Foundational Ideas and Concepts

Only interest and curiosity but no higher mathematical background are needed to enjoy an informal introduction to the following ideas and concepts.

• Space and dimension, manifold
• Sets and functions
• Probability
• Symmetry

 

    Linear Algebra in Rⁿ

A fully developed introduction to the following topics of linear algebra

• Points, vectors; norm, dot product and applications
• Systems of linear equations: Solving geometrically, by row reduction, by Cramer's rule
• Matrices, matrix operations and their properties, linear equations and matrix algebra
• Linear transformations, given by a matrix, key examples, properties
• Determinants, their algebraic properties, their geometrical interpretation as oriented volume, cross product, Cramer's rule
• Subvector spaces, basis, change of basis, orthogonal complement, Gram-Schmidt orthonormalization
• Linear transformations II: linear maps between subspaces of Rn, arbitrary rotations, shear maps
• Eigenvectors and eigenvalues

 

    Differential Calculus of Several Variables

• An intermediate level exposition of differential calculus in several variables.
• Total derivative and partial derivatives
• Rules of differentiation
• Inverse function theorem and its relatives

 

    Complex Numbers

• Introduction, brief history, rules for computing

 

Manifolds

• The Poincare conjecture

 

    Algebraic Topology

• Introduction, Homotopy concept, homotopy equivalent spaces
• Fundamental group

 

Topological Groups

• Examples,  Constructions (new groups from old),  Structural Properties