Our goal is to boost our research output for 2024 by √2 over our previous year. We measure our research output by SNIP points, which are described in our Journals page. Help us reach our target of 40 points for this year.

## Concept

### Scattering amplitude dynamics

We review classical gravitational dynamics derived from quantum field theoretic scattering amplitudes, a tool used in detecting gravitational waves.

### Morse discriminant combinatorics

The rich combinatorics of the recently-computed Newton polytope of the Morse discriminant provides insights on connections between algebra, geometry and combinatorics.

Geometry

### Multiple solutions of enumerative

In enumerative geometry, when constraints, such as lines, are in special positions, several solutions of a problem collide to form a multiple solution.

Topological photonics

### Topological protection

Ideas from topology are used to show the protection of quantum state density matrix elements, revealing new applications of topological photonics.

### Gravitational interactions of massiv

Formulating, for the first time, field theories for massive higher-spin particles for the potential application to black-hole dynamics.

## Drafts

### Immortality unleashed

Ageing does not confer a benefit toward natural selection in a changing environment.

### Irreducibility via Gram determinants

A combination of techniques proves the irreducibility of unitary representations of infinite-dimensional groups based on Gram determinants.

### Dirichlet meets Kauffman

Long the province of statistical physics, the structure of the Kauffman model of genetic computation is uncovered via Dirichlet convolutions.

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Number theory

### Recursive divisor properties

The recursive divisor function is found to have a simple generating function, which leads to a number of new Dirichlet convolutions.

### Learning Gibbs measures

Even when the dynamics of a glass combine into a single metastable state, we can still learn the couplings accurately.

### Singularities of discriminants

Investigating to what extent Whitney's theorem holds true for more general universal polynomials such as A-discriminants by Gelfand, Kapranov and Zelevinsky.

Condensed matter theory

### Odd topological kinks

Topological kinks in matter with non-reciprocal bonds that do not conserve energy can be studied through experiment, simulations and theory.

### Elliptical murmurations

Investigating oscillating patterns for Dirichelt coefficients of low-degree arithmetic L-functions gives insight into this newly-discovered phenomena.

Chiral photon states

### Bound states with π flux

The interplay between the interactions and chirality of bound photons reveals insight into, and can even predict new physics.

### Design and interpretation of transcr

Graph models representing the steps in transcription factor screens reveal the most costly error sources and most promising set of factors.

Algebraic geometry

### Topology of tropical polynomials

Tropical geometric objects share many characteristics with classical algebraic geometry objects. We study this correspondence for the topology of polynomials.

### MSSM vacuum structure

On the vacuum structure of the minimal supersymmetric standard model, which considers only particle states and interactions consistent with reality.

### Quantum transport

Quantum geometric tensor exploration to solve a variational problem of state transport in quantum nets.

Statistics

### Paying for science

An analysis of UKRI grant funding gives a strategy for maximising unrestricted funds, thereby helping research centres cover their costs.

### Combinatoric Topological Strings

We find a physical interpretation, in terms of combinatorial topological string theory, of a classic result in finite group theory theory.

## Arxiv

Theory of innovation

### Recursive structure of innovation

A theoretical model of recursive innovation suggests that new technologies are recursively built up from new combinations of existing ones.

Number theory

### Higher energies

Generalising the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three leads to developments in the theory of the higher energies.

Algebraic geometry

### Slight degenerations

The tools used to study polynomial equations with indeterminate coefficients are extended to some important cases with interrelated ones.

Algebraic geometry

### Schön complete intersections

A uniform approach to a class of varieties is described that includes important types of objects from geometry, optimisation and physics.

Number theory

### Ample and pristine numbers

Parallels between the perfect and abundant numbers and their recursive analogs point to deeper structure in the recursive divisor function.

Machine learning

### Speaking DNA

A family of transformer-based DNA language models can interpret genomic sequences, opening new possibilities for complex biological research.

Algebraic geometry

### Symmetric spatial curves

We study the geometry of generic spatial curves with a symmetry in order to understand the Galois group of a family of sparse polynomials.

Number theory

### Recursive divisor properties

The recursive divisor function has a simple Dirichlet series that relates it to the divisor function and other standard arithmetic functions.

Evolvability

### Flowers of immortality

The eigenvalues of the mortality equation fall into two classes—the flower and the stem—but only the stem eigenvalues control the dynamics.

Condensed matter theory

### Counting free fermions

A link between the statistical properties of free fermions in one dimension when either half- or alternating- states are initially occupied.

Computational linguistics

### Cross-lingual knowledge

Models trained on a Russian topical dataset, of knowledge-grounded human-human conversation, are capable of real-world tasks across languages.

Condensed matter theory

### Non-reciprocal breather

Producing the first examples of breathing solitons in one-dimensional non-reciprocal media allows their propagation dynamics to be analysed.

Algebraic geometry

### Three-manifolds

Pachner graphs for 3-manifold triangulation can be generated and analysed through the lens of network science.

### Gauge interactions

Formulating, for the first time, field theories for massive higher-spin particles for the potential application to black-hole dynamics.

AI-assisted maths

### Triangulating polytopes

Machine learning generates desirable triangulations of geometric objects that are required for Calabi-Yau compactification in string theory.

### Machine learning polytopes

A supervised machine of learning lattice polytopes predicts properties of volume, dual volume, and reflexivity with up to 100% accuracy.

## Submitted home

Synthetic biology

### Cell soup in screens

Bursting cells can introduce noise in transcription factor screens, but modelling this process allows us to discern true counts from false.

