Schedule:
Schedule:
22nd of January: Beatrice Brienza
Strong HKT manifolds
Abstract: A manifold $(M,J_1,J_2,J_3)$ is called \emph{hypercomplex} if each $J_i$ is a complex structure and $\{J_1,J_2,J_3\}$ satisfy the quaternionic relations. A quaternionic Hermitian metric $g$ is called HKT (hyper-K\"ahler with torsion) if $ \nabla^{B}_{1} = \nabla^{B}_{2} = \nabla^{B}_{3} =: \nabla^{B}$, where $\nabla^{B}_{i}$ denotes the Bismut connection associated with $(J_i,g)$. If, in addition, the Bismut torsion of $\nabla^{B}$ is closed, the metric is called strong HKT. Whenever we have a strong HKT metric, the three Hermitian structures $(J_i,g)$ are Bismut Hermitian Einstein, namely, they are pluriclosed and their Bismut Ricci curvature vanishes. In this talk, we discuss some properties of compact strong HKT manifolds and, more generally, of Bismut Hermitian Einstein manifolds. In particular, we describe the geometry of compact simply connected strong HKT manifolds of real dimension $8$. This is joint work with A.~Fino, G.~Grantcharov, and M.~Verbitsky.
29th of January: Mateo Galdeano
The heterotic G2 system with reducible characteristic holonomy
Abstract: The heterotic G2 system is a set of equations that describes compactifications of heterotic supergravity on a 7-dimensional manifold preserving minimal supersymmetry. From a mathematical perspective, it involves interesting constructions in geometry---such as integrable G2-structures and G2-instantons---coupled to each other in a highly non-trivial way. This makes explicit solutions to the system challenging to obtain. In this talk I will introduce the heterotic G2 system and motivate how it can be solved when the characteristic G2 connection of the manifold has reduced holonomy. I will illustrate this by presenting new solutions on a class of manifolds known as 3-(α,δ)-Sasaki. This talk is based on recently published joint work with Leander Stecker (arXiv:2403.00084).
5th of February: Lorenzo Foscolo
12nd of February: Marco Freibert
Left-invariant generalised Einstein pairs
Abstract: Generalised Einstein geometry first arose in the context of compactifications of type II supergravity theories. In these theories, the common NS-NS sector consists of a closed three-form, a Riemannian metric and the scalar dilaton field. All these data can be reformulated as a pair of a generalised Riemannian metric $\mathcal{G}$ and a divergence operator $\div$ on an exact Courant algebroid and the equations of motion for the NS-NS sector may be rephrased as the pair $(\mathcal{G},\div)$ being generalised Einstein and generalised scalar-flat.
In my talk, I will give a detailed introduction to all this terminology and then will talk about the results of a joint article with Vicente Cortés and Mateo Galdeano on the generalised Einstein condition in a left-invariant setting on Lie groups. The results that I will present include a general reduction result of the generalised Riemannian Einstein condition on a Lie algebra to the commutator ideal with which we were able to prove that all generalised Riemannian Einstein pairs on solvable Lie algebras arise from the flat metric solvable Lie algebras described by Milnor. Besides, I will also present classification results for left-invariant generalised Lorentzian Einstein pairs $(\mathcal{G},\div)$ with $\div=0$ in the four-dimensional case and also in the almost Abelian case in arbitrary dimensions.
19th of February: James Stanfield
26th of February: Federico Giusti
5th of March: Giuseppe Barbaro
12th of March: Oskar Schiller
19th of March: Elia Fusi
26th of March: George Papadopoulos