
 9/7/2023: Homotopy Associative Submanifolds in G2manifolds
 9/11/2022: Some examples of extra twisted connected sum G_2manifolds
 4/11/2019: Distinguishing G2manifolds
 9/13/2018: Extra twisted connected sums and their v invariants
 9/12/2017: The extended v invariant — progress and problems
 6/07/2017: Disconnecting the G_{2} moduli space
 9/6/2016: Connected components of the moduli space of G_{2} manifolds
September 7, 2023
TITLE: Homotopy Associative Submanifolds in G2manifolds
ABSTRACT: Associative submanifolds are certain calibrated submanifolds in G2manifolds. There is the hope that counting them will reveal subtle information about the underlying G2structure. On the other hand, certain singular associatives can be resolved in exactly three different ways, so a naive count will be meaningless. In this talk, we will define homotopy associatives as cobordism classes of threedimensional submanifolds that are adapted to the G2structure in a rather weak sense. We will see that a given cobordism class can be interpreted as a homotopy associative in exactly three different ways. This might help us to define a consistent counting scheme even when the naive number of associatives in a given cobordism class changes due to singularities.
September 11, 2022
TITLE: Some examples of extra twisted connected sum G_2manifolds
ABSTRACT: The twisted connected sum construction is one of a few known ways to produce compact G_2manifolds. Extra twisted connected sums form a slight generalisation. They have been used to show that the moduli space of G_2metrics on a given 7manifold can be disconnected, even if one fixes a connected component of the space of topological G_2structures. In this talk, I want to present some more details of the construction. In particular, I want to present quotients and coverings of extra twisted connected sums, as well as a kind of tduality.
April 11, 2019
TITLE: Distinguishing G2manifolds
ABSTRACT: There are several invariants from differential topology that
distinguish smooth 7manifolds and G2structures on them. I will give a
short introduction, then focus on the nu invariant and its analytic
refinement.
September 13, 2018
TITLE: Extra twisted connected sums and their v invariants
ABSTRACT: Joyce’s orbifold construction and the twisted connected sums by Kovalev and CortiHaskinsNordströmPacini provide many examples of compact Riemannian 7manifolds with holonomy (i.e., manifolds). We would like to use this wealth of examples to guess further properties of manifolds and to find obstructions against holonomy , taking into account the underlying topological structures.
The CrowleyNordström vinvariant distinguishes topological structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the vinvariant can assume all of its 48 possible values. This shows that bordism presents no obstruction against holonomy . We also exhibit examples of 7manifolds with disconnected moduli space. Our computation of the vinvariants involves integration of the BismutCheeger ηforms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind ηfunction.
September 12, 2017
TITLE: The extended v invariant — progress and problems
ABSTRACT: We recall the definition of the extended invariant, which distinguishes connected components of the moduli space of compact Riemannian manifolds whose holonomy group is (manifolds for short). All known computations of for manifolds give values that are divisible by three. This implies that all these examples are topologically nullbordant. It is therefore interesting to know if for all manifolds. We will report on results and projects related to this question.
June 7, 2017
TITLE: Disconnecting the G_{2} moduli space
September 6, 2016
TITLE: Connected components of the moduli space of G_{2} manifolds
ABSTRACT: The CrowleyNordström ()invariant distinguishes topological structures on 7manifolds. It takes values in () /48. There is a ()valued extension for manifolds of holonomy . We will introduce both invariants and show how they can be computed for extra twisted connected sums using ()invariants of Dirac operators. This allows us to exhibit examples of 7manifolds where the space of holonomy metrics is disconnected.
We will then talk about some related open questions and problems and sketch possible next steps in our research program.