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INdAM Fellowship Programs in Mathematics and/or Applications cofunded by Marie Skłodowska-Curie Actions



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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

  1. Ó. Maciá, S. G. Chiossi, SO(3)-structures on 8- manifolds, accepted by Ann. Glob. Anal. Geom. (2012)
  2. P.-A. Nagy, S. G. Chiossi, Systems of symplectic forms on four- manifolds, accepted by Ann. Sc. Norm. Sup. Pisa (2011)
  3. P.-A. Nagy, S. G. Chiossi, Complex homothetic foliations on Kähler manifolds, Bull. London Math. Soc., 44 (2012), 113-124
  4. A.Fino, S. G. Chiossi, Nearly integrable SO(3) geometry, Publ. de la RSME vol.10 (2007) 131-136.
  5. A.Fino, S. G. Chiossi, Nearly integrable SO(3) structures on 5- dimensional Lie groups, J. Lie Theory 17 (3), 539-562 (2007)
  6. A.Fino, S. G. Chiossi, G2 geometry, solvable Lie groups and (super)symmetries, Publ. de la RSME vol.10 (2007) 275-279
  7. I. Agricola, A. Fino, S. G. Chiossi, Solvmanifolds with integrable and non-integrable G2 structures, Diff. Geom. and its Appl. 25/2, 125-135, (2007)
  8. A. Fino, S. G. Chiossi, Special metrics in G2 geometry, Rev. Un. Mat. Arg. 47 (1), 35-49, (2006)
  9. A. Fino, S. G. Chiossi, Conformally parallel G2 structures on a class of solvmanifolds, Math. Z. 252 (4), 825-848, (2006)
  10. A. Swann, S. G. Chiossi, G2 structures with torsion from half-integrable nilmanifolds, J. Geom. Phys. 54 (3), 262-285, (2005)
  11. S. Salamon, S. G. Chiossi, The intrinsic torsion of SU(3) and G2 structures, Differential Geometry, Valencia 2001, World Sci. Publishing, 115-133, (2002)

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
  • Séminaire Besse
    École Polytechnique, Paris, France, 2006


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