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.

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.

### Murmurations of L-functions

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.

Algebraic geometry

### Three-manifolds

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

### 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.

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

The geometry of symmetric spatial curves reveals characterisations of general one-parameter families of complex univariate polynomials with fully-symmetric Galois groups.

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.

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.

### Slight degeneration

Newton polytopes are used to study systems of general polynomial equations, which consist of given monomials with generic coefficients. We describe what happens to solutions of these systems when the coefficients slightly degenerate.

### Gauge interactions

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

### 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.

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.

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

An analog of quasi-regular representations can be constructed for an infinite-dimensional group, despite the absence of the Haar measure.

## Submitted away

Representation theory

### Infinitely high parallelotopes

The height of an infinite parallelotope is infinite, an essential ingredient to prove the irreducibility of unitary representations of some infinite-dimensional groups.

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.

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.

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.

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.

### 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.

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.

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.

Number theory

### Elliptic curve murmurations

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

### From words to blocks

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

AI-assisted maths

### ML Sasakian manifolds

Topological quantities for the Calabi-Yau link construction of G2 manifolds are computed, analysed and machine learnt.

### Machine Learning Clifford invariants

To be added

## Published

Condensed matter theory

### Strange kinks

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

### Permuting the roots

We combine topology of braids and tropical geometry to describe the Galois group of a typical rational function composed of prescribed monomials. The same new technique solves several similar multivariate problems.

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.

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.