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Manifolds admitting metrics of positive sectional curvature are conjectured to have a very rigid topological structure. However, this structure is still highly speculative and the strongest results in this direction are known under the assumption of Lie group actions. In this talk I shall try to illustrate this interplay between geometry, symmetry and topology. In particular, using such symmetry assumptions, I will speak about generalisations of a conjecture by Hopf which states that $\mathbb{S}^2\times\mathbb{S}^2$ cannot carry a metric of positive curvature. Presented results come from joint work with Lee Kennard.
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Besides the Dolbeault cohomology, the Bott-Chern cohomology provides a further invariant in the geometry of complex non-Kähler manifolds. It also arises as a natural tool in investigating special Hermitian metrics. On the one side, the Bott-Chern cohomology may encode, in a sense, more informations on the complex and geometric structures. On the other side, it seems, algebraically, less "natural". As an example: the Frölicher inequality for the Dolbeault cohomology has an analogue for the Bott-Chern cohomology, which is stronger in the sense that the equality characterizes the validity of the $\partial\overline{\partial}$-Lemma. As a counter-example: it is not clear how far a theory of formality may be restated in terms of Bott-Chern cohomology: the talk would like to be an attempt to understand this issue.
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The HK/QK correspondence was discovered by Andriy Haydys in 2006. Starting from a (pseudo-)hyper-Kähler manifold endowed with a real-valued function fulfilling certain assumptions, it constructs a quaternionic pseudo-Kähler manifold of the same dimension. I will introduce the correspondence, discuss its compatibility with hyper-Kähler and quaternionic Kähler quotients, and present a few examples.
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The Hitchin flow equations in seven dimensions are certain 1st order pdes whose solutions define 8d Riemannian manifolds with holonomy contained in the exceptional holonomy group Spin(7). Now the irreducible holonomy groups Sp(2) and SU(4) are contained in Spin(7) and may so, in principal, also be obtained by the Hitchin flow.
In fact, it is known that the holonomy group is contained in SU(4) if the Hitchin flow is induced by the hypo flow in seven dimensions.
We study both the Hitchin and the hypo flow in the left-invariant setting on a 7d Lie group and present some conditions on the initial values of these flows which ensure that the obtained Riemannian manifold has holonomy equal to SU(4). We also discuss the case of holonomy equal to Sp(2) and exclude this holonomy for certain initial values.
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The Atiyah-Bott-Berline-Vergne localization formula allows, in presence of a torus action, to localize the integral of certain differential forms to the fixed point set of the action. In this talk I will report on a generalization of this formula for Riemannian foliations in which the role of the fixed point set is played by the closed leaves of the foliation. One can apply the formula to compute various geometric quantities, such as the volume of Sasakian manifolds or secondary characteristic numbers of Riemannian foliations. The talk is based on joint work with Hiraku Nozawa and Dirk Töben.
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On a closed Kähler manifold, the space of all Kähler metrics in a fixed cohomology class has a natural structure of infinite dimensional manifold. On it, several (weak) Riemannian metrics can be assigned and the most studied ones are called $L^2$, Calabi and Gradient (or Dirichlet) metric. I will recall known results about their different geometries and write down and compare the relative geodesic equations as PDEs on the manifold. Finally I will discuss my contribution, joint with S. Calamai and K. Zheng, about the geodesic equation of the gradient metric and of the Ebin metric restricted to the (similarly defined) space of Sasakian metrics.
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A manifold is called rationally elliptic if its rational homotopy groups, form a finite dimensional vector space over $\mathbb{Q}$. In this talk we look at effective actions by tori on rationally elliptic manifolds. In particular, we produce sharp bounds for the dimension of the torus in term of the dimension of $M$, and provide a classification up to rational homotopy equivalence of those manifolds admitting actions of maximal dimension. In particular, we provide further evidence to the conjectured relationship between non-negatively curved and rationally elliptic manifolds.
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The J-flow is a gradient geometric flow of Kähler structures firstly studied by Donaldson from the point of view of moment maps and by Chen in relation to the Mabuchi energy. The aim of this talk is to introduce an odd-dimensional analogoue of this flow in the Sasakian context.
