ABSTRACTS of TALKS (Provisional).




Jose Colletti Negreiros CAIO

Universidade de Campinas

caione@ime.unicamp.br


TITLE... ``On new examples of harmonic maps on flag manifolds".

Abstract: In this talk we intend to construct new examples of harmonic maps between flag manifolds or from closed Riemann surfaces into maximal flag manifolds.The most of the results are in the paper "(1,2)-symplectic structures on flag manifolds", Tohoku Mathematical Journal,vol.52(2),171-183 (2000), written by Xiaohuan Mo and myself. We use in a essential way Cartan's moving frame method. Our approach in entirely based in the ideas contained in "Tournaments, Flags and Harmonic Maps", Math.Ann. 277,249-265 (1987) written by F. E. Burstall and S. Salamon. Our results are intimately relate to the ones in "Applications harmoniques et varietes Kahleriennes", Symp.Math.III, Bologna (1970), written by A. Lichnerowicz.





Louis CRANE

Dept. Mathematics, Kansas State Univ., USA

crane@math.ksu.edu


TITLE... ``Integral geometry on the classical and Quantum hyperboloids".

Abstract: By means of the Gelfand Graev transform, we construct propagators and kernels for the classical and q hyperbolic spaces. Applications to relativistic and q relativistic spin nets are studied





Birte FEIX

IMADA, Odense University

feix@imada.sdu.dk


TITLE... ``Hyperkahler metrics on cotangent bundles".

Abstract: We construct the twistor space and show the existence of twistor lines for a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of any real-analytic Kahler manifold; the hyperkahler structure is compatible with the canonical holomorphic- symplectic structure of the cotangent bundle, extends the given Kahler metric and the circle action by scalar multiplication in the fibres is isometric. We also derive some necessary conditions for completeness of these metrics using work by Anderson, Cheeger and Gromoll, and Yau.


References:


M. T. Anderson, On the topology of complete manifolds of non-negative Ricci curvature, Topology 29 (1990), 41-55.

J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom. 6 (1971), 119-128.

B. Feix: Hyperkähler metrics on cotangent bundles, to appear in J. reine angew. Math.

D. Kaledin, Hyperkähler structures on total spaces of holomorphic cotangent bundles, alg-geom/9710026 (1997).

S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228.





Carlos FLORENTINO

Dep. Matematica I.S.T., Lisboa

cfloren@math.ist.utl.pt


TITLE... ``Coherent State Transform and Theta Functions"

Abstract: B. Hall introduced a Coherent State Transform valid for compact Lie groups and proved its unitarity. We generalize his construction for non-diagonal Laplacians and prove unitarity with respect to the averaged heat kernel measure. As an application, we obtain the space of (abelian) theta functions of every level k on an abelian variety, by applying this transform to distributions supported on Bohr-Sommerfeld orbits of level k of a real polarization of the abelian variety. We apply an analogous construction in the non abelian (higher rank) case as well





Mikhail V. FOURSOV

Universite de Lille-I, France

foursov@math.umn.edu


TITLE... ``Lie theory of PDEs and integrability"

Abstract: Lie theory is a powerful tool for treating both ODEs and PDEs. The existence of infinitely many generalized symmetries appears to be characteristic of soliton equations. Implementing classifications of equations possessing higher generalized symetries allows one to find equations that are likely to be solvable by inverse scattering.

Recently, this theory was generalized to the case of equations taking values in a noncommutative associative algebra and many new noncommutative equations were found.

References:

Olver, P., Applications of Lie Groups to Differential Equations, (Second Edition, 1993), Springer-Verlag, New York.

Olver, P.J. and Sokolov, V.V., Integrable evolution equations on associative algebras, Comm. Math. Phys., 193, 245-268 (1998).

Mikhailov, A.V., Shabat, A.B. and Sokolov, V.V., The symmetry approach to classification of integrable equations , in ``What is integrability?'', (V.E. Zakharov, ed.), Springer-Verlag, Berlin, (1991), pp. 115-184.

