Speaker: Oksana Yakimova MSU, IUM, Russia Title: From weakly symmetric to weakly commutative spaces Abstract: The notion of a weakly symmetric Riemannian homogeneous spaces was introduced by Selberg in his famous paper on the trace formula. It is a generalisation of the notion of a symmetric space. Weakly symmetric spaces have a number of intriguing properties that can be explained in algebraic and geometric context. For instance, any weakly symmetric space is a real form of a complex spherical homogeneous space. Another interesting feature is that these spaces are "commutative", i.e., the algebra of invariant differential operators on such spaces is commutative. And if the associated Poisson algebra is commutative then the space is called weakly commutative. Classifications ofcommutative spaces are not known by now, but we will discuss some promising approaches to the problem. In particular, a criterion for weak commutativity will be proved. weakly symmetric, commutative, and weakly Sasha Smirnov MSU, IUM, Russia Title: The classification of nearly closed orbits of complex semisimple groups in linear projective spaces. Abstract: Suppose $G\subset GL(V)$ is a linear semisimple complex group. Then it also acts on the projective space $\mathbb{P}(V)$. It is well known that if $V$ is irreducible, then there exists a single closed orbit in the space $\mathbb{P}(V)$. In my work I classify the so called nearly closed orbits $G$, meanly such orbits that the closure $\overline{G}$ consists exactly of two orbits.