I will discuss recent joint work with Luca Migliorini on new structures on the singular cohomology of complex projective manifolds.

Introduction to the relation between the representation theory of a finite subgroup G of SL(n, C) and the geometry of a resolution of the quotient Cⁿ/G. I'll also look at the Batyrev-Kontsevich solution to the McKay problem on the level of stringy Hodge numbers.

By strengthening the assumptions on G, certain resolutions of Cⁿ/G can be constructed as moduli of quiver representations and hence the representation theory of G is related to deeper geometric information (K-theory, derived categories). Time permitting, I'll speculate on a link between the approaches of talks I and II.

Birational Yang-Baxter maps ("set-theoretical solutions of the Yang-Baxter equation") are considered. A birational map (x,y)→ (u,v) is called quadrirational, if its graph is also a graph of a birational map (x,v)→ (u,y). We obtain a classification of quadrirational maps on CP¹x CP¹, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.

A classical theorem of PDE translated in the theory of D-modules: the Cauchy-Kovalevski theorem.

Introduction to D-modules. Examples. Note: different time and place

Holonomic D-modules and the Riemann-Hilbert correspondence. Note: different time and place.

An example of application of D-modules to Lie groups: the Harish-Chandra theorem.

While in Riemannian geometry for a manifold to have positive total scalar curvature is a very weak condition, in Kähler geometry this condition already implies that the manifold should be of negative Kodaira dimension. The question we address in this talk is a stronger form of the converse. If M is a projective manifold of negative Kodaira dimension, does it admit a Hodge metric of positive total scalar curvature? We present a possible approach to this question in the case of 3-dimensional rationally connected manifolds.

I shall present a new homological apparatus for the study of secondary geometric invariants. The fundamental objects are called spark complexes. Quasi-equivalent spark complexes yield isomorphic secondary theories. The machinery shall be applied to: 1. Establish the equivalence of many quite different approaches to differential characters ((R, Z)-sparks). 2. Develop an analogous theory of (O, Z)-sparks, with applications to characteristic classes and foliations. 3. Present a generalization of 2. to higher truncations of the Dolbeault complex. This gives a theory of secondary classes which contains Deligne cohomology. 4. Present an arithmetic spark complex which extends the arithmetic Chow groups of Gillet-Soulé.

I shall present a new homological apparatus for the study of secondary geometric invariants. The fundamental objects are called spark complexes. Quasi-equivalent spark complexes yield isomorphic secondary theories. The machinery shall be applied to: 1. Establish the equivalence of many quite different approaches to differential characters ((R, Z)-sparks). 2. Develop an analogous theory of (O, Z)-sparks, with applications to characteristic classes and foliations. 3. Present a generalization of 2. to higher truncations of the Dolbeault complex. This gives a theory of secondary classes which contains Deligne cohomology. 4. Present an arithmetic spark complex which extends the arithmetic Chow groups of Gillet-Soulé.