Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.

The goal of this class is to give an introduction to harmonic analysis, covering a series of classical important results such as:1) Convergence properties of Fourier series2) Interpolation theory3) Hardy-Littlewood Maximal inequality3) Calderón-Zygmund theory4) Hardy and BMO spaces5) Littlewood-Paley decomposition

In this class, we will explore the mathematical theory of fluid dynamics. We will cover classical and modern techniques related to nonlinear partial differential equations (PDEs), including the Euler equations (nonlinear hyperbolic PDEs) and the Navier-Stokes equations (nonlinear parabolic PDEs).

In this course, we will explore the most fundamental and classical topics in Harmonic Analysis, including maximal functions, Marcinkiewicz interpolation, singular integrals, Calderon-Zygmund theory, and Littlewood-Paley theory.After an introductory session led by the instructor, participants will present seminar talks each week.