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.