Current Geometry,
the IX Edition of the International Conference on problems and trends of contemporary geometry. Hotel Oriente, Via Luigi Serio 10, 80069 Vico Equense (Naples), Italy. June 24  26, 2008 This page was originally posted on March 4, 2008. This page is constantly updated (last update June 30, 2008). Please check it frequently for the latest news! 
NEW! The photoalbum of the conference just added.
To contact the Organizing Committee of the Conference, please use the following address:
Documents to the Organizing Committe can be faxed to: +39 089 963303.
Ordinary mail to the Organizing Committe can be sent to:
2008 Current Geometry Organizing Committee
c/o prof. Giovanni Sparano
Dipartimento di Matematica e Informatica
Università degli Studi di Salerno
Via Ponte don Melillo
84084 Fisciano, Salerno
ITALY
An electronic copy of the Conference poster can be downloaded here.
The conference is organized under the auspices of
 Associazione “Diffiety School” (Santo Stefano del Sole (AV), Italy);
 Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II" (Italy);
 Dipartimento di Matematica e Informatica, Università degli Studi di Salerno (Italy);
 Istituto Italiano per gli Studi Filosofici (Italy).
Contents
 Reasons
 Scientific Committee
 Preliminary List of Speakers
 Program
 Titles and Abstracts of the Talks
 Accommodation
 Organizing Committee
 Location
Reasons
The power of synthesis of Geometry, which led in the past to the formulation of"grand unification theories", has got an essential role nowadays, especially
because of the growing fragmentation of knowledge due to scientific progress.
In order to avoid too a big dispersion, geometers need a constant dialogue.
Therefore, a stable experience of personal meetings, apart from telematic
interchanges, cannot be renounced. Current Geometry was born to allow a
periodic update about actual progresses in Geometry (and its applications) on
the international scene.
List of topics:
 Secondary Calculus & Cohomological Physics;
 Physics and Geometry;
 Differential Geometry;
 Algebraic Geometry;
 Geometry of PDEs.
People who wish to give a talk at the conference must apply by filling the registration form. Their request will be discusses by the Scientific Committee.
Scientific Committee
 G. Marmo (Naples, Italy),
 A. Shelekhov (Tver’, Russia)
 A. Vinogradov (Salerno, Italy).
Preliminary List of Speakers
Lists below are under constant updating.
INVITED SPEAKERS:
 G. Barnich (Bruxelle, Belgium);
 F. Bottacin (Salerno, Italy);
 A. Cattaneo (Zürich, Switzerland);
 L. Gatto (Turin, Italy);
 S. Igonin (Utrecht, Netherlands);
 M. Marvan (Opava, Czeck Republic);
 G. Moreno (Naples, Italy);
 N. Netsvetaev (St. Petersburg, Russia);
 P. Severa (Geneva, Switzerland);
 A. Shelekhov (Tver'. Russia);
 J. Slovak (Brno, Czech Republic);
 A.M. Vinogradov (Salerno, Italy);
 L. Vitagliano (Salerno, Italy).
Program
The latest version of the program can be downloaded here.Titles and Abstracts of the Talks
List below will be regularly updated.G. Barnich
 Algebraic
structure of gauge systems: Theory and
applications. 
The general
construction of the BRSTantifield formalism for gauge
systems is reviewed with a special emphasis on the role
played by locality. Applications ranging from anomalies
and counterterms in quantum field theory to symmetries
and consistent deformations in the classical framework
are discussed.

