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

  • AVA. V. Kosyak Submitted

    Group 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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    Modelling the final state of a mobile impurity particle immersed in a one-dimensional quantum fluid after the abrupt application of a force.

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    Infinitely high parallelotopes

    We demonstrate that the height of an infinite parallelotope is infinite if no non-trivial combinations of its edges belong to l2(N)l_2(\mathbb N).

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    The popularity gap

    A cyclic group with small difference set has a nonzero element for which the second largest number of representations is twice the average.

  • Submitted

    Counting recursive divisors

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

  • Submitted

    Infinite dimensional irreducibility

    The criteria of irreducibility of representations of the inductive limit of certain general linear groups acting on three infinite rows.

  • Arxiv

    Recursive divisor properties

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

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  • Submitted

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  • Physics Letters B

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  • Physical Review Letters

    Kauffman cracked

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

  • Advances in Theoretical and Mathematical Physics

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

  • Finite Fields and Their Applications

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

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    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.

  • Bulletin of the London Mathematical Society, in press

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

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

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    We study the geometry of generic spatial curves with a symmetry in order to understand the Galois group of a family of sparse polynomials.

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    The dynamics of the Kauffman network can be expressed as a product of the dynamics of its disjoint loops, revealing a new algebraic structure.

  • Journal of High Energy Physics

    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.

  • Journal of High Energy Physics

    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.

  • Discrete Mathematics, in press

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    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.

  • Communications in Mathematical Physics

    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.

  • Journal of Number Theory

    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.

  • Arxiv

    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.

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

  • Journal of the Institute of Mathematics of Jussieu, in press

    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.

  • Arxiv

    Ample and pristine numbers

    Parallels between the perfect and abundant numbers and their recursive analogs point to deeper structure in the recursive divisor function.

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

  • European Journal of Combinatorics

    Hypercube eigenvalues

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

  • Journal of Physics A

    Exactly solvable random graphs

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

  • Forum of Mathematics, Sigma

    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.

  • Journal of Physics A

    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.

  • SIAM Journal on Discrete Mathematics

    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.

  • ESAIM: Proceedings and surveys

    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.

  • Journal of Physics A

    Entropies of graph ensembles

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

  • Electronic Journal of Probability

    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.

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    Random close packing fractions

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

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    Unbiased randomization

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  • Journal of Physics A

    Tailored random graph ensembles

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  • Physical Review E

    Ever-shrinking spheres

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

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    Random cellular automata

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