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
Special Hermitian metrics on non-compact Calabi-Yau 3-folds
Abstract: We present the construction of infinitely many examples of distinguished non-Kähler Hermitian metrics on non-compact Calabi-Yau 3-folds. These metrics solve a system of equations known as the IIB system, which arises in theoretical physics and is related to recent attempts to define notions of "canonical" metrics on non-Kähler Calabi-Yau manifolds. The examples we construct include infinitely many complete metrics obtained by deforming an asymptotically conical Kähler Ricci-flat metric in the direction of a non-trivial Äppli class and families of solutions on the ordinary double point and its smoothing that enjoy a cohomogeneity one symmetry (i.e. there is a symmetry group that acts with 1-dimensional orbit space). The talk is based on joint work with Mario Garcia-Fernandez.
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
Homogeneous generalized Ricci flows
The generalized Ricci flow, or GRF (first studied by Callan–Friedan–Martinec–Perry and later by Streets), is an analogue of the Ricci flow in the setting of Hitchin’s generalized geometry. It is a certain supersolution of the Ricci flow coupled with the heat flow on 3-forms and subsumes several geometric flows, including the classical Ricci flow and the so-called pluriclosed flow appearing in non-Kähler complex geometry.
In this talk, we discuss the behavior of the GRF on (discrete quotients of) Lie groups. We establish the global existence of the invariant flow on solvmanifolds in arbitrary dimensions—a result that is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton that allows for nontrivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the GRF under certain circumstances. We also prove that bi-invariant metrics on compact semisimple Lie groups are dynamically stable. Our main tool is an adaptation of Lauret’s “bracket flow” to the GRF, together with a new formula for the generalized Ricci curvature in terms of the moment map for the action of a real reductive Lie group on the space of generalized Lie brackets. This work is based on joint work in progress with Elia Fusi (Università di Torino) and Ramiro Lafuente (The University of Queensland).
26th of February: Federico Giusti
Special metrics through reversed conifold transitions
Abstract: Conifold transitions, as conjectured by Reid, appear as central in the study of the moduli of Calabi-Yau threefolds. One of their main features is that they lead outside the Kähler world, hence to obtain a proper understanding of the spaces arising from these transitions, we need to come up with notions of “canonical geometries” in a non-Kähler (but still Calabi-Yau) setting, the most famous of which are identified by the Hull-Strominger system (or the Type IIB system), which also lies at the center of a geometrized version of Reid’s fantasy proposed by Yau. In this talk I will discuss how, in the case of reversed conifold transitions, we have several good ingredients to work on the equations of the Hull-Strominger system, in particular by presenting a joint work with Cristiano Spotti in which we show that the resulting small resolutions always carry Chern-Ricci flat balanced metrics.
5th of March: Giuseppe Barbaro
Toric Generalized K\"ahler--Ricci Solitons
Abstract: We study generalized K\"ahler--Ricci solitons (GKRS). We show that in four dimensions, all GKRS are either described by the generalized K\"ahler Gibbons--Hawking ansatz, or have split tangent bundle, or are A-type toric. This further motivates the study of toric GKRS. In this setting, we establish a local equivalence between toric steady K\"ahler--Ricci solitons and A-type toric GKRS. Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions.
12th of March: Oskar Schiller
19th of March: Elia Fusi
26th of March: George Papadopoulos
2nd of April: Caleb Jonker
9th of April: Udhav Fowdar
16th of April: Leander Stecker
23rd of April: Andres Moreno