Morava $E$-theory is an important cohomology theory in chromatic homotopy theory. Using work of Ando, Hopkins, and Strickland, Rezk described the algebraic structure found in the homotopy of $K(n)$-local commutative $E$-algebras via a monad on $E_*$-modules that encodes all power operations. However, the construction does not see that the homotopy of a $K(n)$-local spectrum is $L$-complete (in the sense of Greenlees-May and Hovey-Strickland). We improve the construction to a monad on $L$-complete $E_*$-modules, and discuss some applications. Joint with Tobias Barthel.

In this talk it is considered the Galois point of view on determining the structure of space of ordering of fields via considering small Galois quotients of absolute Galois groups of Pythagorean fields. We use mainly Galois theoretic, group theoretic and combinatorial arguments. When a simple invariants of order space determine this order space completely ? Some interesting cases when this happen will be described.

Random topology is the study of topological invariants of random topological spaces. In this talk I will briefly survey work on topology of random simplicial complexes, starting with 1-dimensional models, i.e. random graphs. The Erdos-Renyi theorem characterizes the threshold edge probability where the random graph becomes connected, and we now know several different generalizations of this theorem to higher dimensions. In this talk, I'll discuss recent progress in proving such theorems by understanding eigenvalues of random matrices. I will not assume any particular topology or probability prerequisites, and the talk will aim to be self contained. Part of this is joint work with Chris Hoffman and Elliot Paquette.

Lagrangian and totally real submanifolds are two objects deeply related because of their definitions. The tangent space of a totally real manifold meets its complex rotation in only one point, the origin; while the tangent space of a Lagrangian submanifold is orthogonal to its complex rotation. One should notice that there are totally real 3-spheres in $\mathbb C^3$, but these spheres cannot be Lagrangian.

Around 1995 Duval and Sibony introduced a new relation between totally real, Lagrangian, and rationally convex manifolds. They proved that, at least for compact totally real submanifolds, rational convexity is equivalent to be Lagrangian for some appropriate Kaehler form.

Moreover, in a recent paper Cieliebak and Eliashberg have proved that, for $n>2$, the closure of a bounded domain in $\mathbb C^n$ is isotopic to a rationally convex set if and only if it admits a defining Morse function with no critical points of index strictly larger than n. This result implies in particular that there are smooth 3-spheres in $\mathbb C^3$ with a compact and rationally convex tubular neighbourhood. These smooth 3-spheres cannot be Lagrangian.

Techniques developed for transcendental number theory have had many surprising applications in the study of purely arithmetical questions. The aim of the talk will be to discuss this phenomenon.

I will first explain the notion of ergodicity for classical dynamical systems and will go over some of the well known examples of such systems. I will then introduce the notion of quantum ergodicity for quantized Hamiltonian systems and discuss some open problems.