Analysis on non-Archimedean field extensions of the real numbers

实数领域R由于其特殊的性质在数学和科学中起着重要的作用。这个场是阿基米德式的：如果R中的x y等于0 b| y|。R的这个性质对应于测量的常识性经验；但阿基米德公理在普朗克尺度下就失效了，即距离小于1.6 × 10-35米，持续时间小于5.4 × 10-44秒。此外，实数在解释直观的科学概念方面存在缺陷；例如，由于缺乏无穷小，导数作为微分商的思想不能在R内严格地表述。由于连续体的精细结构是无法通过科学手段观察到的，所以阿基米德性不是自然界所需要的，把它抛在脑后可能会为上述问题提供解决方案，并使我们更好地了解宇宙。因此我对R的非阿基米德域扩展很感兴趣。*******我的研究重点一直放在列维-奇维塔域R上，它是实数的最小的非阿基米德域扩展，在有序拓扑中是实闭的和完全的。这个领域足够小，所以它的数字可以在计算机上实现，允许计算应用，其中一个是快速准确地计算实值函数的导数，直到高阶。*******在接下来的五年里，我将扩展我的研究重点，首先将我在列维-奇维塔场的工作推广到任何包含实数的非阿基米德场F，即在有序拓扑中是实闭的和完备的，其哈恩群是阿基米德的。然后，我将在F上进行新的研究问题，这些问题在经典分析、概率论、生物学、理论物理、宇宙学等科学和工程领域有潜在的应用。扩大我的研究范围将使更多的数学家对它更感兴趣，并将为非阿基米德分析的新合作打开大门。我计划的研究跨越了应用数学（如计算应用）和纯数学（如单变量和多变量微积分、泛函分析、拓扑学、复变分析、微分方程解的存在性和唯一性、特殊函数等）的许多领域*******我计划在未来五年内招收具有较强数学背景的本科生、硕士和博士与我一起研究拟议的研究目标。学生将在我的研究小组中接受的培训将使他们准备好领导成功的学术事业（大学教授或学校教师）或在公司中获得成功的专业工作，他们将获得先进的分析和/或计算技能，这将使他们在竞争相同工作的其他候选人中占有优势

The field of real numbers R plays a fundamental role in Mathematics and the sciences due to certain special properties. The field is Archimedean: if x,y in R are such that 0 |y|. This property of R corresponds to common sense experiences of measurement; but the Archimedean axiom breaks down at the Planck scale, i.e. for distances less than 1.6x10-35 m and durations less than 5.4x10-44 s. Moreover, the real numbers have shortcomings in interpreting intuitive scientific concepts; e.g. the idea of derivatives as differential quotients cannot be formulated rigorously within R due to the lack of infinitesimals. Since the fine structure of the continuum is not observable by means of science, Archimedicity is not required by nature, and leaving it behind may provide solutions for the aforementioned problems and allow a better understanding of the universe. Hence my interest in non-Archimedean field extensions of R in general.*******The focus of my research has been on the Levi-Civita field R which is the smallest non-Archimedean field extension of the real numbers that is real closed and complete in the order topology. The field is small enough so that its numbers can be implemented on a computer, allowing for computational applications, one of which is the fast and accurate computation of the derivatives of real-valued functions up to high orders.*******In the next five years, I will expand my research focus by first generalizing my work on the Levi-Civita field to any non-Archimedean field F that contains the real numbers, that is real closed and complete in the order topology, and whose Hahn group is Archimedean. Then I will work on new research problems on F with potential applications in Classical Analysis, Probability, Biology, Theoretical Physics, Cosmology and other fields of Science and Engineering. Enlarging the scope of my research will make it more interesting to a wider audience of mathematicians and will open the door to new collaborations in non-Archimedean Analysis. My proposed research spans many areas of Applied Mathematics (e.g. computational applications) and Pure Mathematics (e.g. one-variable and multi-variable Calculus, Functional Analysis, Topology, Complex Analysis, existence and uniqueness of solutions of differential equations, special functions, etc.)*******I plan to recruit undergraduate, M.Sc. and PhD students with strong mathematical background in the next five years to work with me on the proposed research objectives. The training that the students will receive in my research group will prepare them to lead successful academic careers (professors at universities or teachers in schools) or successful professional jobs in companies where the advanced analytical and/or computational skills they will have acquired will give them an advantage over other candidates competing for the same jobs.***