I will talk about A-hypergeometric systems (also known as GKZ-systems). They can be introduced in order to study families of varieties or exponential sums. Basically, these arise when one takes, for example, a family defined by an equation and puts a different parameter on each monomial in the equation. For example, one has the classical Hasse family of elliptic curves:
X^3 + Y^3 + Z^3 - 3\mu XYZ = 0,
a one-parameter family depending on \mu.
The Picard-Fuchs equation is an ordinary differential equation of the second order in the variable \mu.
From the A-hypergeometric point of view, one would consider the family
\mu_1 X^3 + \mu_2 Y^3 + \mu_3 Z^3 +\mu_4 XYZ = 0,
a four-parameter family depending on \mu_1,...,\mu_4. The analogue of the Picard-Fuchs equation is now a system of PDE's in \mu_1,...,\mu_4. But it is easier to write down this system of PDE's and its solutions at the origin than it is to write down the original Picard-Fuchs equation in \mu and find its solutions at the origin. This will be related to some recent results of Beukers.
Andreatta - Iovita:
Consider a flat family of proper curves over a formal, p-adic disk which degenerates exactly at one point. We use some embedding of the p-adic field in the complexes to base chage the family and obtain a family of curves over a complex disk, degenerating at one point. For the two families we define:
1) The families of universal, unipotent de Rham sheaves associated to the two families.
2) The connection with the unipotent de Rham fundamental groups.
3) The monodromy operators on the Lie algebras of the unipotent de Rham p-adic respectively complex fundamental groups and a comparison between them.
4) T. Oda's theorem and the proof of the result: let K be a finite extension of \Q_p and X a smooth proper curve of genus larger or equal to 2 over K with a semi-stable model over the ring of integers of K. Then X has good reduction if and only if the monodromy operator on the Lie algebra of its unipotent fundamental group vanishes.
Baldassarri: Regular singularities and and Grothendieck-Ogg-Shafarevich formula.
We will explain the meaning of regular singular point from the differential point of view according to Malgrange/Deligne. We then deal with the Grothendieck Ogg-Shafarevich formula.
Chiarellotto - Tsuzuki: The Clemens Schimdt sequence.
The series of talks will start with the definitions of a series of relevant cohomology theories in char p: rigid, convergent, log-congergent, log-rigid and log-crystalline following the work of Shiho. We will then deal with the definition of monodromy in this setting according to the work of Hyodo-Kato. We will complete the tools we need by explaining the realization of the weight/monodromy conjecture in this setting (along the work of Crew). We will end with the proof of the Clemens schimdt sequence.
Illusie: Nearby cycles and monodromy in étale cohomology.
1. Discs and traits.
Analogies. Structure of inertia. Grothendieck's local monodromy lemma.
2. The functors $R\Psi$ and $R\Phi$
Definition. Stalks. Galois action. Tame nearby cycles.
3. General theorems.
Functoriality. Base change. Finiteness. Duality and perversity.
Künneth. Comparison with the complex nearby cycles.
Semistable reduction. Isolated singularities. Quadratic singularities.
5. Grothendieck's local monodromy theorem.
6. The $\ell$-adic weight spectral sequence.
Direct proof of perversity of $R\Psi\Lambda[n]$ in semistable case. Monodromy, kernel, and image filtrations. The weight spectral sequence. Arithmetic case. Main results and conjectures
7. Further developments
Migliorini: Hodge theory of (one-dimensional) degenerations of complex projective varieties
We will deal with the following topics:
-One dimensional degenerations of algebraic varieties: semistable reduction, monodromy, specialization map.
-The monodromy theorem. The monodromy weight filtration.
-Nearby and Vanishing cycle functors.
- Variations of Hodge structures associated with a degeneration.
-The Gauss Manin connection and the limit mixed Hodge structure.
-The Clemens-Schmid exact sequence.