Mathematicians are building a logical toolkit to count solutions to equations across all finite fields at once, not just one prime at a time. This matters because finite fields—arithmetic systems where calculations wrap around like a clock—underpin modern cryptography, coding theory, and digital communications. A classical theorem already shows that the theory of finite fields is decidable, and that formulas reduce to simple geometric forms. But when engineers or cryptographers need uniform behaviour across all primes, the computational approach breaks down. The researchers aim to extend existing model-theoretic results to capture not just whether certain configurations exist, but how many there are, with precise asymptotic counts. The project is fundamental mathematics, not applied research. It draws on deep 20th-century geometry—the l-adic cohomology of Weil, Grothendieck, and Deligne—and aims to connect it to model-theoretic frameworks like simplicity and distality. If successful, the work could provide a unified logical language for incidence bounds and group actions in positive characteristic. Historically, such foundational advances in model theory have later found unexpected applications in number theory and cryptography, though no immediate practical outcome is claimed.
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Gödel showed that mathematics as a whole cannot be given a complete axiomatization. But as Tarski discovered in the case of real algebraic geometry, many rich branches of mathematics are built around an axiomatizable core. Such an axiomatic treatment is of foundational interest, but can also lead to concrete and meaningful new results within the field itself. Model theory is the branch of mathematical logic seeking to extend the range of decidable theories to new areas of mathematics. Examples include Tarski’s theorem on real algebraic geometry, concerning shapes described by polynomial inequalities, the expansion to bounded analytic functions on a bounded domain by Denef and van den Dries, and Wilkie’s incorporation of the exponential function. Applications, notably the Pila-Zannier method in the study of unlikely intersections, have been spectacular. At the same time, model theory develops structural tools to utilize the axiomatic or definable framework effectively. The standard of depth was achieved for the relatively restricted class of stable theories, largely by Shelah; he also proposed much more comprehensive extensions to simple and NIP theories, developed further by Kim-Pillay, Peterzil-Starchenko, Chernikov-Simon, and many others. These theories are often based first on a notion of dimension, and frequently of definable measure. Our intertwined projects all aim to deepen or extend these notions in a number of geometrically rich settings. A) Each individual finite field - capturing arithmetic modulo a prime - is accessible to computation. But this breaks down when one wants uniformity in the prime. A classical theorem of Ax says nevertheless that the theory of finite fields is decidable, and all formulas reduce to certain geometrically simple ones. Further, by results of Chatzidakis-van-den-Dries-Macintyre, this theory not only knows whether configurations with certain properties exist, but even counts them, asymptotically, via a definable dimension and measure theory. We aim to extend this to a structure more precise than a measure, giving also the next-order asymptotics. The necessary structures are known in geometry (l-adic local systems and constructible sheaves and their cohomology, constructed by Weil, Grothendieck, Deligne in one of deepest and most beautiful episodes of 20th century mathematics), but have not been treated model-theoretically; this requires extensions of the existing mathematical results. We expect the resulting theory to be simple and thus connect to deep areas of model theory, that may further resonate with the geometric landscape. B) Pseudofinite dimensions are relevant beyond pseudofinite fields. The Pila-Zannier methods evoked earlier depend on the Pila-Wilkie theorem, which roughly asserts that the set of rational points of bounded complexity satisfying definable but non-algebraic conditions has pseudofinite dimension zero. Meanwhile, asymptotic incidence bounds on sizes of finite configurations translate to bounds on pseudofinite dimensions. Model theory exposes a logical source, namely distality, of many such incidence bounds, and provides a toolbox for obtaining algebraic consequences, in particular via group reconstruction theorems. We aim to extend the reach of both arms, pushing distality to the realm of positive characteristic fields and employing further model-theoretic tools to treat group actions. C) In the NIP setting, definable measures are very well behaved. By work of EH, Peterzil, Pillay, Simon, they have compact symmetry groups. We discuss a very intriguing conjecture that these compact groups are actually made up from the very classical class of Lie groups over local fields.
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