# 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?

### Recursively divisible numbers

Generalizing the divisor function to find a new kind of number that can be recursively divided into parts, for use in design and technology.

### Bootstrap percolation

Advancing the mathematical theory of bootstrap percolation, where active cells on a lattice with few active neighbours cease to be active.

### Structure of how things relate

Creating mathematical tools for characterizing the structure of ideal graphs and irregular networks, and the behaviour of processes on them.

### Spectre of hypercubes

Exploring the spectral properties of subgraphs of the hypercube and Hamming graphs for insights into coding theory and models of evolution.

### Intelligence of graphs

Predicting the behaviour of graphs and processes on them by treating topological patterns as constraints on a random graph ensemble.