Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

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 <|x| <|y| then we can find a positive integer n such that n|x| > |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.