Foursov. M.V., On integrable evolution equations in commutative and noncommutative variables, PhD thesis, University of Minnesota, 1999.





Marina GREBENYUK

University, Kiev, Ukraina

ahha@i.com.ua


TITLE... ``About the three-component distribution of affine space."

Abstract: We study the three-component distribution of affine space which consist of the basic distribution of the first kind of the r-dimensional linear elements and of the equipping distribution of the first kind of the m-dimensional linear elements and of the equipping distribution of the first kind of the hyperplane elements.

We were built the projective normals of the first kind which determinate bunches of the normals of the first kind of the three-component distribution by an inner invariant method in the differential vicinities of the second and third orders of the forming element of the distribution.





Louis H. KAUFFMAN

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

kauffman@uic.edu


TITLE... ``Virtual Knot Theory, Free Fermions and the Alexander Polynomial"

Abstract: Virtual knot theory ( see L. Kauffman. "Virtual Knot Theory", European J. Comb. 1999, Vol. 20,pp. 663-690) is an extension of classical knot theory to the study of abstract Gauss codes and their representations as knot diagrams with virtual crossings in the plane. Most knot invariants extend to virtuals, but there are many surprises such as virtual knots that have non-integer fundamental group but unit Jones polynomial, and non-trivial virtuals that have integer fundamental group and unit Jones polynomial. This talk will discuss such examples and in particular a relationship with the generlaized Alexander polynomials of Jaeger, Saleur and Kauffman (F. Jaeger, L. Kauffman, H. Saleur. The Conway Polynomial in ${\rm I\!R}^{3}$ and Thickened Surfaces, J. Comb. Theory. 1994).





Igor KANATCHIKOV

Department of Analytical Mechanics and Field Theory, Institute of Fundamental Technological Research, Polish Academy of Sciences and Tallinn Technical University, Tallinn, Estonia

ikanat@ippt.gov.pl, kai@fuw.edu.pl


TITLE... ``Polysymplectic Structure, Poisson-Gerstenhaber brackets, and geometric quantization in field theory"

Abstract: A recent work [1,2,3] on polysymplectic formulation and graded Poisson-Gerstenhaber brackets on differential forms in field theory is reviewed. The formalism is presented as a finite dimensional manifestly covariant generalization of the Hamiltonian formalism to field theory, A possible application to quantization of field theories (c.f. [4]), in particular, to geometric quantization, is outlined. A generalization of the Konstant-Soureau-Segal prequantization formula is presented and an interpretation of the polysymplectic form as a curvature of Quillen's superconnection is put forward.


References


1. I. Kanatchikov, Rept.Math.Phys. 41 (1998) 49-90, hep-th/9709229

2. I. Kanatchikov, Rept.Math.Phys. 40 (1997) 225, hep-th/9710069

3. I. Kanatchikov, hep-th/9612255, in GROUP21, Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras, vol. 2, eds. H.-D. Doebner e.a. (World Sci., Singapore, 1997) p. 894

4. I. Kanatchikov, Nucl.Phys.Proc.Suppl. 88 (2000) 326-330, gr-qc/0004066





Olga S. KOUZNETSOVA

Volgograd State University

astra1987@mail.ru


TITLE... ``The geometry of univalent polynomials"

Abstract: We consider the problem of deciding whether a polynomial will be univalent in the unit disk (we study also the case when the disk can be removed by so-called quadrature domain). This question is of great importance in various areas of complex analysis and its applications. But the full description of the univalent polynomials is known for the degrees 2 and 3 only.

We consider new approach for the problem concernig on the geometric point of view. Namely, we using the polynomial star-likeness property of the set of all univalent polynomials and study its geometric and extremal structure.

Some applications to the complex moments theory will be also discussed.





Petr KULISH

U.C.E.H. Universidade do Algarve, Campus do Gambelas, 8000 Faro, Portugal

(on leave of absence from the Steklov Mathematical Institute, St.Petersburg, Russia )

pkulish@ualg.pt


TITLE... ``Reflection Equation and Link Invariants in Handlebody"

Abstract: The Yang-Baxter equation is actively used to describe link invariants in three dimensional space. The braid group in a handlebody has additional relation, which coincides with the reflection equation, used in the quantum inverse scattering problem to study integarble systems with non-periodic boundary conditions. Some solutions of the reflection equation can be used to construct link invariants in handlebody.

