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November 21
Spiro Karigiannis - Flows of G2 structures (Abstract)
A G2-structure is a special type of 3-form on an oriented 7-manifold, which determines a Riemannian metric in a nonlinear way. The best class of such 3-forms are those which are parallel with respect to their induced Levi-Civita connections, which is a fully non-linear PDE. More generally, the torsion of a G2-structure is a 2-tensor which quantifies the failure of a G2-structure to be parallel. It is natural to consider geometric flows of G2-structures as a means of starting with a G2-structure with torsion and (hopefully) improving it in some way along the flow. I will begin with an introduction to all of these ideas, and try to survey some of the results in the field. Then I will talk about recent joint work with Dwivedi and Gianniotis to study a large class of flows of G2-structures. In particular, we explicitly describe all possible second order differential invariants of a G2-structure which can be used to construct a quasi-linear second order flow. Then we find conditions on a subclass of these general flows which are amenable to the deTurck trick for establishing short-time existence and uniqueness.
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November 7
Frederik Benirschke (University of Chicago) - An introduction to bi-algebraic geometry, with a view towards strata of differentials (Abstract)
I will introduce some of the main ideas in bi-algebraic geometry, focusing on the case of algebraic tori and Abelian varieties. Afterward, we explore these ideas for a natural bi-algebraic structure on the moduli space of marked points on the sphere that arises from integrating differential forms.
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October 31
Lucia Martin Merchan - Hodge decomposition for Nearly Kähler 6-manifolds (Abstract)
In this talk we discuss the paper of M. Verbitsky (arXiv:math/0510618) where he finds Kähler identities for Nearly Kähler 6-manifolds. From that, he deduces a Hodge decomposition in the compact case, as well as some restrictions on their refined Betti numbers.
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October 24
Amanda Petcu - Partial progress on a conjecture of Donaldson by Fine and Yao (Part 2) (Abstract)
Given a compact hypersymplectic manifold X4, Donaldson conjectured that the hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler structure. Fine and Yao consider a manifold with closed G2-structure that is set up as 𝕋3 × X4. They examine the G2-Laplacian flow under in this setting and give a flow of hypersymplectic structures which evolve according to the equation ∂t ω = d(Q d*(Q-1 ω), where ω is the triple that gives the hypersymplectic structure and Q is a 3 × 3 symmetric matrix that relates the symplectic forms ωi to one another. Lotay—Wei have established long time existence of the G2-Laplacian flow provided the velocity of the flow remains bounded. Fine—Yao use this extension theorem in their setup and manage to improve it by proving long time existence of the hypersymplectic flow provided the torsion tensor T remains bounded. Furthermore, one can relate the scalar curvature and torsion tensor of manifold with closed G2-structure and thus they conclude long time existence for the hypersymplectic flow provided the scalar curvature remains bounded. In this talk we will go over some details from this paper by Fine—Yao.
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October 17
Aleksandar Milivojevic - Formality and non-zero degree maps (Abstract)
I will talk about a recent result with J. Stelzig and L. Zoller showing that formality is preserved under non-zero degree maps. Namely, if the domain of a non-zero degree map is formal, then so is the target. Some geometric applications are the formality of closed orientable manifolds with large first Betti number admitting a metric with non-negative Ricci curvature, and alternative proofs of the formality of positive quaternion-Kähler manifolds, and of singular complex varieties satisfying rational Poincaré duality.
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October 3
Spiro Karigiannis - A curious system of second order nonlinear PDEs for U(m)-structures on manifolds (Abstract)
Compact Kähler manifolds possess a number of remarkable properties, such as the Kähler identities, the ∂∂-lemma, and the relation between Betti numbers and Hodge numbers. I will discuss an attempt in progress to generalize some of these ideas to more general compact U(m)-manifolds, where we do not assume integrability of the almost complex structure nor closedness of the associated real (1,1)-form. I will present a system of second order nonlinear PDEs for such a structure, of which the Kähler structures form a trivial class of solutions. Any compact non-Kähler solutions to this second order system would have properties that are formally similar to the above-mentioned properties of compact Kähler manifolds, including relations between cohomological (albeit non-topological) data. This is work in progress with Xenia de la Ossa (Oxford) and Eirik Eik Svanes (Stavanger).
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September 26
Michael Albanese - Generalised Kähler-Ricci Solitons (Abstract)
The notion of a Kähler-Ricci soliton arises from the study of the Kähler-Ricci flow. They can only exist on certain manifolds, namely Fano manifolds. In this restricted case, there is an equivalent formulation which is no longer equivalent in the non-Fano case - these are called Generalised Kähler-Ricci solitons. I intend to discuss both Kähler-Ricci solitons and Generalised Kähler-Ricci solitons, as well as some differences between them.
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September 19
Anton Iliashenko - A special class of harmonic map submersions with calibrated fibres (Abstract)
Say we have a conformally horizontal submersion between two Riemannian manifolds of dimensions n and k respectively. If the domain admits a closed (n-k)-calibration form, then we can define a special type of maps which we call conformally calibrated (or Smith) maps. We will show that these maps are k-harmonic and we will see a couple of examples.
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September 12
Amanda Petcu - Partial progress on a conjecture of Donaldson by Fine and Yao (Abstract)
Given a compact hypersymplectic manifold X4, Donaldson conjectured that the hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler structure. Fine and Yao consider a manifold with closed G2-structure that is set up as 𝕋 × X4. They examine the G2-Laplacian flow under in this setting and give a flow of hypersymplectic structures which evolve according to the equation ∂t ω = d(Q d*(Q-1 ω), where ω is the triple that the hypersymplectic structure and Q is a 3 × 3 symmetric matrix that relates the symplectic forms ωi to one another. Lotay—Wei have established long time existence of the G2-Laplacian flow provided the velocity of the flow remains bounded. Fine—Yao use this extension theorem in their setup and manage to improve it by proving long time existence of the hypersymplectic flow provided the torsion tensor T remains bounded. Furthermore, one can relate the scalar curvature and torsion tensor of manifold with closed G2-structure and thus they conclude long time existence for the hypersymplectic flow provided the scalar curvature remains bounded. In this talk we will go over some details from this paper by Fine—Yao.