Real and complex numbers, vectors, limits, sequences, series, power series, functions, continuity, differentiation and integration in one variable

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.

Differential and Integral calculus in many variables, vector analysis.

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.

Introduction to calculus in one dimension. Building simple models and analysing them mathematically.Functions of one variable: the notion of a function, of the derivative, the idea of a differential equation, complex numbers, Taylor polynomials and Taylor series. The integral of a function of one variable.

Basics about multidimensional analysis.Ordinary differential equations as mathematical models to describe processes (continuation from Analysis A).Numerical, analytical and geometrical aspects of differential equations.

Introduction of mathematics as the universal language for scientific facts:The lecture aims on one hand at learning and exercising the mathematical trade and in the other hand at applying the learnt concept to medical, biological, chemical and mechanical problems.

Consolidation and extension of mathematics as the universal language for scientific facts:The lecture aims on one hand at learning and exercising the mathematical trade and in the other hand at applying the learnt concept to medical, biological, chemical and mechanical problems.

Examples of partial differential equations. Linear partial differential equations. Separation of variables. Fourier series, Fourier transform, Laplace transform. Applications to solving commonly encountered linear partial differential equations (Laplace's Equation, Heat Equation, Wave Equation).

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.