p-adic bi-analytic Geometry (25rit033)

The Banff International Research Station will host the "p-adic bi-analytic Geometry" workshop in Banff from February 9 to February 16, 2025.

Since the early days of geometry, mathematicians have pondered the notion of irrationality and transcendence. For example: can $\pi$, the ratio of the circumference of a circle to its diameter, be expressed as a fraction with whole number numerator and denominator? (Is $\pi$ a rational number?) If not, is it at least the root of a polynomial with whole number coefficients? (Is $\pi$ algebraic instead of transcendental?) The Lindemann-Weierstrass theorem, a deep result in modern mathematics, shows that $\pi$ is transcendental, but for most real numbers arising from nature and geometry it is still a very difficult question to determine whether they are transcendental or algebraic (although we have precise conjectural expectations in many cases!).

This project is concerned with a related question of $p$-adic transcendence in $p$-adic geometry. Here, for $p$ a prime number, the $p$-adic numbers are an alternative to the real numbers that are constructed by measuring distance not in the usual way but instead by using divisibility properties of integers. Although one cannot imagine the associated $p$-adic geometry easily with pictures, it satisfies many of the same formal properties as the usual geometry of our world, making it a powerful tool for mathematical reasoning about integers and divisibility. This Research in Teams project focuses on developing a geometric transcendence theory for certain special shapes in $p$-adic geometry that have previously played a key role in important work in mathematics relating rational solutions of polynomial equations to the analysis of symmetries arising in sound waves and other signals (a part of the Langlands program in number theory). By developing a deeper understanding of their geometry and transcendence, the project members hope to arrive at a better understanding of these unusual shapes and their central role in number theory.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.