The second Scientific Revolution

• 1. Measure and Lebesgue integral

• 2. The theory of functions

• 3. Applications and modern developments

• 1. Topological spaces and metric spaces

• 2. Linear spaces of functions

• 3. Hilbert spaces

• 4. Functional spaces, weak topology and compact operators

• 1. Sufficiency theory

• 2. Carathéodory theory.

• 3. Hilbert ‘direct’ method

• 4. Existence theorem and multiple-integral problems

• 5. The Hilbert-Haar theorem

• 6. Plateau-De Giorgi Problem

• 1. Cauchy problem

• 2. Qualitative theory

• 3. Perturbation methods

• 4. Stability

• 5. Limit problems

• 6. Periodic solutions

• 1. The origins of the modern theory of partial differential equations and Poincaré’s work

• 2. Hilbert’s program

• 3. Bernètejn and the beginning of a priori estimates

• 4. Solvability linear elliptic equations of second order.

• 5. The theory of Leray-Schauder

• 6. Hadamard and the classification of PDEs and their boundary problems

• 7. Weak solutions

• 8. The theory of Schwartz distributions

• 9. Methods in Hilbert spaces

• 10. The maximum principle and applications; De Giorgi-Nash estimates

• 11. Nonlinear evolution equations, fluid flows and gas dynamics

• 12. Nonlinear PDE and nonlinear functional analysis

• 13. Calculation of PDE solutions, numerical analysis and computer science