F. Bottacin
 Differential
forms on moduli spaces of
sheaves. 
Since their
introduction, moduli spaces of sheaves have attracted
much interest. From the geometric point of view they
are rather natural objects to study, since they
classify isomorphism classes of vector bundles or, more
generally, coherent sheaves on a fixed variety X. In
more recent years they have also found important
applications in some areas of mathematical physics
(e.g., integrable systems, string theory). In this talk
we shall briefly review the construction of moduli
spaces of sheaves and their infinitesimal deformation
properties. We shall describe how some "geometric
structures" on a moduli space M of sheaves on a variety
X are often induced by analogous structures on X. One
of the first examples of this general phenomenon was
discovered by Mukai in 1984: he proved that the
existence of a holomorphic symplectic structure (i.e.,
closed holomorphic 2form) on a smooth projective
surface X determines, in a natural way, a holomorphic
symplectic structure on any moduli space of stable
sheaves on X. The existence of a symplectic structure
on some moduli spaces of sheaves can lead, in some
cases, to the construction of integrable systems. As an
example, we review the construction of an important
class of integrable systems, known as Hitchin's
systems. Finally, we shall present some new results
concerning the construction of closed holomorphic
nforms, for any n, on moduli spaces of sheaves on
higher dimensional varieties.

A. Cattaneo
 Geometry
and Topological Field Theories.

Graded
manifolds are a generalization of manifolds where the
algebra of functions instead of being strictly
commutative is commutative in the graded sense (graded
manifolds are actually a refined notion of
supermanifolds in which the grading is given by an
integer). Most geometric constructions on manifolds can
be extended to graded manifolds if they are formulated
algebraically in the proper way. In particular, one can
put a structure of (usually infinite dimensional)
graded manifold on the space of maps between graded
manifolds and various structure on the source and
target manifold may be naturally lifted to the map
manifold. As shown by Alexandrov, Konstevich, Schwarz
and Zaboronsky, appropriate structures on the source
and target manifolds give then rise on the map space to
the structure needed to define a topological field
theory in the BatalinVilkovisky formalism. The target
structures correspond to many interesting geometric
structures on an ordinary manifold, like, e.g.,
Poisson, Courant, Generalized Complex structures.
Boundary conditions for the topological theory are then
associated to interesting substructures of the above,
like, e.g., coisotopic submanifolds or Dirac
structures. The graded manifold point of view of these
classical structures has a further advantage besides
its relation with topological field theories (which are
the starting point for the quantization of these
structures): namely, reduction can be approached more
naturally and more systematically. From this viewpoint
one gets, e.g., an ameliorated version of the
MarsdenRatiu reduction of Poisson manifolds and of the
BursztynCavalcantiGualtieri reduction of Generalized
Complex structures. In this talk I will give a short
introduction to graded manifolds and outline some of
the above applications.

