Speaker 
Title and Abstract 
Filippini, Sara 
An introduction to Hodge theory
We give an introduction to the basic concepts of Hodge theory,
including the notion of a pure Hodge structure and the Hodge filtration.
We then discuss the uses of this theory in the study of cohomology,
including the Hodge decomposition and the Lefschetz decomposition.

Garcia–Raboso, Alberto 
Introduction to nonabelian Hodge theory: Higgs bundles and local
sytems I
Hodge theory can be extended to cohomology with coefficients
in nonabelian groups. For GLr, this results in a correspondence between
flat vector bundles (which, by the RiemannHilbert correspondence, are
the same as local systems), and socalled Higgs bundles. Over smooth
projective varieties, the latter are not only holomorphic, but in fact
algebraic, objects. We will discuss this correspondence and how it is
useful (among other things) for constructing variations of Hodge structure.

Harder, Andrew 
The Kuga–Satake construction
There is a well known construction of Kuga and Satake which
embeds the transcendental Hodge structure of any algebraic K3 surface
into the second cohomology group of an abelian variety. I will give
an overview of this construction and show how it can be turned into
an explicit geometric correspondence in some situations.

Kerr, Matt 
Algebraic and arithmetic properties of period maps
The three talks will cover MumfordTate groups and boundary
components, as well as limits of normal
functions and generalized AbelJacobi maps.

Laza, Radu 
Classical period domains
I will discuss the classification of Hermitian symmetric
domains, the connection between HSD’s and VHS, and some examples
of moduli spaces uniformized by HSD’s.

Peters, Chris 
Period domains and their differential geometry revisited
Griffiths’ period domains classify polarized Hodge structures;
they have a pure Lietheoretic description as reductive domains. This
can be used to translate differential geometric properties on associated
bundles into properties for associated Liealgebras. In particular,
this gives an explanation for the curvature properties for the natural
invariant metric on such domains. These results were all known in the
seventies of the last century and due to Griffiths and Schmid. They
obtained them making heavy use of detailed Lietheory. The proposed
approach avoids this. Mixed Hodge structures can also be described by
period domains, but these are no longer reductive. The transformation
group acting transitively on such a domain is no longer semisimple
and the natural metric in general is no longer invariant. This complicates
the curvature calculations. Nevertheless, in special cases which are
of interest in geometric applications one can deduce some properties
analogous to what happens in the pure case. The description of mixed
period domain was also known for some time and is due to Usui and Kaplan.
The curvature calculations were started by Pearlstein. In a joint work
in progress we are extending this result and give some applications.
It is my intention to explain this in 2 lectures focusing mainly on
the pure case.

Rayan, Steve 
Introduction to nonabelian Hodge theory: Higgs bundles and local
systems II
We will discuss interesting geometric aspects of the moduli
spaces for the objects introduced in Part I
in Garcia–Raboso’s talk.

Ruddat, Helge 
Degenerations of Hodge structures
This talk concerns the behaviour of geometric variations
of Hodge structures near singular fibers in a family as studied by SchmidSteenbrink.
We define the canonical extension of a vector bundle with connection
from the punctured disk to the disk and then extend variations of Hodge
structures by extending the GaussManin connection. The limiting object
to be filled in is a mixed Hodge structure. We define nearby and vanishing
cycles sheaves and state various properties of these.

Schütt, Matthias 
[1] Picard numbers of quintic surfaces
The Picard number is a nontrivial invariant of an algebraic
surface which captures much of its inner structure. It is a fundamental
problem which Picard numbers occur within a given class of surfaces.
For the prototype example of quintics in P3, I will show that all numbers
1 and 45 indeed occur as Picard numbers. The main technique consists
in arithmetic deformations.
[2] 64 Lines on quartic surfaces
In a 1943 paper, Benjamino Segre claimed that a smooth complex
quartic surface contains at most 64 lines. However, his arguments turn
out to be incomplete, and at some places wrong. I will present joint
work with S. Rams which uses elliptic fibrations to give a complete
proof of the corresponding statement over any field of characteristic
other than 2 and 3.

Thompson, Alan 
Variations of Hodge structure and the period map
We begin by defining the period map, which relates families
of Kahler manifolds to the families of Hodge structures defined on their
cohomology, and discuss its properties. This will lead us to the more
general definition of a variation of Hodge structure and the GaussManin
connection.
