Mathematicians are developing new tools to measure the gaps between data points—whether those points are days of heavy rainfall, prime numbers, or matching letters in DNA sequences. This matters because current methods only capture average gap sizes or the most extreme gaps, but miss the patterns in where clusters of large, medium, or small gaps occur. For example, heavy rainfall tends to cluster: a rainy day makes another more likely. Understanding these gap clusters could improve predictions in weather, energy spectra, and genetic sequence matching. If successful, the research will create a unified mathematical framework—a "random measure"—that captures both average and extreme gap behaviour as the number of data points grows. This is fundamental science with no immediate practical application. However, similar advances in probability and dynamical systems have historically underpinned improvements in climate modelling, telecommunications, and DNA analysis. A deeper understanding of gap statistics could eventually sharpen forecasts in hydrology, refine energy grid design, or accelerate pattern-matching in genomics.
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Characterising and quantifying random and deterministic behaviours in data, which can be as diverse as rainfall data or the distribution of prime numbers, is a fundamentally important topic. This proposal is based in Ergodic Theory and Dynamical Systems, but motivated by these problems as well as questions around quantifying energy spectra, questions in sequences seen in Number Theory, and by matching sequences of symbols in DNA. Such quantifications can be understood in terms of the average long-term random-type signatures in the data, along with possible rigid clustering behaviour (for example a day of heavy rainfall is more likely if the previous day had heavy rainfall, so we would expect clustering here alongside more random phenomena). In this proposal we focus on the sizes of gaps between data points, which can generally be called gap statistics. We will use state-of-the-art mathematical tools, and to develop new ones, to more fully understand gap statistics, capturing the locations and structure of clusters of gaps of a given type (large, medium or small) for the first time. In problems in energy spectra and number theory the gaps in the data has been previously looked at on the average scale, while in the setting of sequences of symbols (which has applications for DNA), where the size of gaps correspond to how similar two sequences are to each other, gap statistics has previously been looked at on the extreme scale (long strings of symbols matching each other). The location of these gaps and their clustering has not been considered previously. Recently tools from probability, used in a dynamical systems context have become better suited to capturing both of these scales in a random measure, a tool which can incorporate detailed asymptotic behaviour if convergence can be proved, i.e., taking the number of data points to be larger and larger and rescaling the expected average or extreme gap accordingly and seeing if there is a limiting measure. Proving such convergence and fully characterising these measures will be a significant challenge, but developments in both the dynamical and number theoretic setting mean that now is the time to create and benefit from synergies in theories across these different areas of mathematics and lay the groundwork for a significant multifaceted research programme.
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