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
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Fall 2024
Sagun Chanillo
Subtitle:
Minimal Surface Theory
Course Description:
Minimal Surfaces are area minimizing surfaces and therefore have vanishing mean curvature. Topics we will cover:
1. Computation of first variation of area.
2. Monotonicity formula.
3. Bernstein theorem, that complete graphs that are area minimizing in R^3 are flat planes. Global Isothermal coordinates.
4. Weierstrass parametrization via meromorphic functions of classical minimal surfaces.
5. Construction of minimal surfaces, Catenoids, Enneper surface, Scherck surface(which is periodic) using the Weierstrass parametrization.
6. Using Runge's theorem in complex analysis and the work of Jorge and Xavier and Nadirashvillli to construct a bounded, complete, minimal surface---a conjecture of ST Yau.
7. Omission of points by the Gauss map of complete, minimal surfaces. The theorems of Osserman, Fujimoto and Xavier.
8. Second variation formula for the Area functional.
9. Stability theorems for minimal surfaces, the theorems of D. Fischer-Colbrie and R. Schoen and do Carmo-Peng.
10. Setting up the Plateau problem and solving it. Morrey's \epsilon conformality theorem.
11. Theorems on total curvature of Cohn-Vossen and Huber, application to Minimal surfaces and relation to winding number of the non-compact ends. Costa's surface.
12. Survey of the regularity theory of minimal surfaces. Theorems of Osserman and others.
Text:
I have notes for this course. No textbook.
Prerequisites:
501, 502, 503.
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Spring 2024
Maxime Van de Moortel
Subtitle:
Nonlinear wave equations
Course Description:
Wave equations are ubiquitous in the description of physical phenomena ranging from compressible fluids (sound waves) to General Relativity (gravitational waves).
The course will propose an introduction to the mathematical theory of nonlinear wave equations, with a focus on global existence and asymptotic dispersive behavior for large time.
Time permitting and depending on students' interests, we will also discuss more advanced applications such as the stability of Minkowski spacetime in General Relativity or the decay of wave equations on black holes.
Text:
The course will be based on Jonathan Luk's lecture notes on nonlinear wave equations, available online here: https://web.stanford.edu/~jluk/NWnotes.pdf
Prerequisites:
Basic knowledge of functional analysis (such as Sobolev embedding) will be assumed.
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Spring 2022
Avy Soffer
Course Description:
Selected Examples from Modern Analysis:
This will be an invitation to the field of math Analysis.
Examples from various directions will be described. Hilbert Space manifestations in Physics, PDE, and other sciences. The Spectral Theorem in action. The notion of Commutator in Analysis, Geometry, Lie Groups.
Calculus in Infinite Dimensions. Waves.
Text:
Functional Analysis, by Reed-Simon part I
Prerequisites:
Real Analysis
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Fall 2019
Avy Soffer
Subtitle:
Introduction to Spectral Theory
Text:
Reed&Simon , Functional Analysis I, Operator Theory part 4, by Simon.
Prerequisites:
Real Analysis, Complex Analysis, Linear Algebra
Course Description:
This is an introductory course to those interested in the aspects of spectral theory and its applications. It may be of interest to those who work in PDE, Math-Phys, Analytic Number Theory and Analysis on Manifolds.
I begin with a review of the basics of Hilbert spaces and then discuss the theory of linear operators on Hilbert spaces.
The needed results from Functional Analysis will be detailed in class. Topics include d then are the spectral theorem and its applications, compact operators, trace class and Hilbert Schmidt operators, unbounded self-adjoint operators.
Schedule of Sections:
16:640:511 Schedule of Classes