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?
LCP
Related papers
AV
SubmittedGroup representation irreducibility
A general approach to proving the irreducibility of representations of infinite-dimensional groups within the frame of Ismagilov's conjecture.
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Exactly solvable random graphs
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Random close packing fractions
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Unbiased randomization
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New mathematical tools quantify the topological structure of large directed networks which describe how genes interact within a cell.
Ever-shrinking spheres
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Random cellular automata
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