# Mathematics that unifies

Understanding the relations between different branches of pure mathematics, and creating overarching theories that bind them together.

Mathematics is central to theoretical research, and advancing pure mathematics advances science itself. This is especially true for mathematics that unifies seemingly unconnected fields.

Dualities play a key role in how we form insights. Examples include the Langlands programme, monstrous moonshine, the ADE classification and dualities across quantum field and string theories. We investigate extensions of these, and develop their consequences. Can we develop other dualities, especially ones that exploit our intuition for geometry and arithmetic?

To bring rigour to network science, we investigate graphical notions of geometry and topology compatible with their continuum analogues. We develop a theory of statistical inference suitable for high dimensions, which is the basis for much of artificial intelligence. There are compelling arguments for a computational understanding of the universe, and we seek mathematical insights into the scope and limits of computation, particularly the output of simple rules.

Embedding pure mathematics within theoretical science keeps theorists abreast of the latest tools, and inspires mathematicians to take up new directions. Can we re-conceive mathematics as a core part of science, rather than an adjunct, thereby ensuring it is favoured and funded in proportion to its value?

## Related papers

### 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.

### The popularity gap

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.

### Counting recursive divisors

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

### Infinite dimensional irreducibility

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

### Recursive divisor properties

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

### 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.

### Genetic polytopes

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

### Exponential Kauffman scaling

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

### Mahler measuring amoebae

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

### Ungrouped machines

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

### 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.

### On John McKay

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

### 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.

### Random Chowla conjecture

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

### 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.

### AI for arithmetic curves

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

### Single-input Boolean networks

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

### Gauge theory and integrability

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

### Algebra of melting crystals

Certain states in quantum field theories are described by the geometry and algebra of melting crystals via properties of partition functions.

### Set additivity and growth

The additive dimension of a set, which is the size of a maximal dissociated subset, is closely connected to the rapid growth of higher sumsets.

### Mahler measure for quivers

Mahler measure from number theory is used for the first time in physics, yielding “Mahler flow” which extrapolates different phases in QFT.

### Recursively divisible numbers

Recursively divisible numbers are a new kind of number that are highly divisible, whose quotients are highly divisible, and so on, recursively.

### Transitions in loopy graphs

The generation of large graphs with a controllable number of short loops paves the way for building more realistic random networks.

### QFT and kids’ drawings

Groethendieck's “children’s drawings”, a type of bipartite graph, link number theory, geometry, and the physics of conformal field theory.

### Energy bounds for roots

Bounds for additive energies of modular roots can be generalised and improved with tools from additive combinatorics and algebraic 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.

### Bursting dynamic networks

A mathematical model captures the temporal and steady state behaviour of networks whose two sets of nodes either generate or destroy links.

### Hypercube eigenvalues

Hamming balls, subgraphs of the hypercube, maximise the graph’s largest eigenvalue exactly when the dimension of the cube is large enough.

### Exactly solvable random graphs

An explicit analytical solution reproduces the main features of random graph ensembles with many short cycles under strict degree constraints.

### Erdős-Ko-Rado theorem analogue

A random analogue of the Erdős-Ko-Rado theorem sheds light on its stability in an area of parameter space which has not yet been explored.

### Spin systems on Bethe lattices

Exact equations for the thermodynamic quantities of lattices made of d-dimensional hypercubes are obtainable with the Bethe-Peierls approach.

### Maximum percolation time

A simple formula gives the maximum time for an n x n grid to become entirely infected having undergone a bootstrap percolation process.

### Random graphs with short loops

The analysis of real networks which contain many short loops requires novel methods, because they break the assumptions of tree-like models.

### Entropies of graph ensembles

Explicit formulae for the Shannon entropies of random graph ensembles provide measures to compare and reproduce their topological features.

### Percolation on Galton-Watson trees

The critical probability for bootstrap percolation, a process which mimics the spread of an infection in a graph, is bounded for Galton-Watson trees.

### Random close packing fractions

Lognormal distributions (and mixtures of same) are a useful model for the size distribution in emulsions and sediments.

### Unbiased randomization

Unbiased randomisation processes generate sophisticated synthetic networks for modelling and testing the properties of real-world networks.

### Tailored random graph ensembles

New mathematical tools quantify the topological structure of large directed networks which describe how genes interact within a cell.

### Ever-shrinking spheres

Techniques from random sphere packing predict the dimension of the Apollonian gasket, a fractal made up of non-overlapping hyperspheres.

### Random cellular automata

Of the 256 elementary cellular automata, 28 of them exhibit random behavior over time, but spatio-temporal currents still lurk underneath.