Analysis on non-Archimedean field extensions of the real numbers

真实的数域R由于某些特殊的性质在数学和科学中起着基础性的作用。域是阿基米德的：如果R中的x，y使得0| y|. R的这一性质符合测量的常识;但阿基米德公理在普朗克尺度下失效，即距离小于1.6x10-35 m，持续时间小于5.4x10-44 s。此外，真实的数在解释直观的科学概念方面有缺点;例如，由于缺乏无穷小，导数作为微分导数的想法不能在R中严格地表达。由于连续统的精细结构无法通过科学手段观察到，因此自然界不需要Archimedicity，将其留在后面可能会为上述问题提供解决方案，并允许更好地理解宇宙。因此，我对R的非阿基米德域扩展很感兴趣。我的研究重点一直在列维Civita领域R这是最小的非阿基米德领域的真实的号码，是真实的封闭和完整的顺序拓扑扩展。该域足够小，因此其数字可以在计算机上实现，从而允许计算应用，其中之一是快速准确地计算高阶实值函数的导数。在接下来的五年里，我将扩大我的研究重点，首先推广我的工作Levi-Civita领域的任何非阿基米德领域F包含的真实的数字，即真实的封闭和完整的顺序拓扑，其哈恩群是阿基米德。然后，我将致力于新的研究问题，在经典分析，概率论，生物学，理论物理，宇宙学和其他科学和工程领域的潜在应用。扩大我的研究范围将使更广泛的数学家更感兴趣，并将为非阿基米德分析的新合作打开大门。我拟从事的研究涵盖应用数学（例如计算应用）和纯数学（例如单变量和多变量微积分、泛函分析、拓扑学、复分析、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、硕士生和博士生，与我一起完成所提出的研究目标。学生将在我的研究小组中接受的培训将使他们做好准备，领导成功的学术生涯（大学教授或学校教师）或在公司中成功的专业工作，他们将获得先进的分析和/或计算技能，这将使他们比竞争相同工作的其他候选人更具优势。

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.***