I. Shkredov

Expanding the known multiplicative properties of large difference sets yields a new, quantitative proof on the structure of product sets.

On some multiplicative properties of large difference sets

Multiplicativity of sets

A. V. Kosyak

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

The height of an infinite parallelotope is infinite

Infinitely high parallelotopes 

V. F. Lev

A finite nonempty subset A of a cyclic group, with small enough |A–A|, contains a nonzero element with at least (2+o(1))|A|²/|A–A| representations as a difference of two elements. 

The popularity gap

O. Gamayun

M. Panfil

F. Taha Sant'Ana

Explicit computation of injection and ejection impurity’s Green’s function reveals a generalisation of the Kubo-Martin-Schwinger relation. 

Kubo-Martin-Schwinger relation for an interacting mobile impurity

Mobile impurity

P. Dechant

Y. He

E. Heyes

E. Hirst

Investigating cluster algebras through the lens of modern data science reveals an elegant symmetry in the quiver exchange graph embedding.

Cluster algebras: network science and machine learning

AI for cluster algebras

T. Fink

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

Number of ordered factorizations and recursive divisors

Counting recursive divisors

A. Ashmore

B. A. Ovrut

By approximating the basis of eigenfunctions, we computationally determine the harmonic modes of bundle-valued Laplacians on Calabi-Yau manifolds.


Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

Bundled Laplacians

P. Moree

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

Irreducibility of the Koopman representations for the group GL0(2∞,R), acting on three infinite rows

Infinite dimensional irreducibility

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

Properties of the recursive divisor function and the number of ordered factorisations

Recursive divisor properties

R. Aoude

A. Ochirov

How gravitational waves are absorbed by a black hole is understood, for the first time, through effective on-shell scattering amplitudes.

Gravitational partial-wave absorption from scattering amplitudes

Absorption with amplitudes

A. Losev

M. Shifman

The beta function for a class of sigma models is not found to be geometric, but rather has an elegant form in the context of algebraic data.

Peculiarities of beta functions in sigma models

Peculiar betas

D. Zharikova

D. Kornev

F. Ignatov

M. Talimanchuk

D. Evseev

K. Petukhova

V. Smilga

D. Karpov

Y. Shishkina

D. Kosenko

M. Burtsev

A new open-source platform is specifically tailored for developing complex dialogue systems, like generative conversational AI assistants.

DeepPavlov dream: platform for building generative AI assistants

DeepPavlov dream

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

Monolingual and cross-lingual knowledge transfer for topic classification

Cross-lingual knowledge

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

Some applications of representation theory to the sum–product phenomenon

Representation for sum-product

V. Fishman

Y. Kuratov

M. Petrov

A. Shmelev

D. Shepelin

N. Chekanov

O. Kardymon

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


GENA-LM: A family of open-source foundational models for long DNA sequences

Speaking DNA

P. Berglund

V. Jejjala

A. Lukas

Genetic algorithms, which solve optimisation problems in a natural selection-inspired way, reveal previously unconstructed Calabi-Yau manifolds.

New Calabi-Yau manifolds from genetic algorithms

Genetic polytopes

A. Hutsalyuk

B. Pozsgay

M. B. Zvonarev

The spin-spin correlation function of the Hubbard model reveals that finite temperature spin transport in one spatial dimension is diffusive.

Finite temperature spin diffusion in the Hubbard model in the strong coupling limit

Spin diffusion

F. Sheldon

Surprisingly, the number of attractors in the critical Kauffman model with connectivity one grows exponentially with the size of the network.

Number of attractors in the critical Kauffman model is exponential

Exponential Kauffman scaling

S. Chen

A. Nestor

A. Zahabi

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

Mahler measuring the genetic code of amoebae

Mahler measuring amoebae

A new connection between continued fractions and the Bourgain–Gamburd machine reveals a girth-free variant of this widely-celebrated theorem.

On a girth-free variant of the Bourgain–Gamburd machine

Ungrouped machines

G. Caldarelli

E. Arcaute

M. Barthelemy

M. Batty

C. Gershenson

D. Helbing

S. Mancuso

Y. Moreno

J. J. Ramasco

C. Rozenblat

A. Sánchez

J. L. Fernández-Villacañas

A complexity-science approach to digital twins of cities views them as self-organising phenomena, instead of machines or logistic systems.


The role of complexity for digital twins of cities

Complex digital cities

B. Hanson

M. Rudnev

D. Zhelezov

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.

The sum-product problem for integers with few prime factors

Sum-product with few primes

This obituary celebrates the life and work of John Keith Stuart McKay, highlighting the mathematical miracles for which he will be remembered.



John Keith Stuart McKay: 1939-2022

On John McKay

A. Bulatov

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


Scaling transformer to 1m tokens and beyond with RMT

BERT enhanced with recurrence

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

Some new results on the higher energies I

Higher energies

O. Klurman

M. W. Xu

The distribution of partial sums of a Steinhaus random multiplicative function, of polynomials in a given form, converges to the standard complex Gaussian.

On the random Chowla conjecture

Random Chowla conjecture

A. Esterov

L. Lang

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

Bernstein-Kouchnirenko-Khovanskii with a symmetry

Symmetric spatial curves

A new, simple approach to the critical Kauffman model with connectivity one sharpens the bounds on the number and length of attractors.

A simple solution of the critical Kauffman model with connectivity one

Landau meets Kauffman

K. Lee

T. Oliver

AI can predict invariants of low genus arithmetic curves, including those key to the Birch-Swinnerton-Dyer conjecture—a millennium prize problem.

Machine learning invariants of arithmetic curves

AI for arithmetic curves

A new, simpler approach to the critical Kauffman model with connectivity one reveals that it has more attractors than previously believed.

Exact dynamics of the critical Kauffman model with connectivity one

Single-input Boolean networks

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

Dealing with diffuse contaminants in single-droplet sequencing of transduced cells

Cell soup in screens

L. Cangemi

M. Chiodaroli

H. Johansson

P. Pichini

E. Skvortsov

Effective field theories for Kerr black holes, showing the 3-point Kerr amplitudes are uniquely 
predicted using higher-spin gauge symmetry.

Kerr black holes enjoy massive higher-spin gauge symmetry

Kerr black holes symmetry

P. Villegas

T. Gili

A. Gabrielli

Applying diffusion-based graph operators to complex networks identifies the proper spatiotemporal scales by overcoming small-world effects.

Laplacian renormalization group for heterogeneous networks

Network renormalization

M. Cheung

J. Li

Cluster variables in Grassmannian cluster algebras can be classified with HPC by applying the tableaux method up to a fixed number of columns.

Clustering cluster algebras with clusters

Clustered cluster algebras

N. Moshchevitin

B. Murphy

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

On Korobov bound concerning  Zaremba's conjecture

Bounding Zaremba’s conjecture

A. Kolchinsky

F. Caravelli

Balancing memory from linear components with nonlinearities from memristors optimises the computational capacity of electronic reservoirs.

Computational capacity of  LRC, memristive and hybrid reservoirs

Optimal electronic reservoirs

J. Bao

The algebra of a toric quiver gauge theory recovers the Bethe ansatz, revealing the relation between gauge theories and integrable systems.

A note on quiver Yangians and ℛ-matrices

Gauge theory and integrability

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

Flowers of immortality

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

Regulatory motifs: structural and functional building blocks of genetic computation

Structure of genetic computation

J. Ipiña

A neural network learns to classify different types of spacetime in general relativity according to their algebraic Petrov classification.

Machine-learning the classification of spacetimes

AI classifies space-time

Disentangling links in networks

Clusters of neurons