(Ref.: P.P.Kulish and R.Sasaki "Covariance properties of reflection equation algebras", Prog. Theor. Physics, 89, N 3 (1993) 741-761; and recent preprints POMI (St.Petersburg) of P.P.Kulish and A.M.Nikitin on this subject )





Trinidad Perez LOPEZ

Departamento de Geometria y Topologia. Universidad de Santiago de Compostela.

trinipl@usc.es


TITLE... ``Harmonic-Killing, pluriharmonic and
$\alpha$-pluriharmonic vector fields"

(joint work with C.T.J. Dodson and M.E. Vázquez-Abal)

Abstract: We have considered the harmonicity of local infinitesimal transformations associated to a vector field on a (pseudo)-Riemannian manifold to characterise intrinsically a class of vector fields that we have called harmonic-Killing vector fields. We have extended this study to other properties, such as the pluriharmonicity and the $\alpha$-pluriharmonicity ($\alpha$ harmonic 2-form) of the local infinitesimal transformations, obtaining characterisations of these kind of vector fields.

Different properties have been considered for the integral flows corresponding to vector fields. For instance, when the corresponding 1-parameter group of local transformations consists of isometric maps, affine maps or conformal maps, a vector field is called respectively Killing, affine-Killing or conformal. However, harmonicity has only been used before to study other aspects of vector fields. Yano ([5]), defined harmonic vector fields as those having harmonic associated 1-form. Several authors ([3], [4]), use the harmonicity of the section induced on the tangent bundle with different lift metrics: Sasaki, complete, ... .

We give a new intrinsic characterization for a class of vector fields through the harmonicity of the local transformations arising from their integral curves. An easily-used necessary and sufficient intrinsic condition on a vector field is obtained for its 1-parameter group of local transformations to consist of harmonic maps. This notion, of harmonic-Killing vector field, gives rise to new examples of harmonic maps in pseudo-Riemannian geometry, especially for compact manifolds and we link these to known results. The approach emphasises the importance of the complete lift metric for tangent bundles in the study of harmonicity.

We provide the relationship among Killing, affine-Killing, conformal and harmonic-Killing vector fields and show the characterization of these kinds of vector fields with respect to the sections which they define. Finally we obtain that a vector field is a Jacobi field along the identity map if and only it is a harmonic-Killing vector field, which is a special case of a theorem of Ferreira's ([2]).

With the objective of defining and characterising new types of vector fields, we consider other different properties of the local infinitesimal transformations associated to a vector field on a (pseudo)-Riemannian manifold. We study harmonic-Killing vector fields in Kähler manifolds, obtaining that in the compact case such vector fields coincide with the holomorphic ones. We consider next the vector fields for which 1-parameter groups of local transformations consist of pluriharmonic or $\alpha$-pluriharmonic maps; we call such vector fields pluriharmonic or $\alpha$-pluriharmonic vector fields, respectively. We end by obtaining intrinsic characterisations and giving relations among the new types of vector fields.




[1] J. EELLS, L. LEMAIRE, Two reports on harmonic maps, World Scientific, Singapore-New Jersey-London-Hong Kong, 1995.

[2] M.J. FERREIRA, A characterization of Jacobi fields along harmonic maps. Internat. J. Math. 4 (1993) 545-550. [cf. also Aplicaçoes ramificadas conformes de superficies de Riemann e problemas variacionais, Thesis, University of Lisbon, (1985)].

[3] E. GARCfIA-RfIO, L. VANHECKE, M.E. V´AZQUEZ-ABAL, Harmonic endomorphism fields, Illinois J. Math., 41 (1997), 23-29.

[4] O. NOUHAUD, Transformations infinitésimales harmoniques, C. R. Acad. Sc. Paris, 274 (1972), 573-576.

