Manifold calculus of functors was introduced and developed by T. Goodwillie and M. Weiss in order to study spaces of embeddings. In a few words the goal of their method is to understand how from the spaces Emb(U,N) of smaller open subsets U of M we can describe the space Emb(M,N) of embeddings of the entire manifold M into N. Naively it is sometimes called "patching method". I will describe briefly the ideas of this theory and also explain some recent advances which gives a connection with the theory of operads.

When does an "infinite tower of exponentials" converge? To clarify, for positive real numbers $a,b, \ldots$ let us set $E_a(x) = a^x$, and $E(a,b, \ldots, c) = E_a \circ E_b \circ \cdots \circ E_c (1)$, so $E(a) = a$, $E(a,b) = a^b$, $E(a,b,c) = a^{b^c}$. The question then is: for what sequences $\{a_n\}$ of positive real numbers does the sequence $\{E(a_1, \ldots, a_n)\}$ converge?

Starting with an action $G\times X\to X$ we analyze the maximal $G$-rational subalgebra $\mathscr{O}_K$ of $k(X)$ and use it to obtain the action $G_K\times U_K\to U_K$ where $K = k(X)^G$, and $U_K$ is a certain quasi-affine variety over $K$ with $\mathscr{O}(U_K) = \mathscr{O}_K$. This gives us a generic "homogeneous" picture of the original action.

We also analyze the maximal $G$-rational subalgebra of $k[X]_\mathfrak{p}$, where $\mathfrak{p}$ is a height-one $G$-prime of $k[X]$. We use these results to assess the behavior of the canonical map $\pi : U\to U/G$ for a sufficiently small $G$-invariant, open subset $U$ of $X$.

Finally we use ${\textit{observable}}$ $G$-actions over $k$ to construct the functor $K\mapsto H^1(K,G/H)$, from finitely generated fields over $k$ to ${\textit{Sets}}$. From there we define the ${\textit{essential dimension}}$ of a homogeneous space $G/H$, whenever $H\subset G$ is a pair of connected, reductive groups.