This is an obligatory course for first year students. It is taught 6 hours per week. It covers the introduction of coordinates in Geometry, algebra of vectors, the classical theorems of Geometry (Menelaus, Pappus, Ceva, Desargues), inner product, exterior product, mixed triple product, lines, planes, parallelism and orthogonality, areas, volumes, conic sections and the basic theorems (Dedenlin, Apollodorus), ending with a complete study of the second order equation.

(Athens, every year to Physics students, 3 hrs/week) Systems of linear equations, matrices, vector spaces and linear maps, eigenvalues/vectors/spaces, linear systems of differential equations with constant coefficients.

(Athens, undergraduate, every year, 3 hrs/week) Fundamental group, covering spaces, Singular Homology. The online book by A. Hatcher (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) is quite useful.

(Athens, MSc, 4 hrs/week) Discussed the orbit method. Kirillov's book is a good guide on the subject.

(Athens, seminar) Introduced K-theory of C*-algebras, pseudodifferential calculus, Fredholmness, ellipticity and analytic index, Lie groupoids, and finally gave the proof of the Atiyah-Singer index theorem.

(University of Crete, seminar) Introduced topological K-theory and explained its relation with algebraic K-theory and C*-algebras. Presented Both periodicity, 6-terms exact sequence and Kunneth theorem.

(University of Crete, undergraduate, 3 hrs/week)