Fall 2026
Mariusz Mirek
Subtitle:
Exponential sum estimates in analysis and number theory
Course Description:
In this graduate-level course titled "Exponential Sum Estimates in Analysis and Number Theory," we will delve into classical exponential sum estimates and their applications in estimating the Riemann zeta function. The course will begin with a comprehensive overview of these fundamental concepts, followed by a focused discussion on decoupling estimates for the moment curve.
We will decoupling estimates to derive the Vinogradov mean value theorem, which will provide the basis for the latest results concerning Weyl's inequality, the van der Corput test, and the Lindelöf conjecture. If time allows, we will also explore recent breakthroughs by Guth and Maynard, highlighting their new large value estimates for Dirichlet polynomials.
This course aims to equip students with both the theoretical understanding and practical tools essential for utilizing exponential sum estimates in research related to modern analysis and analytic number theory.
Text:
No textbook.
Prerequisites:
Real and complex analysis at the level of 501 and 503 respectively
Schedule of Sections:
16:640:511 Schedule of Classes
Previous Semesters:
- Fall 2026 - Mirek, Mariusz
- Fall 2024 - Chanillo, Sagun
- Spring 2024 - Van De Moortel, Maxime
- Spring 2022 - Soffer, Avraham
- Fall 2019 Prof. A. Soffer