[5] K. YANO, Integral Formulas in Riemannian Geometry , Marcel Dekker, Inc., New York, 1970.

[6] K. YANO, S. ISHIHARA, Tangent and Cotangent Bundles, Marcel Dekker, New York, 1973.

[7] K. YANO, T. NAGANO, On geodesic vector fields in a compact orientable Riemannian manifold, Comment. Math. Helv. 35(1) (1961), 55-64.





Marco MACKAAY and Roger PICKEN

Area Departamental de Matematica, UCEH

and Departamento de Matemática, Instituto Superior Técnico

mmackaay@ualg.pt, picken@math.ist.utl.pt


TITLE... ``Bundles, gerbes and holonomy"

Abstract: In this talk I wish to outline the results of [1], namely the correspondence between bundles with connection and holonomy maps, as well as the analogous correspondence for (abelian) gerbes with connection, in the Chatterjee-Hitchin language [2], and their holonomy maps. I will also attempt to present a TQFT-viewpoint on this subject.

Barrett (and later Caetano and Picken) showed that up to gauge equivalence principal G-bundles with connection over a connected manifold M, correspond bijectively, via the notion of holonomy, to smooth group homomorphisms from the "thin" fundamental group of M into G. The thin fundamental group is defined like the ordinary fundamental group, but instead of dividing out by all smooth homotopies we only divide by those which are thin. In Caetano and Picken's setup a smooth homotopy is thin if its rank is at most 1 (Barrett used a slightly different definition). Caetano and Picken defined also the higher thin homotopy groups, which are Abelian, and asked for a geometrical interpretation of smooth group homomorphisms from them into U(1). Recently Picken and I showed that if M is 1-connected, then smooth group homomorphisms from the second thin homotopy group of M into U(1) correspond bijectively to U(1)-gerbes with gerbe connections. In my talk I will explain this result and comment on the general case for which M is not necessarily 1-connected.

[1] M. Mackaay and R. Picken, The holonomy of gerbes with connections, math.DG/0007053

[2] N. Hitchin, Lectures on special Lagrangian submanifolds, in School on Differential Geometry (1999), Abdus Salam International Centre for Theoretical Physics, math.DG/9907034





Emilia MEZZETTI

University of Trieste

mezzette@univ.trieste.it


TITLE... ``Highly Tangent Lines to Projective Hypersurfaces"

Abstract: In a joint paper with Dario Portelli [1], a complete classification has been given of threefolds X in the complex projective space of dimension 4, which are covered by lines. In particular, it results that, if the Fano scheme of lines on X is generically reduced of dimension two, then the maximum number of lines contained in X passing through a general point of X is bounded by 6. Moreover, if the bound 6 is attained, then X is a cubic threefold. I would like to discuss this result and possible generalizations to higher dimension.


[1] E. Mezzetti - D. Portelli: On threefolds covered by lines, to appear on "Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg", arXiv:math.AG/0003213





Dmitri MILLIONSCHIKOV

Moscow State University, Dept. of Mech. and Math.

million@mech.math/msu.su


TITLE... ``Cohomology of Nilmanifolds, Massey operations and symplectic structures"

Abstract: Nilmanifolds are widely used for construction of symplectic manifolds with no Käler structure. We study the manifolds Mn constructed by means of nilpotent "n-jet" Lie algebra $\bar L_n,$with free generators $e_1, e_2, \dots, e_n$ and ther bracket

\begin{displaymath}[e_i,e_j]
=\left\{\begin{array}
{r}
(j-i)e_{i+j}, i+j \le n;\\ 0, i+j \gt n. \\ \end{array} \right. \end{displaymath}

Calculation of cohomology of these manifolds is closely related to the Gontcharova theorem in the theory of infinitedimensional Lie algebras, namely Theorem. Stable Betti numbers of H*(Mn) are Fibonacci numbers. Also this result is related to the famouse Euler identity in Combinatorics.

