The function that measures, for instance, the position of a pendulum as time passes, is under ideal conditions (such as no friction) very smooth, without sudden jumps, and it belongs in fact to a class of very smooth functions called "analytic".
Analytic functions emerge naturally in a wide variety of contexts, from abstract mathematics to models of realworld phenomena. Such functions are well behaved locally; for functions of time, as in the example of the pendulum, it means well behaved in sufficiently short time intervals. But wildly different behaviours can emerge when considered globally, i.e. after longer and longer periods of time, such as exponential growth (e.g. the size of a debt accumulating interest over time), exponential decay (the radioactivity of radioactive waste), or oscillation (as per a pendulum without friction).
Model theory, a branch of mathematical logic, provides a sharp divide between analytic functions that oscillate and functions that do not, and a further distinction between oscillating functions. A real analytic function lies in an "ominimal structure" if every set of real numbers defined by a firstorder formula involving the function is made of finitely many points and intervals, thus no oscillation may appear. On the other hand, a complex analytic function, which must oscillate as soon as it is transcendental, lies in a "quasiminimal structure" if every set of complex numbers defined by a first order formula is either countable, or its complement is countable.
The aim of this fellowship is to shed light on the ominimality and quasiminimality of classes of functions of interest in mathematics. The ominimality of real exponentiation, known since the 1990s, had and still has a major impact across mathematics, from number theory to analysis. It is still an open problem whether there are ominimal functions that grow much faster than exponentially (called transexponential), which would have implications on dynamical systems, such as around Hilbert's 16th problem on polynomial vector fields. The quasiminimality of complex exponentiation is still one of the big problems in model theory, 25 years after it was first conjectured, and a positive answer is likely to have farreaching consequences as did the ominimality of real exponentiation.
I will use Conway's surreal numbers (an extension of real numbers with infinite and infinitesimal numbers, encompassing both reals and ordinals) to investigate Hardy fields, which are classes of real nonoscillating functions, and transseries, which are formal asymptotic expansions meant to represent them; and in particular tackle the problem of the existence of ominimal transexponential functions. I will do so by strengthening the recently discovered connections between surreal numbers, transseries and Hardy fields; creating and analysing the model theory of transexponential functions on surreal numbers; introducing a framework that ties together functions on surreal numbers, nonoscillating functions and ominimality.
Furthermore, I will investigate the quasiminimality of complex exponentiation and analogous structures. I will do so by proving instances of exponentialalgebraic closure, which predicts when systems of polynomialexponential equations should have complex solutions, and extending the results to other exponential functions arising from abelian varieties and their extensions, paving the way for a universal quasiminimal structure containing all the exponential functions of commutative algebraic groups.