L. Gatto
 Cohomologies
on grassmannians via derivations on exterior
algebras. 
Let G(k,n)
be the complex grassmannian variety parameterizing
kplanes in C^{n}. Its integral singular
cohomology ring H*(G(k,n)) has been widely investigated
along the last two centuries. Furthermore, some
geometrically relevant "deformations" of it have been
defined and studied in the last few decades. In his
celebrated paper "The Verlinde Algebra and the
cohomology of Grassmannians", E. Witten introduced and
begun the study of the quantum cohomology ring of
G(k,n), which is a suitable quantum deformation of
H*(G(k,n)). The Tequivariant cohomology of G(k,n),
instead, is the deformation one considers when the
grassmannian comes equipped with a certain action of a
torus T: it has been extensively studied in an
important work by Knutson and Tao via the beautiful
combinatorics of puzzles.
The aim of the lecture will be to show that all these different kind of cohomology theories living on grassmannians can be treated in a unified way within a new, more general and more powerful formalism (in spite of being very elementary) regarding derivations of the exterior algebra of a free module over a commutative Zalgebra. Such a description is also related with some recent important work by D. Laksov and A. Thorup, which will be briefly discussed. 
S.
Igonin 
Analogues
of coverings and the fundamental group in the category
of PDEs. 
We describe
a new geometric invariant of PDEs: the fundamental Lie
algebra of a system of PDEs.
This algebra is somewhat analogous to the fundamental group in topology and is responsible for Lax pairs and Backlund transformations in soliton theory. Its key property is that finiterank KrasilshchikVinogradov coverings of PDEs are in onetoone correspondence with vector field representations of the fundamental algebra. We compute these Lie algebras for several integrable systems. They turn out to be isomorphic to certain infinitedimensional Lie algebras of matrixvalued functions on rational, elliptic, and hyperelliptic curves. 
M.
Marvan 
Zero
curvature representations, horizontal gauge cohomology,
and recursion operators. 
Zero
curvature representations play important role both in
integrability theory and geometry of PDE. Many
equations of geometric origin come with a canonical
zero curvature representation. Such equations are
integrabile if we can embed the zero curvature
representation into a oneparameter family. This might
be called the spectral parameter problem.
We show that the first gauge cohomology group associated with a zero curvature representation contains obstructions to the solution of the spectral parameter problem. We shall present a method to compute the group, which allows us to bound the class of integrable equations from above. Next, we shall discuss a method to insert the parameter by using a recently observed relation between zero curvature representations and inverse recursion operators. 
G. Moreno
 Cohomological
theory of transversality conditions in the Calculus of
Variations. 
It is
wellknown that the EulerLagrange equations can be
easily obtained as an extension of the standard
differential of a function, thanks to the Green
formula, also known as integration by parts. In this
way, however, the coordinate invariance of the
EulerLagrange equations does not look natural. In
fact, the Calculus of Variations lies on weak
conceptual foundations, since it is not able to define
in a proper mathematical way the elementary objects of
its own study, the socalled "variational integrals".
The CSpectral Sequence is a cohomological theory associated with the spaces of infinite jets, invented by A.M. Vinogradov in the late seventies, to give a solid conceptual foundation to the Calculus of Variations. In this theory, the "variational integrals" appear as the elements of bidegree (0,n) of the term E_{1}, from which one can obtain the corresponding EulerLagrange equations just by applying the differential d_{1}. The coordinate invariance of the EulerLagrange equations is not even under question, since there are no coordinates involved. Every ingredient of the Calculus of Variations, including the Green formula itself and the theory of integration over arbitrary manifolds, is described in terms of homological algebra. This erases the analytical heritage from the whole framework, paving the way to farreaching generalizations. The abstractedness of a cohomological approach to the Calculus of Variations is countered by a great versatility. In this talk I propose a nice experiment: if we replace the category of manifolds by the category of manifolds with boundary, the whole machinery, being completely functorial, does not see the difference, and it works untroubled producing an interesting output: a cohomological theory of the variational problems with free boundary, which I called the relative CSpectral Sequence. Such a theory cannot have any analogous in elementary Calculus, since points have no boundary, and this is confirmed by the fact that the EulerLagrange equations (called relative) obtained from the relative CSpectyral Sequence, are graded objects. A nice discovery is that the transversality conditions and the standard EulerLagrange equations constitute its nontrivial components, thus explaining why the two of them always arise together. 
N.
Netsvetaev  Topology
of complex algebraic varieties. 
Abstract: NA.

P.
Severa 
Integration
and differentiation in the world of Linfinity algebras
and differential graded
manifolds. 
The
correspondence between Lie algebras and Lie groups has
a rich generalization, with Lie algebras replaced by
differential graded manifolds (possibly with additional
structure) and Lie groups replaced by Kan simplicial
manifolds and similar objects. We shall discuss both
the integration procedure based on simple ideas of
Sullivan's rational homotopy theory, and
differentiation based on computing differential forms
on contravariant functors.

