The Courant-Herrmann conjecture

The Courant

On Regular Courant Algebroids

For any regular Courant algebroid, we construct a characteristic class à la Chern-Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Ševera class (in H DR(M)). On the other hand, when the Courant algebroid is a quadratic Lie algebra g, it coincides with the class of the Cartan 3-form (in H(g...

Transitive Courant algebroids

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describ...

Courant-Fischer and Graph Coloring

4.1 Eigenvalues and Optimization I cannot believe that I have managed to teach three lectures on spectral graph theory without giving the characterization of eigenvalues as solutions to optimization problems. It is one of the most useful ways of understanding eigenvalues of symmetric matrices. To begin, let A be a symmetric matrix with eigenvalues α1 ≥ α2 ≥ · · · ≥ αn, and corresponding orthono...

Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent ∆ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie br...