After recalling the Rost nilpotence property and its meaning, I will show that for 3-dimensional smooth projective varieties over a field of characteristic 0, one has only 'to care' about 0-cycles. Using the Brown-Gersten-Quillen spectral sequence, one can get one's hands on the zero cycles of a 3-dimensional toric model allowing a proof of Rost nilpotence for such varieties.

I shall prove the Harer-Zagier formula and Wigner's semicircle law for random matrices. We shall then move to evaluation of matrix integrals via graphs on surfaces.

We will introduce equivalent ways to define toric varieties and discuss their relationship.

I am going to give a basic definition of Hochschild and cyclic cohomology first, and then I will try to give the local index formula in a more general case, if time is enough, I will give the proof of local index formula for the even case.

Given the category of modules over an algebra, we would like to know when it is a monoidal category with the usual tensor product of vector spaces. This leads to the notion of Hopf algebra.

I will present detailed calculations of the cohomology rings of several small groups, together with the action of the Steenrod squares on said cohomology rings.