Page 121 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 121
TOPICS IN COMPLEX AND QUATERNIONIC GEOMETRY (MS-74)
Slice regular functions and orthogonal complex structures in eight
dimensions
Caterina Stoppato, caterina.stoppato@unifi.it
Università di Firenze, Italy
Coauthors: Riccardo Ghiloni, Alessandro Perotti
The theory of slice regular functions, introduced by Gentili and Struppa in 2006, is a successful
quaternionic analog of the theory of holomorphic functions of one complex variable. It includes
new interesting phenomena, due to the noncommutative setting.
This theory has been applied to the problem of classifying Orthogonal Complex Structures
on open dense subsets R4 \ Λ of R4. Traditionally, this problem had been addressed with
a toolset limited to quaternionic linear fractional transformations: only the case when Λ has
Hausdorff dimension less than 1 and the case when Λ is a circle or a straight line could be
addressed. Then the class of injective quaternionic slice regular functions became available
as a tool for classification, which made other cases approachable. The construction of this
new toolset required a detailed study of the differential topology of quaternionic slice regular
functions. This study was a joint work with Gentili and Salamon published in 2014.
The talk will look at the theory of octonionic slice regular functions, introduced by Gentili
and Struppa in 2010, through the lens of differential topology. This study has an independent
interest, because of the peculiar features of the nonassociative setting of octonions. It leads
to a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions.
Moreover, it opens the path for a possible use of slice regular functions in the study of almost-
complex structures in eight dimensions.
Special geometries with torus symmetry
Andrew Swann, swann@math.au.dk
Aarhus University, Denmark
A survey of recent results on torus symmetry for metrics with special holonomy, particularly G2
and Spin(7), and related geometries such as nearly Kähler six-manifolds. The known explicit
examples of these geometries all have a large compact symmetry group and in particular an
action of some torus. Studying the orbit structure and using ideas such as multi-moment maps,
one gets pictures related to the Delzant description for toric manifolds, and/or certain geometric
flows.
∂-Harmonic forms on compact almost Hermitian manifolds
Adriano Tomassini, adriano.tomassini@unipr.it
Università di Parma, Italy
Let M be a smooth manifold of dimension 2n and let J be an almost-complex structure on M .
Then, J induces on the space of forms A•(M ) a natural bigrading, namely
A•(M ) = Ap,q(M ) .
p+q=•
119
Slice regular functions and orthogonal complex structures in eight
dimensions
Caterina Stoppato, caterina.stoppato@unifi.it
Università di Firenze, Italy
Coauthors: Riccardo Ghiloni, Alessandro Perotti
The theory of slice regular functions, introduced by Gentili and Struppa in 2006, is a successful
quaternionic analog of the theory of holomorphic functions of one complex variable. It includes
new interesting phenomena, due to the noncommutative setting.
This theory has been applied to the problem of classifying Orthogonal Complex Structures
on open dense subsets R4 \ Λ of R4. Traditionally, this problem had been addressed with
a toolset limited to quaternionic linear fractional transformations: only the case when Λ has
Hausdorff dimension less than 1 and the case when Λ is a circle or a straight line could be
addressed. Then the class of injective quaternionic slice regular functions became available
as a tool for classification, which made other cases approachable. The construction of this
new toolset required a detailed study of the differential topology of quaternionic slice regular
functions. This study was a joint work with Gentili and Salamon published in 2014.
The talk will look at the theory of octonionic slice regular functions, introduced by Gentili
and Struppa in 2010, through the lens of differential topology. This study has an independent
interest, because of the peculiar features of the nonassociative setting of octonions. It leads
to a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions.
Moreover, it opens the path for a possible use of slice regular functions in the study of almost-
complex structures in eight dimensions.
Special geometries with torus symmetry
Andrew Swann, swann@math.au.dk
Aarhus University, Denmark
A survey of recent results on torus symmetry for metrics with special holonomy, particularly G2
and Spin(7), and related geometries such as nearly Kähler six-manifolds. The known explicit
examples of these geometries all have a large compact symmetry group and in particular an
action of some torus. Studying the orbit structure and using ideas such as multi-moment maps,
one gets pictures related to the Delzant description for toric manifolds, and/or certain geometric
flows.
∂-Harmonic forms on compact almost Hermitian manifolds
Adriano Tomassini, adriano.tomassini@unipr.it
Università di Parma, Italy
Let M be a smooth manifold of dimension 2n and let J be an almost-complex structure on M .
Then, J induces on the space of forms A•(M ) a natural bigrading, namely
A•(M ) = Ap,q(M ) .
p+q=•
119
