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.

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).

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;
- Les Tables de Multiplication en ligne;
- 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.

- G. Marmo (Naples, Italy),
- A. Shelekhov (Tver’, Russia)
- A. Vinogradov (Salerno, Italy).

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).

The latest version of the program can be downloaded here.

G. Barnich - Algebraic structure of gauge systems: Theory and applications.

The general construction of the BRST-antifield 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 2-form) 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 n-forms, 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 Batalin-Vilkovisky 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 Marsden-Ratiu reduction of Poisson manifolds and of the Bursztyn-Cavalcanti-Gualtieri 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 k-planes in Cn. 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 T-equivariant 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 Z-algebra. 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 finite-rank Krasilshchik-Vinogradov coverings of PDEs are in one-to-one 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 infinite-dimensional Lie algebras of matrix-valued 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 one-parameter 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 well-known that the Euler-Lagrange 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 Euler-Lagrange 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 so-called “variational integrals”.

The C-Spectral 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 bi-degree (0,n) of the term E1, from which one can obtain the corresponding Euler-Lagrange equations just by applying the differential d1. The coordinate invariance of the Euler-Lagrange 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 far-reaching 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 C-Spectral 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 Euler-Lagrange equations (called relative) obtained from the relative C-Spectyral Sequence, are graded objects. A nice discovery is that the transversality conditions and the standard Euler-Lagrange equations constitute its non-trivial 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 L-infinity 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 three-web, which consists of three foliations. The totality of such structures must be investigated up to local diffeomorphisms. For instance, since any three-web make it possible to define a binary operation (called coordinate loop) of the third foliation, one can classifies three-webs in terms of their coordinate loops (W. Blaschke, Tomsen, K. Reidemeister, G. Bol, S.S. Chern, M. Akivis, 1926—1969). Moreover, any three-web 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 three-webs 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 three-web one can associate a G-structure and ask whether such a structure is closed (Akivis, Fedorova, Shelekhov). The concept of closed G-structures 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 three-webs 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) Rho-tensors 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 G0. 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 G0-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 Euler-Lagrange 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.

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, 23-26), 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.

Non-invited 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.

Hotel price policy aims at encouraging guests to choose full meals accommodation. Upon timely request, the Hotel can also arrange free take-away meals for those people who want to spend some time off the conference.

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).

- M. Bächtold (Zürich),
- A. De Paris (Naples),
- C. Di Pietro (Salerno),
- G. Moreno (Naples),
- G. Sparano (Salerno),
- L. Vitagliano (Salerno).

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 north-western 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 world-famous 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 (1740-1813).

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 non-stop buses leaving every 20-30 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.

Multiplications

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