Sharmila Gunasekaran (Radboud University)

Rigidity of quasi-Einstein manifolds

Extreme black holes possess event horizons at zero temperature, referred to as degenerate Killing horizons. These horizons are exclusively delineated by a specific limiting procedure, defining a near-horizon geometry or, more broadly, a quasi-Einstein equation which governs their properties. Solutions to this equation manifest as triples (M, g, X), where M represents a closed manifold (the horizon), g denotes a Riemannian metric, and X is a 1-form. 

In the scenario where X is a closed 1-form, it signifies the static case, encompassing near-horizon geometries of static extreme black holes. It was known that when the quasi-Einstein constant is zero, X must vanish. In collaboration with Eric Bahuaud, Hari K Kunduri, and Eric Woolgar, we have extended this characterization to the case of non-vanishing quasi-Einstein constant. When the constant is positive, X must be exact and when the constant is negative, either X vanishes or is not exact, in which case it is a product of negative Einstein manifold and a circle (finalized by Will Wylie).

Recently, Dunajski-Lucietti established that the near horizon geometry arising from a Kerr-(A)dS black hole is unique on a 2-sphere. I will discuss some partial deformability results for this metric.

Yagna Dutta (Leiden University)

Twists of intermediate Jacobian fibration

Given an elliptic fibration of a K3 surface, one can reglue the fibres of the elliptic fibration differently to obtain different K3 surfaces. These regluings are governed by a group scheme over the base of the elliptic fibration. My plan for the talk is to tell this story. Moving from curves to 3-folds, we find some very interesting group schemes related to the intermediate Jacobian of a cubic 3-fold. I will report on a joint work in progress with Mattei and Shinder where we consider the family of cubic 3-folds obtained as the hyperplane sections of a fixed smooth cubic 4-fold. The total space this time is a hyperKähler manifold. HyperKähler manifolds are nothing but higher dimensional analogues of K3 surfaces, resulting in impressive parallels with elliptic fibrations of K3 surfaces.

Renee Hoekzema (Free University Amsterdam)

The striped cylinder cobordism category

Cobordism categories describe the algebraic gluing structure of manifolds, and they are central in the functorial description of topological quantum field theories (TQFTs). We consider a new “nested” variation of a cobordism category where manifolds come with embedded submanifolds and cobordisms with subcobordisms. An example is the category of cylinders with lines. In this talk I will describe the algebraic structure associated with this striped cylinder cobordism category. This algebraic structure has links to Temperley-Lieb algebras as well as bearing similarity to the simplicial and the cyclic category, which are involved in the definition of the (cyclic) bar construction. We define a new cylindrical bar construction, a novel algebraic construction for self-dual objects in a strict monoidal category.