Let k be a positive integer. A k-differential on a Riemann surface C is a section of the k-th tensor power of the canonical bundle of C. I will review what is known about the space of holomorphic k-differentials in the case when C is compact. I will state some new results for the case when C is non-compact.

An $A_{\infty}$-algebra generalizes an associative algebra, by requiring the binary operation to only be associative up to the derivative of a ternary operation. There is then a 4-ary operation satisfying some relation between these two, and we can continue to get the structure of an operad. Beginning with an augmented differential graded associative (dga) algebra, we use the Eilenberg-MacLane bar construction to get a dga coalgebra with some nice properties. We use this to define a general $A_{\infty}$-algebra, and then manipulate it as a chain complex to find the homotopy.

After introducing rapidly the context of diffeology, I'll try to review some of the main constructions and show, through examples, how this differential geometry treat objects which do not belong to the classical category of manifolds. I will show also how using diffeology some classical theorems are dramatically simplified, and extended, in particular in de Rham calculus. I may also, if I have the time, introduce the program of symplectic diffeology: results, questions and perspectives.

Let $X$ be a complex affine variety with an action of a torus $T$, and an attractive fixed point $x_0$. We say that $X$ is a rational cell if $H^{2n}(X,X-\{x_0\})=\mathbb{Q}$ and $H^{i}(X,X-\{x_0\})=0$ for $i\neq 2n$, where $n={\rm dim\,}_{\mathbb{C}}(X)$. These objects appear naturally in the study of group embeddings. A fundamental result in equivariant cohomology asserts that the transgression ${\bf Eu}_T \in H^{2n}(BT)$ of a generator of $H^{2n}(X,X-\{x_0\})$ splits into a product of singular characters, ${\bf Eu}_T={\chi_1}^{k_1}\ldots {\chi_m}^{k_m}$. This characteristic class is by definition the Equivariant Euler class of $X$ at $x_0$. Loosely speaking, one could think of $X$ as a sort of $T$-vector bundle over a point. My goal in this talk is to make this claim precise, and to show why one could hope to build similar elements in equivariant $K$-theory, i.e. Bott classes, by using localization and completion techniques. This is work in progress.