A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) or pluriclosed if the torsion of the associated Bismut connection associated is closed. I will present some general results on pluriclosed metrics in relation to symplectic geometry, the pluriclosed flow and Kähler-like curvature conditions.

The subject of this talk is a series of results on Hermitian, Complex and Symplectic geometries on the flag manifolds of noncompact semi-simple Lie groups (real or complex). Equations of Differential Geometry in coset spaces of Lie groups are reduced to algebraic equations by homogeneity. Thus many questions are stated and solved in the realm of algebraic properties of the isotropy representation at the tangent space at the origin. In case of the flag manifolds these representations are usually dealt with a combinatorics involving the algebraic structure of the root systems and their Weyl groups.

In this talk some questions and results in this direction will be worked out. Best results are obtained for flag manifolds of the complex groups because the isotropy representations are better behaved. Results on real flag manifolds are more sparse and technically harder. In the complex case it will be presented a more detailed account on the \((1,2)\)-symplectic Hermitian metrics on the maximal (full) flag manifolds.

Let \(M\) be a compact differentiable manifold and let \(\mathcal{M}\) denote the space of all unit volume Riemannian metrics on \(M\). Back in 1915, Hilbert proved that the critical points of the simplest curvature functional, given by the total scalar curvature \(Sc:\mathcal{M}_1\rightarrow {\Bbb R}\), are precisely Einstein metrics.

In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be \(G\)-invariant for some compact Lie group \(G\) acting transitively on \(M\). As an application, we will give the stability and critical point types of all Einstein metrics on flag manifolds with \(b_2(M)=1\).

We will speak about a joint project, with {\it Antonio J. Di Scala} and {\it Francisco Vittone}, for the study of the structure of irreducible homogeneous Riemannian manifolds \(M^n= G/H\) whose curvature tensor has a non-trivial nullity. In a recent paper we developed a general theory to deal with such spaces. By making use of this theory we were able to construct the first non-trivial examples in any dimension. The key fact is the existence of a non-trivial transvection \(X\) at \(p\) (i.e.\((\nabla X)_p = 0\)) such that \(X_p\notin \nu_p\) (the nullity subspace at \(p\)), but the Jacobi operator of \(X_p\) is zero. The nullity distribution \(\nu\) is highly non-homogeneus in the sense that no non-trivial Killing field lie in \(\nu\) (an so \(\nu\) is not given by the orbits of an isometry subgroup of \(G\)). One has that the Lie algebra \(\mathfrak g\) of \(G\) is never reductive. The co-nullity index \(k\) must be always at least \(3\) and if \(k=3\), then \(G\) must be solvable and \(H\) trivial. The leaves of the nullity foliation \(\mathcal F\) are closed and isometric to a Euclidean space. Moreover, the stabilizer of a given leaf acts effectively on the quotient space \(M/\mathcal F\) and so it does not admit a Riemannian \(G\)-invariant metric. Our main new result is that the so-called {\it adapted} tranvections lie in an abelian ideal \(\mathfrak a\) of \(\mathfrak g\). The distribution \(q\mapsto \mathfrak a .q + \nu _q\) is integrable and does not depend on the presentation group \(G\) and so it defines a geometric foliation \(\hat {\mathcal F}\) on \(M\) (by taking the clausure of the leaves). Moreover, the nullity \(\nu\) is parallel along any leaf of \(\hat {\mathcal F}\) and the projection to the quotient space \(M/ \hat {\mathcal F}\) is a Riemannian submersion. We intend to relate the geometry of \(M\) to that of this quotient.

We prove that a curve is a soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space if and only if its geodesic curvature is given as the inner product between its tangent vector field and a vector of the 3-dimensional Minkowski space. We prove that there are three classes of such solutions and for each fixed vector there exits a 2-parameter family of soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant

In this talk we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a family of \(6\)-dimensional Lie group \(H\). We also investigate which of these metrics are Hermitian and classify the corresponding complex structures. This talk is based on a joint work with Silvio Reggiani, based on ``The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups'', arXiv:2011.02854

Gromov and Gallot showed that for a fixed dimension \(n\) there exists a number \(\varepsilon(n)>0\) so that any \(n\)-dimensional riemannian manifold \((M,g)\) satisfying \(\textrm{Ric}_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)\) has first Betti number smaller than or equal to \(n\). In the equality case, \(\textrm{b}_1(M)=n\), Cheeger and Colding showed that then \(M\) has to be bi-Holder homeomorphic to a flat torus. This part can be seen as a stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension have to be a torus.

The proofs of Gromov and, Cheeger and Colding rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of \(M\) and then study the geometry of the covering. In this talk we will generalize these results to the case of \(RCD(K,N)\) spaces, which is the synthetic notion of riemannian manifolds satisfying \(\text{Ric} \geq K\) and \(\text{dim} \leq N\). This class of spaces include Ricci limit spaces and Alexandrov spaces.

Chow and Hamilton introduced the notion of cross curvature and the associated geometric flow in 2004. Several authors have built on their work to study the uniformisation of negatively curved manifolds, Dehn fillings, and other topics. Hamilton conjectured that it is always possible to find a metric with given positive cross curvature on the three-sphere and that such a metric is unique. We will discuss several results that support the existence portion of this conjecture. Next, we will produce a counterexample showing that uniqueness fails in general.