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# Simon G. Chiossi

INdAM-COFUND Outgoing fellow between 2012-07-01 and 2014-06-30

### Project information

#### Title

SO(3): The geometry of SO(3) manifolds

#### Outgoing host organisation Return host organisation

Philipps-Universitaet Marburg Politecnico di Torino

Biegenstrasse 10 – 12 Corso Duca degli Abruzzi, 24

35037 Marburg, Germany 10129 Torino, Italy

http://www.uni-marburg.de/ http://www.polito.it/

#### Abstract

Geometrical objects can be studied – and many of their features understood – by the way certain transformations modify them. For instance, the ‘perfect’ symmetry of a sphere in 3-dimensional space depends on the fact that it is not changed by rotations: the round sphere, in other words, is invariant under the action of the Lie group *SO(3)*. Although this is a very simple group, the theory of its actions is nonetheless extremely yielding.

This project proposes to investigate a situation analogous to the one above, in which *SO(3)* acts infinitesimally on a smooth manifold of larger dimension. Said more precisely, the goal is to study certain actions of the Lie group *SO(3)* on the tangent spaces of smooth *5*-, *7*-, *8*– and *4k*-dimensional manifolds, shedding light on the topological, algebraic, and geometrical aspects.

The study is expected to produce rich classes of examples and lead to a broad classification (both local and global) based on structure theorems. Evidence will be brought to the fore linking the *SO(3)*-theory to other subjects, high-energy particle physics included. This will furnish a different understanding of quaternionic manifolds, explicit constructions of manifolds with exceptional geometry, generalised Einstein metrics and the like.

### Fellow information

#### Research interests

My interests are mainly spread within the realms of

geometry, topology, analysis and algebra, especially the intersections

thereof (à la Klein). They dwell upon two areas, vaguely described

by

- Geometric structures on Riemannian, almost

Hermitian and quaternionic manifolds:

special holonomy, existence of Einstein metrics,

Kähler manifolds, Riemannian and holomorphic foliations, complex

surfaces, minimal models and formality, harmonic differential forms;

supersymmetric models in string theory, complexes of differential

operators (eg Dirac, twistor, Dolbeault, et c.). - Lie theory:

representations of Lie groups and algebras,

orbits of differential forms and spinors, Lie groups actions, geometry

and topology of homogeneous spaces, nilpotent and solvable Lie groups;

deformations of invariant structures (eg complex, symplectic, …)

#### Previous positions, awards

*2008-2012*: ‘Senior research fellowship’, Politecnico di Torino, Italy*2005-2008*: ‘Wiss. Mitarbeiter’, Humboldt-Universität zu Berlin, Germany*2003-2005*: ‘Adjunkt’, Syddansk Universitet, Odense, Denmark*2003*: EDGE ‘Young Researcher’, Syddansk Universitet, Odense, Denmark

#### Publications, preprints, other works

- Ã. MaciÃ¡, S. G. Chiossi,
*SO(3)-structures on 8- manifolds*, accepted by Ann. Glob. Anal. Geom. (2012) - P.-A. Nagy, S. G. Chiossi,
*Systems of symplectic forms on four- manifolds*, accepted by Ann. Sc. Norm. Sup. Pisa (2011) - P.-A. Nagy, S. G. Chiossi,
*Complex homothetic foliations on Kähler manifolds*, Bull. London Math. Soc., 44 (2012), 113-124 - A.Fino, S. G. Chiossi,
*Nearly integrable SO(3) geometry*, Publ. de la RSME vol.10 (2007) 131-136. - A.Fino, S. G. Chiossi,
*Nearly integrable SO(3) structures on 5- dimensional Lie groups*, J. Lie Theory 17 (3), 539-562 (2007) - A.Fino, S. G. Chiossi,
*G*, Publ. de la RSME vol.10 (2007) 275-279_{2}geometry, solvable Lie groups and (super)symmetries - I. Agricola, A. Fino, S. G. Chiossi,
*Solvmanifolds with integrable and non-integrable G*, Diff. Geom. and its Appl. 25/2, 125-135, (2007)_{2}structures - A. Fino, S. G. Chiossi,
*Special metrics in G*, Rev. Un. Mat. Arg. 47 (1), 35-49, (2006)_{2}geometry - A. Fino, S. G. Chiossi,
*Conformally parallel G*, Math. Z. 252 (4), 825-848, (2006)_{2}structures on a class of solvmanifolds - A. Swann, S. G. Chiossi,
*G*, J. Geom. Phys. 54 (3), 262-285, (2005)_{2}structures with torsion from half-integrable nilmanifolds - S. Salamon, S. G. Chiossi,
*The intrinsic torsion of SU(3) and G*, Differential Geometry, Valencia 2001, World Sci. Publishing, 115-133, (2002)_{2}structures

#### Conferences, schools, other events

*Geometric structures on manifolds and their applications*

Marburg, Germany, 2012*Geometry in Bicocca*

Milan, Italy, 2012*Workshop on hyperKähler geometry and related topics*

Bonn, Germany, 2012*Geometric structures on complex manifolds*

Moscow, Russia, 2011*New Trends in Differential Geometry*

L’Aquila, Italy, 2011*Gauge Theory and Complex Geometry*

Leeds, United Kingdom, 2011*Progressi Recenti in Geometria Reale e Complessa*

Levico Terme, Italy, 2010*Conference on Geometry and Topology of Foliations*

CRM Bellaterra, Spain, 2010*Conference on Kähler and related geometries*

Nantes, France, 2009*A St. John Geometry Day*

Torino, Italy, 2009*Workshop on Dirac operators and special geometry*

Marburg, Germany, 2009*Incontro di geometria Riemanniana*

Torino, Italy, 2008*Holonomy Groups and Applications in String Theory*

Hamburg, Germany, 2008*SÃ©minaire Besse*

École Polytechnique, Paris, France, 2006