We show that, under a division condition, the tangential Cauchy-Riemann cohomology of a compact Lie group with a left-invariant CR structure can be computed on a suitable maximal torus. As a consequence, we conclude that the tangential Cauchy-Riemann cohomology is finite-dimensional. We also show that, for a class of CR structures, this division condition is necessary for the total cohomology to be finite-dimensional. The proof combines Fourier analysis on compact Lie groups, highest-weight representations and Lie algebra cohomology. This not only generalizes but provides completely new proofs for the analogous result due to Pittie and for its extensions to Levi-flat CR structures, obtained by Jacobowitz and Jahnke.

A compact symplectic manifold $(M, \omega)$ is called positive monotone if its first Chern class is a positive multiple of $[\omega]$ in the second de Rham group $H^2(M)$. A smooth Fano variety is a complex manifold, whose anti-canonical line bundle is ample. Such a variety can be endowed with a symplectic form such that it becomes a positive monotone symplectic manifold. For this reason, positive monotone symplectic manifolds are considered the symplectic counterparts of smooth Fano varieties.

In the field of symplectic geometry, a general outstanding issue is understanding in what context positive monotone symplectic manifolds differ from Fano varieties. In low dimensions, namely two and four, it has been proven by Gromov, Taubes, McDuff, and Ohat-Ono that any positive monotone symplectic manifold is symplectomorphic to a Fano variety. Starting from dimension twelve, work by Fine and Panov provides examples of positive monotone symplectic manifolds that are not even homotopy equivalent to a Fano variety.

In this talk, I will explain what is known about the differences between Fano varieties and positive monotone symplectic manifolds endowed with a Hamiltonian action of a compact torus $T$. In particular, I will present new results for the case where the complexity of the action is one, i.e., $\frac{1}{2}\dim(M)-\dim(T)=1$.

This talk is based on joint work with Liat Kessler, Silvia Sabatini, and Daniele Sepe.

Since the groundbreaking work of M. Gromov in the 1980s many tools have been developed for distinguishing open symplectic domains. However, until recently, similar questions in the contact geometric setup were largely open. For instance, it was not known whether there are open domains in the standard contact vector space of dimension $>3$ which are diffeomorphic but not contactomorphic to it (in dimension $3$ it is known that all of them are). In my lecture I will discuss Floer theoretic tools for answering this type of questions. As one application I will construct a continuous family of pairwise non-contactomorphic open balls in the standard contact ${\mathbb R}^5$. The lecture is based on a joint work in progress with K. Ajij, Mahan Mj, Dishant Pancholi and L. Polterovich.

I will first introduce the bigraded cohomology for real algebraic varieties developed by Johannes Huisman and Dewi Gleuher. This is a cohomology theory that refines the equivariant cohomology "à la Kahn-Krasnov" of the complex points of a real variety, the latter often being preferred (by the algebraic geometers) in the cohomological study of real algebraic varieties. Since the construction of this bigraded cohomology and its associated characteristic classes relies on the sheaf exponential morphism, I will explain how to produce an arithmetic (or algebraic) variant of these cohomology groups, whose main advantage is toeliminate topological or transcendental conditions. I will conclude by comparing these two versions of bigraded cohomology.