Title of the Course: An Introduction to Zeta- and L-Functions: Riemann, Weil, and Beyond
Instructor: Mr. James Douglas Boyd
Institution: Institute for the Future of Mathematics (IFUMA)
Dates: 12-18 August 2024
Prerequisites: Some familiarity with at least some of the following will be helpful: introductory number theory, complex analysis, abstract algebra, linear algebra.
Level: Most appropriate for graduate and advanced undergraduate students, but all are invited.
Abstract:
Together, zeta- and L-functions are often understood as providing a “door” to contemporary number theory. Although fascinating in their own right, zeta- and L-functions have continued to find nuanced applications in various areas of number theory and geometry: thus, they offer a door to a house with many rooms. This course is intended to provide an introduction to such functions and sample the different kinds of roles that they play across interrelated fields of contemporary mathematics.
The course will begin with an introduction to the Riemann zeta function and the Riemann hypothesis, as well as the motivation for the latter (i.e. the prime-counting function). Students will then be given an introduction to the greater Selberg Class of functions that generalize the properties of the zeta function. Students will be introduced to the Weil Conjectures as an algebro-geometric generalization of the Riemann hypothesis. Finally, case studies in the application of L-functions to class field theory, such as those of Artin and Iwasawa, will be discussed.
Language: English