical mechanics with gravity and quantum information. Prof. Ginestra Bianconi proposes a new “Gravity from Entropy” action for gravity, with a Lagrangian given by the geometric quantum relative entropy. This framework recovers Einstein’s equations in the low-energy, small-curvature limit and predicts modified gravity beyond.

[Ginestra Bianconi](https://webspace.maths.qmul.ac.uk/g.bianconi/) is a professor at Queen Mary University of London, an APS Fellow and member of the European Academy of Sciences. She is recognised for pioneering work in statistical mechanics and theoretical physics, especially on how topology shapes dynamics in complex networks.

Ginestra Bianconi illustrates her novel approach to quantum gravity, which is grounded in statistical mechanics and information theory.

Gravity from entropy

The history of “artificial intelligence” is the history of an overhyped intellectual brand that has only very recently come to signify a set of deployable technologies with broad application and clear, if somewhat horrifying, purposes. Since its debut in 1955 the AI brand has been attached to a rotating cast of technologies with only loose connections to each other or to cognition, none of which has yet come close to delivering on the promise of creating computer systems with human-like intelligence. One AI insider characterized the story of AI as “the history of failed ideas.” Yet in the process of failing, early AI researchers made vital but incidental contributions to the development of computer technology and computer science.

In this seminar, Thomas Haigh explores where the AI brand came from, why it was so attractive to researchers and sponsors, and how artificial intelligence institutionalised as a subfield of computer science through research labs, curricula, textbooks and professional associations. Haigh also documents continuities and discontinuities between our own moment and earlier cycles of AI hype and disillusionment.

## Event information

The event takes place on Tuesday 7 October 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/n04qXZ).

Thomas Haigh traces the rise of AI as an overhyped brand, from failed ideas to today’s powerful technologies and their unsettling impact.

AI’s history of hype

Artificial Intelligence: the brand that wouldn’t die

The story of science is often told as a series of flashes of insight, enjoyed by heroic figures such as the Royal Institution’s Sir Humphry Davy and Michael Faraday. Yet in reality, innovation is rarely so simple or dramatic.

Drawing on examples from the history of the Royal Institution, as well as insights from his recent book, *Like: The Button that Changed the World*, London Institute Trustee Martin Reeves makes an original case for how scientific breakthroughs really happen. He demonstrates that most are not the result of eureka moments, but a combination of iteration, collaboration and, strangest of all, serendipity. With the aid of vivid demonstrations in the tradition of the Royal Institution, he explains why this does not just mean surrendering to chance. On the contrary, he hones in on the mathematical basis for serendipity—hidden patterns in the history of discovery—and shares the strategies that innovative businesses and research centres use to harness it.

## Event information
This event takes place in the historic lecture theatre of the Royal Institution. Tickets for the talk may be purchased [here](https://www.rigb.org/whats-on/science-serendipity-how-innovation-really-works). London Institute guests, who don’t need to buy tickets, are invited to join the speaker for drinks after the talk in our private rooms on the second floor of the Royal Institution.

<a href="https://www.youtube.com/embed/wGe1WRN4wEA?si=YWmt0wnUGZPRD5Jb">Watch the event</a>

In the Royal Institution’s lecture theatre, Martin Reeves reveals the startling truth about how innovation happens in science and technology.

How innovation works

The science of serendipity: how innovation really works

Prof. Mark Gross discusses the notion of scattering diagrams, a powerful means of encoding often very complex wall-crossing data in a wide range of geometric problems. Originally developed in the foundational work of Kontsevich and Soibelman, scattering diagrams have since emerged in many different contexts, including mirror symmetry, cluster algebras, wall-crossing for moduli spaces of sheaves and theoretical physics.

[Prof. Mark Gross](https://www.dpmms.cam.ac.uk/person/mg475) is a pure mathematician at Cambridge University. He did his PhD at UC Berkeley and has held posts at Cornell, Warwick, and UC San Diego. He is renowned for his work on mirror symmetry which has profoundly influenced algebraic, tropical and differential geometry.

Prof. Mark Gross explores how scattering diagrams are a powerful tool for capturing wall-crossing data in many diverse mathematical contexts.

Surveying scattering

Wall-crossing and scattering diagrams in geometry and beyond

Symmetry is a powerful organising principle, enabling remarkably precise, all-order predictions about the behaviour of quantum systems. Dr Christian Copetti explains how generalised symmetries act on boundary conditions and defects, and uses these ideas to constrain key features of (1+1)d scattering amplitudes, such as crossing symmetry, and shows how ultraviolet defects may be screened by bulk degrees of freedom.

[Dr Christian Copetti](https://www.maths.ox.ac.uk/people/christian.copetti) is a postdoc at Oxford University. He did his PhD at IFT in Madrid and then held a research position at SISSA in Trieste. His recent work spans generalized symmetries, defect dynamics, S-matrix bootstrap, and topological constraints in quantum field theories. 