References

1. D.Millionschikov, Cohomology of Nilmanifolds and Gontcharova Theorem, http://at/yorku/ca/cgi-bin/amca/cadq-01.

2. D.B.Fukhs, Cohomology of the infinitedimensional Lie algebras, Consultant Bureau, New York, 1987.

3.I.K.Babenko, I.A.Taimanov, On nonformal simply connected symplectic manifolds, SFB 2888, Preprint 358, 1998.





Ivailo M. MLADENOV

Bulgarian Academy of Sciences

mladenov@bgcict.acad.bg


TITLE... to be announced.

Abstract: to be announced.





Oleg I. MOKHOV

Centre for Nonlinear Studies, Landau Institute for Theoretical Physics, Moscow, Russia

Department of Mathematics, University of Paderborn, Germany

mokhov@uni-paderborn.de, mokhov@landau.ac.ru


TITLE... ``Compatible flat metrics"

Abstract: We solve the problem of description of nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) for the general N-component case (see [1]-[2]). The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics are found and integrated. The integrating these equations is based on reducing to a special nonlinear differential reduction of the Lame equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method). The notion of flat pencils of metrics is introduced by Dubrovin and very important for the theory of Frobenius manifolds, the theory of associativity equations in two-dimensional topological field theory and the theory of integrable systems of hydrodynamic type.

The author's publications on the subject:


[1] O.I.Mokhov. Compatible and almost compatible pseudo-Riemannian metrics. Preprint. 2000.
arXiv:math.DG/0005051.

[2] O.I.Mokhov. On integrability of the equations for nonsingular pairs of compatible flat metrics. Preprint. 2000. arXiv: math.DG/0005081.

[3] O.I.Mokhov. On compatible Poisson structures of hydrodynamic type. Uspekhi Matemat. Nauk. 1997. V. 52, No. 6. P. 171-172. English translation in: Russian Math. Surveys. 1997. V. 52. No. 6. P. 1310-1311.

[4] O.I.Mokhov. On compatible potential deformations of Frobenius algebras and associativity equations. Uspekhi Matemat. Nauk. 1998. V. 53, No. 2. P. 153-154. English translation in: Russian Math. Surveys. 1998. V. 53. No. 2. P. 396-397.

[5] O.I.Mokhov. Compatible Poisson structures of hydrodynamic type and associativity equations. Trudy Matemat. Inst. imeni V.A.Steklova Akad. Nauk. 1999. V. 225. P. 284-300. English translation in: Proceedings of the Steklov Institute of Mathematics. 1999. V. 225. P. 269-284.

[6] O.I.Mokhov. Compatible Poisson structures of hydrodynamic type and the equations of associativity in two-dimensional topological field theory. Reports on Mathematical Physics. 1999. V. 43, No. 1/2. P. 247-256.

[7] O.I.Mokhov. Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Uspekhi Matemat. Nauk. 1998. V. 53, No. 3. P. 85-192. English translation in: Russian Math. Surveys. 1998. V. 53. No.3. P. 515-622.





Rita PARDINI

Universitaí di Pisa

pardini@dm.unipi.it


TITLE... ``Bicanonical map of surfaces of general type with pg=0"

Abstract: I would like to report on joint work with Margarida Mendes Lopes on the bicanonical map of surfaces of general type with pg=0. Most results are contained in papers [MP1 and [MP2] of references. I enclose the abstracts of both papers.

Abstract of [MP1]: Let S be a minimal surface of general type with pg(S)=0 for which the bicanonical map $\phi: S\to{\rm I\!P}^{K^2_S}$ is a morphism. Then $\deg\phi\le 4$ by [ML], and if it is equal to 4 then $K^2_S\le 6$by [MP2]. We prove that if K2S=6 and $\deg\phi=4$ then S is a Burniat surface. We show moreover that minimal surfaces with pg=0, K2=6 and bicanonical map of degree 4 form a 4-dimensional irreducible connected component of the moduli space of surfaces of general type.

