Terence Tao- The Circle Method from the Perspective of Higher Order Fourier Analysis

Explore higher order Fourier analysis in this comprehensive lecture by Terence Tao at the Hausdorff Center for Mathematics. Delve into the intricacies of controlling multilinear averages that are beyond the reach of conventional linear Fourier analysis methods. Discover how nilsequences replace linear phase functions in this advanced theory. Gain insights into revisiting the linear circle method from a higher order perspective, with a focus on downplaying the role of Fourier identities. Learn about major and minor arcs, the complexity of problems, the lambda3 form, inverse theorems, and generalized nominal inequalities. Investigate U2 identity, structure theorems, equidistribution theory, and bracket quadratic analysis. Explore connections to group theory, the Heisenberg manifold, and densification techniques in this hour-long, in-depth exploration of advanced mathematical concepts.

Introduction
The circle method
Fourier analysis
Major and minor arcs
Problems with the circle method
Higher order Fourier analysis
Complexity
Identity
Revisiting the circle method
Finding the complexity of the problem
The lambda3 form
The inverse theorem
Proof by identities
Generalized nominal inequality
U2 identity
Structure theorem
Equidistribution theorem
Integer part operation
Bracket quadratic analysis
Equidistant theory
Heisenberg manifold
Group theory
Densification