Introduction to the differential and integral calculus in one real variable: fundaments of mathematical thinking, numbers, sequences, basic point set topology, continuity, differentiable functions, ordinary differential equations, Riemann integration.

Introduction to differential and integral calculus in several real variables, vector calculus: differential, partial derivative, implicit functions, inverse function theorem, minima with constraints; Riemann integral, vector fields, differential forms, path integrals, surface integrals, divergence theorem, Stokes' theorem.

The focus of this class is on Direct Methods in the Calculus of Variations. We will address the following topics:1) existence of minimizers to classical variational problems;2) regularity of minimizers to scalar and vectorial problems;3) regularity theory for elliptic PDEs with measurable and smooth coefficients.

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