Christian Copetti from Oxford University shows how generalised symmetries guide quantum systems, shaping boundary conditions and defects.

Defects and scattering

Symmetry constraints on defects and scattering amplitudes

Prof. Darryl Holm shows how geometric mechanics uses Lie–Noether symmetries of Hamilton’s principle to derive momentum maps and Euler–Poincaré equations, with variables that evolve by coadjoint motion. This motion imparts common features to their solutions, demonstrating how geometric mechanics offers a way to study symmetry breaking and the multi-scale challenges arising in modern investigations of dynamics in nature.

[Prof. Darryl Holm](https://profiles.imperial.ac.uk/d.holm/about) is a professor of applied mathematics at Imperial College London. His main research interests lie in nonlinear science and most of his work is based on Lie symmetry reduction of Hamilton's principle. He is particularly interested in emergent singular phenomena.

Prof. Darryl Holm explores how geometric mechanics links symmetry-breaking to dynamics, revealing patterns in nature’s complex systems.

Mechanics in a box

Geometric mechanics in a box

Madhuparna Das from the University of Exeter explores a key challenge in the field of artificial intelligence: the automation of mathematical discovery. She introduces HypothesiX, an AI agent that creates novel mathematical conjectures by blending formal reasoning from Lean’s Mathlib with the creative power of Large Language Models. She demonstrates its potential in pure mathematics and explores additional applications.

[Madhuparna Das](https://experts.exeter.ac.uk/35716-madhuparna-das) is an EPSRC Doctoral Fellow at the University of Exeter, supervised by Prof. Nigel Byott. Before her PhD, she did a masters by research at Exeter under Prof. Kyle Pratt and Dr Gihan Marasingha. She works on probabilistic number theory and additive combinatorics.

Madhuparna Das presents an AI agent automating novel mathematical conjecture discovery by integrating Lean’s Mathlib with LLMs’ creativity.

Automated conjectures

Automated conjecture formulation

Prof. Misha Rudnev explains how the long-standing Erdős distinct distance conjecture was resolved in 2010 by Larry Guth and Nets Katz using a combination of tools that were well beyond earlier approaches. Reviewing the methodology, which drew on 19th-century geometry and early 20th-century algebraic topology, Prof. Rudnev discusses some results and a plethora of still wide open questions that followed from their work.

[Prof. Misha Rudnev](https://research-information.bris.ac.uk/en/persons/misha-rudnev) is a professor of mathematics at the University of Bristol. He studied applied mathematics and physics at the Moscow Engineering Physics Institute and did his PhD at Caltech. His research focuses on problems in geometric, arithmetic and additive combinatorics.

Prof. Misha Rudnev from the University of Bristol shows how algebraic geometry helped settle a famous discrete geometry conjecture of Erdős

Erdős and topology

A few helpful interventions from baby algebraic geometry to “hard Erdős questions”

Artificial intelligence is no longer confined to crunching numbers—it is beginning to propose original theorems, sketch proofs and reveal new and unexpected patterns in the abstract world of mathematics. Speaking in the iconic lecture theatre of the Royal Institution, Prof. Yang-Hui He—one of the pioneers of AI-assisted discovery— explores how machine learning is transforming both the practice and philosophy of mathematical research.

Drawing on his own leading-edge work, alongside insights from a recent gathering in Berkeley where mathematicians challenged a powerful AI chatbot to solve problems designed to resist automation, Prof. He traces the emergence of a new form of mathematical exploration. Describing his AI-assisted discovery, with colleagues, of “murmuration” patterns in elliptic curves, Prof. He shows how, by fusing human intuition with computational power, this partnership is reshaping research and redefining the creative boundaries of the field.

## Event information
This event takes place in the historic lecture theatre of the Royal Institution. Tickets for the talk or the live stream may be purchased [here](https://www.rigb.org/whats-on/mathematics-rise-machines). London Institute guests are invited to join the speaker for drinks after the talk in our private rooms on the second floor of the Royal Institution. 

In the Royal Institution’s lecture theatre, Prof. Yang-Hui He reflects on how growing AI-human collaboration is shaping the future of maths.

Maths in the age of AI

Mathematics: the rise of the machines?

Prof. Zhengcheng Gu from the Chinese University of Hong Kong gives an analytical construction of the fixed point tensors of a 2D rational CFT and provides a concrete example by using several models to compute the scaling dimensions explicitly. He shows that the construction of the fixed point tensors is closely related to a strange correlator, where the holographic picture and generalised symmetries naturally emerge.

Prof. Zhengcheng Gu describes the construction of a fixed-point tensor network and its generalised symmetries in conformal field theory.

CFT tensor network

Fixed-point tensor network construction and generalised symmetry for conformal field theory

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.

Where two worlds meet

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.

Geometric categories for continuous gauging

2 PM, 7 Apr 2026