Evolvability

### I want to be forever young

The mortality equation governs the dynamics of an evolving population with a given maximum age, offering a theory for programmed ageing.

Number theory

### Counting recursive divisors

Three new closed-form expressions give the number of recursive divisors and ordered factorisations, which were until now hard to compute.

Representation theory

### Infinite dimensional irreducibility

The criteria of irreducibility of representations of the inductive limit of certain general linear groups acting on three infinite rows.

## Submitted away

Combinatorics

### In life, there are few rules

The bipartite nature of regulatory networks means gene-gene logics are composed, which severely restricts which ones can show up in life.

Combinatorics

### Structure of genetic computation

The structural and functional building blocks of gene regulatory networks correspond, which tell us how genetic computation is organised.

Combinatorics

### Biological logics are restricted

The fraction of logics that are biologically permitted can be bounded and shown to be tiny, which makes inferring them from experiments easier.

Combinatorics

### Representation for sum-product

A new way to estimate indices via representation theory reveals links to the sum-product phenomena and Zaremba’s conjecture in number theory.

Number theory

### Bounding Zaremba’s conjecture

Using methods related to the Bourgain–Gamburd machine refines the previous bound on Zaremba’s conjecture in the theory of continued fractions.

Linear algebra

### Infinitely high parallelotopes

We demonstrate that the height of an infinite parallelotope is infinite if no non-trivial combinations of its edges belong to $l_2(\mathbb N)$.

Combinatorics

### Quadratic residues

Additive combinatorics sheds light on the distribution of the set of squares in the prime field, revealing a new upper bound for the number of gaps.

Algebraic geometry

### Permuting the roots

The Galois group of a typical rational function is described and similar problems solved using the topology of braids and tropical geometry.

### Continuous quivers

Continuous quivers enable exact Wilson loop calculation, reveal an emergent dimension, and raise tantalising questions on dual strings.

Machine learning

### BERT enhanced with recurrence

The quadratic complexity of attention in transformers is tackled by combining token-based memory and segment-level recurrence, using RMT.

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### Free energy and learning

Using the free energy principle to derive multiple theories of associative learning allows us to combine them into a single, unifying framework.

Representation theory

### Group representation irreducibility

A general approach to proving the irreducibility of representations of infinite-dimensional groups within the frame of Ismagilov's conjecture.

Number theory

### Sum-product with few primes

For a finite set of integers with few prime factors, improving the lower bound on its sum and product sets affirms the Erdös-Szemerédi conjecture.

Algebraic geometry

### Sparse singularities

Geometric properties, including delta invariants, are computed for singular points defined by polynomials with indeterminate coefficients.

Number theory

### Reflexions on Mahler

With physically-motivated Newton polynomials from reflexive polygons, we find the Mahler measure and dessin d’enfants are in 1-to-1 correspondence.

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### From words to blocks

Combining a language model with reinforcement learning enables object construction in a Minecraft-like environment from natural language instructions.

## Published

AI-assisted maths

### On AI-driven discovery

Reviewing progress in the field of AI-assisted discovery for maths and theoretical physics reveals a triumvirate of different approaches.

Condensed matter theory

### Strange kinks

A new non-linear mechanical metamaterial can sustain topological solitons, robust solitary waves that could have exciting applications.

Statistical physics

### Multiplicative loops

The dynamics of the Kauffman network can be expressed as a product of the dynamics of its disjoint loops, revealing a new algebraic structure.

Statistical physics

### Landau meets Kauffman

Insights from number theory suggest a new way to solve the critical Kauffman model, giving new bounds on the number and length of attractors.

Quantum field theory

### PCM in arbitrary fields

The first exact solution for the vacuum state of an asymptotically free QFT in a general external field found for the Principal Chiral Model.

Neurocomputing

### Spiky backpropagation

The training algorithm for digital neural networks is adapted and implemented entirely on an experimental chip inspired by brain physiology.

Condensed matter theory

### A kicked polaron

Modelling the final state of a mobile impurity particle immersed in a one-dimensional quantum fluid after the abrupt application of a force.

Quantum field theory

### Peculiar betas tamed

Inconsistencies between two approaches to deriving beta functions in two-dimensional sigma models are resolved by adding heavy superpartners.

String theory

### World in a grain of sand

An AI algorithm of few-shot learning finds that the vast string landscape could be reduced by only seeing a tiny fraction to predict the rest.

Gravity

### QFT illuminates black holes

Classical Kerr amplitudes for a rotating black hole derived using insights from recent advances in massive higher-spin quantum field theory.

Algebraic geometry

### Analysing amoebae

Genetic symbolic regression methods reveal the relationship between amoebae from tropical geometry and the Mahler measure from number theory.

Machine learning

### Beyond attention

Investigating recurrent memory augmentation of pre-trained transformer models reveals the scope for storage in memory of up to 2 million tokens.

Number theory

### Elliptical murmurations

Certain properties of the bivariate cubic equations used to prove Fermat’s last theorem exhibit flocking patterns, machine learning reveals.

Condensed matter theory

### Spin-charge separation

A transformation for spin and charge degrees of freedom in one-dimensional lattice systems allows direct access to the dynamical correlations.

Gravity

### Root-Kerr from higher-spin theory

Two approaches that provide local formulae for Compton amplitudes of higher-spin massive objects in the quantum regime and classical limit.

AI-assisted maths

### Computing Sasakians

Topological quantities for the Calabi-Yau link construction of G2 manifolds are computed and machine learnt with high performance scores.

AI-assisted maths

### Clifford invariants by ML

Coxeter transformations for root diagrams of simply-laced Lie groups are exhaustively computed then machine learned to very high accuracy.