We will discuss the basics of what is now known as "generalized geometry", a subject that took form after the works of Hitchin and Gualtieri around 10 years ago. Generalized geometry provides a broad framework in which various geometric structures, such as symplectic and complex, can be seen in a unified manner. This brings new insights into known geometries, reveals new structures, and has interesting ties with physics. Topics discussed in this minicourse will include Courant algebroids and Dirac structures, pure spinors, generalized complex and Kahler geometry, along with the main geometric constructions, examples, and recent aspects of the subject.
Non-commutative geometry is the geometry of associative algebras. One way to give a meaning to this idea was indicated by Kontsevich and Rosenberg, who proposed to associate to an associative algebra the sequence of spaces of all its n-dimensional representations, for each n. The geometry of these spaces, called representation schemes, should encode the non-commutative geometry of the given associative algebra. This idea works well if the representation schemes are smooth, which is the case if the associative algebra is smooth in the sense of Cuntz and Quillen. In the general case derived representation schemes, recently introduced by Berest, Khachatryan and Ramadoss, are more appropriate. In these lectures I will introduce this notion and the related notions of derived character schemes and of representation homology. I will discuss some simple examples, applications to combinatorial identities and some open problems and conjectures.
The notion of Batalin—Vilkovisky algebra first appeared in mathematical physics (renormalization theory by Batalin—Vilkovisky) and geometry (Poisson manifolds by Koszul). But it also plays a role in topology (double loop spaces by Getzler and string topology by Chas-Sullivan) and mirror symmetry (Frobenius manifolds by Barannikov-Kontsevich—Manin). this notion is actually quite simple since made up of a commutative product and degree one operator of differential order 2. However, all the above mentioned BV-algebras structures admit a chain complex or homology groups for underlying space. Therefore, there is a need for a homotopy theory for BV-algebras. A state of the art of this topic will be given together with applications to these fields.
In 1970s K. Saito discovered a remarkable polynomial flat structure on the orbit space of a Coxeter group. Dubrovin showed that it is almost dual to a natural logarithmic Frobenius structure on the complement to the corresponding reflection hyperplane arrangement. I will discuss the general theory of such structures, which naturally appear in the theory of the WDVV equation prominently featured in 2D topological field theory and Seiberg-Witten’s approach to N=2 SUSY Yang-Mills theory.