Abstract of [MP2]: A minimal surface S of general type with pg(S)=0 satisfies $1\le
K^2\le 9$ and it is known that the image of the bicanonical map $\phi$ is a surface for $K_S^2\geq 2$, whilst for $K^2_S\geq
5$, the bicanonical map is always a morphism. In this paper it is shown that $\phi$ is birational if KS2=9 and that the degree of $\phi$ is at most 2 if KS2=7 or KS2=8. By presenting two examples of surfaces S with KS2=7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with KS2=8 is, to our knowledge, a new example of a surface of general type with pg=0. The degree of $\phi$ is also calculated for two other known surfaces of general type with pg=0, KS2=8. In both cases the bicanonical map turns out to be birational.

References:

[ML] M. Mendes Lopes, ``The degree of the generators of the canonical ring of surfaces of general type with pg=0, Arch. Math., 69, (1997), 435-440.

[Mp1] M. Mendes Lopes, R. Pardini, ``A connected component of the moduli space of surfaces with pg=0", Topology (to appear, Eprint: Eprint: Math.AG/9910012).

[MP2] M. Mendes Lopes, R. Pardini, ``The bicanonical map of surfaces pg=0 and $K^2\ge 7$", Bulletin of the London Mathematical Society (to appear Eprint: mathAG/9910074)

[Pe] C. Peters, ``On certain examples of surfaces with pg=0 due to Burniati", Nagoya Math. J., Vol. 166 (1977), 109-119.





Jean-Baptiste POMET

INRIA, France

pomet@sophia.inria.fr


TITLE... ``Dynamic equivalence of control systems"

Abstract: This talk will present control systems as underdetermined systems of ODEs, and the notion of Dynamic equivalence between control systems, that is very close to what E. Cartan called absolute equivalence [1] between differential systems.

The (unsolved) problem of deciding when a control system is equivalent to linear controllable one, or a "free" one, or a "differentially flat" one according to [2] will be presented.


References (non exhaustive):


[1] Elie Cartan, "Sur l'equivalence absolue de certains systemes d'equations differentielles et sur certaines familles de courbes", Bull. de la Soc. Math. de France, 42:14-48, 1914.

[2] Michel Fliess, Jean Levine, Philippe Martin, and Pierre Rouchon, "Flatness and defect of nonlinear systems: Introductory theory and examples", Int. J. of Control , 61(6):1327-1361, 1995.

[3] Bronislaw Jakubczyk, "Remarks on equivalence and linearization of nonlinear systems, NOLCOS92... Marcel Dekker, New-York, 1992.

[4] Jean-Baptiste Pomet, "A differential geometric setting for dynamic equivalence and dynamic linearization", in B. Jakuczyk et al editors, Geometry in Nonlinear Control and Differential Inclusions, vol. 32 of Banach Center Publications, pages 319-339, 1995.





Lev SABININ and Larissa SBITNEVA

Russian Friendship of Nations University and University of Quintana Roo

lsabinin@balam.cuc.uqroo.mx, lsabinin@correo.uqroo.mx


TITLE... ``Horizonts of non-smooth Geometry and Analysis (Search for a new Mathematics for Applied Sciences)"

Abstract: This seminar will be devoted to some new ideas in order to replace "smooth models" of Universe by more natural. continuous and algebraic models. In the first part we consider a new vision of Geometry according to Kikkawa and Sabinin. This vision is based on nonassociative algebra (loops and quasigroups theory).. Some applications to relativity will be discussed. In the second part we consider some way to develop Mathematical analysis a bit "beyond Leibnitz" (Repagular formalism). This is an attempt to introduce generalized functions with values at points using Quantum machanics philosophy.


1. L.Sabinin, Smooth quasigroups and Loops. Monograph, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, xvi-250

2. L. Sabinin, Methods of Nonassociative algebra in Diffrential geometry, Differential Geometry and Applications, Proceedings of the 7-th International conference (Satellite Conference of ICM in Berlin. Brno. Czech Republic.1998) 1999, pp.419-427. Masaryk University. Brno Czech Republic.