A.
Shelekhov 
Geometry
and algebra of functions of two
variables. 
A smooth
function z=f(x,y)
of two variables defines on its domain
M
a geometric
structure called threeweb,
which consists of three foliations. The totality of
such structures must be investigated up to local
diffeomorphisms. For instance, since any threeweb make
it possible to define a binary operation (called
coordinate
loop) of the
third foliation, one can classifies threewebs in terms
of their coordinate loops (W.
Blaschke, Tomsen, K. Reidemeister, G. Bol, S.S. Chern,
M. Akivis, 1926—1969). Moreover,
any threeweb gives rise to a connection on
M,
whose structure equations are PDEs called the
structure
equations of the web (Chern,
1936 and Akivis, 1969). One can then characterize
certain classes of threewebs in terms of the curvature
and torsion tensors of the associated connection.
Remarkable classes are those given by the
regular
webs (A.
Shelekhov, 2005), and the web consisting of 3 families
of straight lines, which is essential in the problems
of nomography
(V.V.
Goldberg, V. Lycagin, M. Akivis, 2005). Finally, to any
threeweb one can associate a Gstructure
and ask whether such a structure is closed
(Akivis,
Fedorova, Shelekhov). The concept of closed
Gstructures
was introduced by M. Akivis in 1975 as an important
generalization of Lie groups, due to its physical
applications. In this context A. Malcev proved in 1953
that analytic Moufang loops are in fact special classes
of threewebs and A. Shelekhov proved that such loops
can be embedded into a Lie group.

J. Slovak
 Invariant
calculus for Parabolic
Geometries. 
Motivated
by the rich geometry of conformal Riemannian manifolds
and the already classical development of geometries
modelled on homogeneous spaces G/P with G semisimple
and P parabolic, the Weyl structures and the preferred
connections were introduced in this general framework
by Andreas Cap and myself a few years ago. In
particular, the notions of scales, closed and exact
Weyl connections, and (Schouten's) Rhotensors were
extended, and straightforward generalizations of
classical normal coordinates in affine geometry have
been discussed. In this setting, the Weyl connections
on a parabolic geometry of type G/P correspond to
reductions of the parabolic structure group P to its
reductive part G_{0}. Following the conformal
Riemannian example, the differential invariants of
parabolic geometries of any fixed type G/P are
considered as affine invariants of the Weyl
connections, expressed by means of the algebraic
G_{0}invariant operations, and with values
independent of the particular choices.
After a brief review of the main ingredients of the parabolic geometry theory, an exposition of a calculus for the differential invariants is presented, together with an analogy to the classical 'first invariant theorem' in Riemannian geometry. Exactly as in the case of the conformal Riemannian calculus due to Schouten and Wünsch, the approach rather describes a much larger class of expressions distinguished by very special algebraic transformation behaviour. All differential invariants are then special cases of the latter expressions and thus the difficulties in their treatment are heavily reduced. At the same time, there are general vehicles to create and treat such invariants, the BGG and general tractor calculi in particular. 
A.M.
Vinogradov  Differential
Calculus over Commutative Algebras: state of art and
perspectives. 
Abstract: TBA.

L.
Vitagliano  On the
Geometry of the Covariant Phase
Space. 
The
covariant phase space (CPS) of a lagrangian field
system is the solution space of the associated
EulerLagrange equations. It is, in principle, a nice
environment for covariant quantization of a lagrangian
field theory. Indeed, it is manifestly covariant and
possesses a canonical (functional) presymplectic
structure W (as first noticed by Zuckerman in 1986)
whose degeneracy (functional) distribution is naturally
interpreted as Lie algebra of gauge transformations.
The CPS has been often described by functional analytic
methods. I will describe it geometrically (and
homologically) in the framework of Secondary
Calculus.

