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Previous Talks:
Weeks
1 &
2:
Nathan George, Why
Generalized Complex Structures?
An introduction
to the linear algebra of GCS's and their descriptions as a
complex Dirac structure and maximal isotropic subspaces of the
direct sum of a vector space with its dual. We
constructed a basis for a Dirac structure on a real vector
space and showed how this allows us to identify it with the
orthogonal group via a generalized Cayley transform.
Finally, we looked at the Clifford algebra on the direct sum
space and showed how the spinor description follows naturally
as an action of the Clifford algebra on the exterior algebra
of the dual. Main references are Chapter 2 of
(1),
as well as parts of
(2)
and (3).
Weeks 1 & 3:
Santiago Canez, Spinors
Provided an introduction of spin structures
and spin bundles and constructed the spinor bundle for the
generalized tangent bundle. In conclusion, we introduced the
Courant bracket and discussed how the integrability of
sub-bundles can be explained in terms of spinors. Ref:
(1),
(4),
and (5).
Weeks 3 & 4:
Alan Weinstein,
Integrability and Lie algebroids
Introduced complex
Lie algebroids (of which generalized complex structures are a
special case) and recalled the definition of (real,
holomorphic and complex) Lie algebroids, briefly explaining
how real and holomorphic Lie algebroids are the infinitesimal
objects associated to smooth and holomorphic groupoids. The
rest of the talk, based on material in
(7), was
devoted to examples of complex Lie algebroids on real
manifolds and their “integration” to “holomorphic stacks”.
Week 5:
Santiago Canez,
Generalized Complex Structures
Introduced
the main objects of study this semester – generalized complex
structures. After defining these structures on vector spaces,
we transported the notion to manifolds and discussed how
Courant integrability fits in. I will then discuss how
symplectic and complex structures both give rise to
generalized complex structures, and give some examples of
generalized complex structures on manifolds which admit no
known symplectic or complex structure. Finally, I will
introduce the notion of a twisted generalized complex
structure. Ref: Chapter 4 of
(1),
(4) and
(5).
Week 6: Benoit Jubin,
Generalized Kahler Structures
Introduced
a Riemannian metric on the generalized tangent bundle and
defined generalized Kähler structures, which can be seen as a
further reduction of the structure group of the generalized
tangent bundle to U(n) × U(n). We then discussed their
relation with bi-Hermitian geometries introduced in
supersymmetric models and gave some examples of generalized
Kähler four-manifolds. Ref: Chapter 6 of
(1).
Week 7:
Rajan Mehta,
Supermanifolds, Lie
algebroids, and Courant algebroids
Will
present an introduction to the "diffrential algebra"
approach to Lie algebroids, as well as an
interpretation of this approach in terms of
supermanifolds. Then we will describe a
characterization, due to Roytenberg, of Courant
algebroids as degree 2 symplectic Q-manifolds, with
primary focus on the standard Courant algebroid TM \oplus
T*M. Ref:
(6),
(8),
and (13).
Week 8:
None, Conference at MSRI
Week 9: None, Spring
Break
Week 10: None
Week 11:
Christian Blohmann,
Twisted Courant Brackets
Explained the twist of the Courant bracket by a
3-form and its relation to central extensions of the
tangent Lie algebroid. Commented on the geometric
realization of these central extensions by line
bundles and bundle gerbes.
Ref:
(1)
and (5).
Week 12:
Rajan Mehta,
The homological view of Lie bialgebroids
The main purpose of this talk is to describe in
detail Roytenberg's supergeometric "commuting
Hamiltonian" approach to Lie bialgebroids. One
nice aspect of this point of view is that it makes
the generalization to "quasi" structures very
natural. Then we'll look at how Dirac
structures fit into the picture.
Ref:
(6),
(8), (9),
and (13).
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