By extending the canonical trace of Kontsevich-Vishik to Connes' pseudodifferential operators on noncommutative tori, we study various spectral invariants associated to elliptic operators in this setting. We also consider a family of Cauchy-Riemann operators over noncommutative 2-torus and using the machinery of canonical trace, we compute the curvature form of the associated Quillen determinant line bundle.

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.