Η Κ-Θεωρία ανήκει στην κλάση των λεγόμενων γενικευμένων (ή εξωτικών) θεωριών ομολογίας και η χρησιμότητά της στα περίπου 50 χρόνια ζωής της έχει αποδειχθεί ήδη πολύ μεγάλη σε πολλούς τομείς των μαθηματικών όπως στην τοπολογία, την ολική ανάλυση, τη θεωρία αριθμών, την άλγεβρα, την γεωμετρία κλπ. Ενδεικτικά αναφέρουμε ότι αποτελεί το ένα από τα δύο βασικά συστατικά των περίφημων Θεωρημάτων Δείκτου που αποτελούν αναμφίβολα το πιο κομβικό αποτέλεσμα στα μαθηματικά του δεύτερου μισού 20ου αιώνα αφού ενώνει τρεις διαφορετικούς τομείς των μαθηματικών: την Ανάλυση, την Γεωμετρία (Τοπολογία) και την Άλγεβρα.

This survey paper is an expanded version of lectures given at the Clay Mathematics Academy ; see this http URL These lectures were intended to very young (and motivated) college students with little background. Therefore, they are accessible to a mathematician of any speciality willing to understand the subject. A much more complete introduction to K-theory may be found in the “Handbook of K-theory”, recently edited by Springer.

This is an expository paper which aims at explaining a physical point of view on the K-theoretic classification of D-branes. We combine ideas of renormalization group flows between boundary conformal field theories, together with spacetime notions such as anomaly cancellation and D-brane instanton effects. We illustrate this point of view by describing the twisted K-theory of the special unitary groups SU(N).

This expository paper is an introductory text on topological K-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the material presented here is distilled from Atiyah’s classic “K-Theory” text, as well as his series of seminal papers “The Index of Elliptic Operators” with Singer. Additional topics include equivariant K-theory, the G-index theorem, and Bott’s paper “The Index Theorem for Homogeneous Differential Operators”. It also includes an appendix with a proof of Bott periodicity, as well as sketches of proofs for both the standard and equivariant versions of the K-theory Thom isomorphism theorem, in terms of the index for families of elliptic operators. A second appendix derives the Atiyah-Hirzebruch spectral sequence. This text originated as notes from a series of lectures given by the author as an undergraduate at Princeton. In its current form, the author has used it for graduate courses at the University of Oregon.

These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the topological and algebraic K-theory, K-theory of C^*-algebras, and K-homology. I then discuss elementary properties of cyclic cohomology using the Cuntz-Quillen version of the calculus of noncommutative differential forms on an algebra. As an example of the relation between the two theories we describe the Chern homomorphism and various index-theorem type statements. The remainder of the notes contains some more detailed calculations in cyclic and reduced cyclic cohomology.