Accommodation
Arrival day is Monday, June 23, and departure day is Friday, June 27.
Each invited speaker will be provided with a double room at the hotel Oriente. A double room can be used by a single person or be shared with an accompanying person. A double room will be booked by default for each invited speaker, for four nights (June, 2326), by the Organizing Committee, which kindly asks them to fill up the hotel rooms' booking form. Invited speakers can prolong their stay beyond these terms, or invite other people, on their own charge. Hotel Oriente guarantees the same special rates for the extra days and the extra people, provided that the interested participants fill up the special hotel rooms' booking form.
Invited speakers are entitled to reimbursement for travel expenses. They can buy their own tickets and then, during their stay at the conference, fill the reimbursement form provided by the organizers (IBAN number is an essential data). Invited speakers are kindly requested not to misplace the tickets and the boarding cards and to have them sent to the Organizing Committee upon the end of their travel, to successfully carry out the reimbursement procedure. Alternatively, invited speakers can ask to the Organizing Committee to buy tickets for them, specifying their travel preferences.
Breakfast, lunch and supper will be served within the hotel walls to all invited speakers and accompanying persons. Rooms are covered by wireless network for internet access.
Scientific activities will take place in the hotel's own conference rooms.
The invited speakers, the accompanying persons, and all those who want to attend to the conference must register at the designated registration desk, located in the hotel, upon their arrival. All the registered persons can take part to scientific activities and enjoy the coffee breaks. Anyone who wish to attend to the conference should be so kind and inform the Organizing Committee in advance about his/her coming via electronic or ordinary mail.
Noninvited speakers and all those who would like to spend one ore more full day at the hotel must fill the registration form. Lodging and meals expenses are on their own charge. People should obviously feel free to choose other hotels or different lodging solutions at their will, but must take care of booking their own rooms. Also participants who do not lodge at the Hotel are kindly requested to fill the registration form.
Hotel Oriente's special rates for participants to the conference are:
 Double room and breakfast: 50 EUR per person.
 Double room, breakfast and one meal: 60 EUR per
person.
 Double room and full meals: 70 EUR per person.
 Double room single usage and breakfast: 80 EUR per
person.
 Double room single usage, breakfast and one meal: 100
EUR per person.
 Double room single usage and full meals: 110 EUR per
person.
 One meal: 25 EUR per person.
People who intend to book rooms at the hotel Oriente are kindly suggested to stress that they are going to participate to the conference in order to benefit of the special rates. Booking requests addressed to the hotel Oriente can be sent to the Organizing Committee by means of the hotel rooms' booking form.
The hotel offers classy inner environments as well as a nice terrace from which it is possible to behold the Bay of Naples while tasting typical Italian dishes.
The hotel is few meters off the train station "Vico Equense" (look at the map).
Organizing Committee
 M. Bächtold (Zürich),
 A. De Paris (Naples),
 C. Di Pietro (Salerno),
 G. Moreno (Naples),
 G. Sparano (Salerno),
 L. Vitagliano (Salerno).
Location
Vico Equense takes
its name from Latin words Vicus  a noun
meaning a group of houses located close to a town
or hamlet  and Aequus  an adjective
meaning "flat", referring to the plateau where the
original settlement was located. Today Vico
Equense has got ≈ 20.000 inhabitants. The
territory stretches itself along the northwestern
side of the peninsula of Sorrento, the strip of
land which separates the two bays of Naples and
Salerno. Off its tip, few kilometers ashore, one
finds the worldfamous island of Capri.
The picture on the left displays a view of the peninsula of Sorrento followed by the island of Capri, from the mount "Sant'Angelo tre Pizzi" (Saint Angel three tips). The Organizing Committee will arrange a guided excursion to the mount if enough participants are interested in it. From the top (40°39' N 14°30' E, 1444 meters above the Sea) it is possible to see both the bays of Naples and Salerno. 
The artistic vision of Vico Equense below is due to the English painter William Marlow (17401813).
Vico Equense can be easily reached by taking the regional train "Circumvesuviana" (literally "circling the Vesuvio") from Naples' main train station "Napoli Centrale", towards destination "T. Annunziata  Sorrento" (the azure line in the below map).
Airport "Capodichino" (NAP) (6 km off Naples center) is connected to Naples' main train station "Napoli Centrale" by nonstop buses leaving every 2030 minutes.
Vico Equense can be also reached directly from the airport "Capodichino" by buses operated by Curreri Viaggi. Details about schedule can be found on the operator web site.