Analysis on non-Archimedean field extensions of the real numbers

由于实数R的某些特殊性质，它在数学和科学中起着基础性的作用。这个领域是阿基米德的：如果R中的x，y是0 m，持续时间小于5.4x10-44 S。此外，实数在解释直观的科学概念方面存在缺陷；例如，导数作为微商的想法不能在R中得到严格的表述，因为没有无穷小。由于连续体的精细结构不能通过科学手段观察到，自然并不需要阿基米德性，而将其抛在脑后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，我一般对R的非阿基米德域扩张感兴趣。 我的研究重点是Levi-Civita域R，它是实数的最小非阿基米德域扩张，它在序拓扑中是实闭的和完备的。这个域足够小，以至于它的数值可以在计算机上实现，允许计算应用，其中之一是快速而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，首先将我在Levi-Civita域上的工作推广到任何包含实数的非阿基米德域F，该非阿基米德域F在序拓扑中是实闭的且其Hahn群是阿基米德群。然后对F在经典分析、概率、生物学、理论物理、宇宙学等科学与工程领域具有潜在应用的新问题进行研究。扩大我的研究范围将使其对更广泛的数学家受众更感兴趣，并将为非阿基米德分析领域的新合作打开大门。我的研究涵盖了应用数学(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、复分析、微分方程解的存在唯一性、特殊函数等)的许多领域。 我计划招收本科生，理科硕士。和具有较强数学背景的博士生在未来五年内就提出的研究目标与我共同努力。在我的研究小组中，学生们将接受的培训将使他们为成功的学术生涯(大学教授或学校教师)或在公司成功的专业工作做好准备，在这些公司，他们将获得的高级分析和/或计算技能将使他们相对于竞争相同工作的其他候选人具有优势。

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 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.