Model theory of analytic functions

例如，测量钟摆随时间推移的位置的函数，在理想条件下（如没有摩擦）是非常光滑的，没有突然的跳跃，事实上，它属于一类称为“解析”的非常光滑的函数。从抽象数学到现实世界现象的模型，解析函数自然地出现在各种各样的环境中。这样的函数在局部表现良好；对于时间的函数，如钟摆的例子，它意味着在足够短的时间间隔内表现良好。但是，当从全球角度考虑时，也就是在越来越长的时间之后，可能会出现完全不同的行为，例如指数增长（例如，随着时间的推移积累利息的债务规模），指数衰减（放射性废物的放射性）或振荡（如没有摩擦的钟摆）。模型理论是数学逻辑的一个分支，它对振荡函数和非振荡函数之间的解析函数进行了明确的区分，并进一步区分了振荡函数。一个实解析函数如果由涉及该函数的一阶公式所定义的每一组实数都由有限多个点和区间构成，则该函数处于“o-极小结构”，因此不可能出现振荡。另一方面，如果由一阶公式定义的每一组复数是可数的，或者它的补是可数的，则复解析函数处于“拟极小结构”，该函数一超越就必须振荡。这个奖学金的目的是阐明在数学中感兴趣的函数类的极小性和拟极小性。自20世纪90年代以来，人们就知道了实数幂的0极小性，从数论到分析，它一直对数学产生重大影响。是否存在比指数增长快得多的0 -极小函数（称为转指数）仍然是一个开放的问题，这将对动力系统产生影响，例如关于多项式向量场的希尔伯特第16个问题。复幂的准极小性仍然是模型理论中的一个大问题，在它首次被推测25年后，一个积极的答案可能会产生深远的影响，就像实幂的零极小性一样。我将使用Conway的超现实数（实数的无限和无穷小数的扩展，包括实数和序数）来研究Hardy域，这是一类实数非振荡函数，以及transseries，这是为了表示它们的形式渐近展开；特别是处理o极小转幂函数的存在性问题。我将通过加强最近发现的超现实数、跨列和哈代场之间的联系来做到这一点；非实数上的转幂函数模型理论的建立与分析引入一个框架，将超现实数上的函数、非振荡函数和极小性联系在一起。此外，我将研究复指数和类似结构的拟极小性。我将通过证明指数代数闭包的实例来做到这一点，它预测多项式指数方程系统何时应该有复解，并将结果扩展到由阿贝尔变量及其扩展产生的其他指数函数，为包含交换代数群的所有指数函数的通用拟极小结构铺平道路。

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 real-world 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 "o-minimal structure" if every set of real numbers defined by a first-order 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 "quasi-minimal 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 o-minimality and quasi-minimality of classes of functions of interest in mathematics. The o-minimality 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 o-minimal 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 quasi-minimality 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 far-reaching consequences as did the o-minimality 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 non-oscillating functions, and transseries, which are formal asymptotic expansions meant to represent them; and in particular tackle the problem of the existence of o-minimal 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, non-oscillating functions and o-minimality.Furthermore, I will investigate the quasi-minimality of complex exponentiation and analogous structures. I will do so by proving instances of exponential-algebraic closure, which predicts when systems of polynomial-exponential 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 quasi-minimal structure containing all the exponential functions of commutative algebraic groups.