3. L. Sabinin, Quasigroups, Geometry and Nonlinear Geometric Algebra, Acta Aplicanda Mathematicae, 50 (1998)45-66, MR 99h: 20106

4. L. Sabinin, L. Sabinina, L. Sbitneva, On the notion of Gyrogroup, Aequationes mathematicae 56 (1998) no. 1-2, 11-17, MR 99i:83004

5. L. Sabinin, A. Nesterov, Smooth Loops, generalized coherent states and geometric phases. International Journal of Theoretical Physics 36 (1997), no.9, 1091-1989, MR 98i:81095





Elena SAFIULINA

Institute of Pure Mathematics, Tartu University

elensfn@math.ut.ee


TITLE... `` About minimal semiparallel surfaces in pseudo-Euclidean spaces."

Abstract: Let Esn be a n-dimentional pseudo-Euclidean space with s negative coefficients in the canonical forms of the metric quadratic forms. A submanifold Mm in Esn is called semiparallel (or semisymmetric, extrinsically), if $\overline
R(X,Y)h=0$ (i.e. the integrability condition of the system $\overline\nabla h=0$, which characterizes a parallel ( or locally symmetric, extrinsically) submanifold). Here $\overline R$ is the curvature operator of $\overline\nabla\;\; (\overline\nabla=\nabla \oplus \nabla^\perp)$ and h is the second fundamental form. In the present report is given classification of minimal semiparallel surfaces, which are space-like ( i.e. have positive definite inner metric ) in pseudo-Euclidean spaces Esn. For all semiparallel space-like surfaces in pseudo-Euclidean spaces Esn was proved:

Theorem.[2] Let M2 be a semiparallel space-like surface in $E_s^n\;\;(s=1,2,3)$. There exists an open and dense part U of M2 such that the connected components of U are of the following types: (i) open parts of totally umbilical M2 in $E^n_s\;\;(s=1,2,3)$ (in particular, of totally geodesic M2); (ii) surfaces with flat $\overline\nabla$; (iii) isotropic surface with nonflat $\nabla^{\perp}$ and with ${\parallel H \parallel}^2=3K$, where K is the Gaussian curvature and H is the mean curvature vector.

A surface of the type (i) is minimal iff this M2 is a totally geodesic. The type (iii) is a second order envelope of Veronese surfaces. It is known that for n=5 such an envelope is a single Veronese surface in E5 (or E35) [3] belonging to a hypersphere S4 (or H24, respectively) and is minimal here. Consideration of surfaces of the type (ii) shown that there are exist minimal semiparallel space-like surfaces withflat$\overline\nabla$.

Proposition. Let M2 be a minimal semiparallel space-like surface with flat $\overline\nabla$ in Esn. Thus it is either 1)a hyperbolic paraboloid in $E_{0,1}^3\subset E_1^3$,or 2)a surface in $E_{0,1}^3\subset E_1^3$, with equation $x={1\over 2}h_{11}(u)^2+{1\over 2}h_{22}(v)^2+h_{12}uv+c_1u+c_2v$, where all coefficients are some constant vectors, it has two families of parabola generators, or 3)a surface in $E_{0,2}^4\subset E_2^4$, with equation $x={1\over 2}h_{11}(u)^2+{1\over 2}h_{22}(v)^2+h_{12}uv+c_1u+c_2v$, where all coefficients are some constant vectors, it has two families of parabola generators, or 4)a 2nd order envelope of a family, consisting of the surfaces of one of the classes 1)-3) in Esn.


REFERENCES.

1. A. Wolf, Space of constant curvature, Univ. California. Berkeley, 1972.


2. E. Safiulina, Parallel and semiparallel space-like surfaces in pseudo-Euclidean spaces, (in print).


3. Ü. Lumiste, Isometric semiparallel immersions of two-dimensional Riemannian manifolds into pseudo-Euclidean spaces. In New Development in Differential Geometry, (Szenthe, J., ed.), Kluwer Ac. Publ., Dordrecht, 1999, 243-264.





Marcos SALVAI

FAMAF, Universidad Nacional de Cordoba, Argentina

salvai@mate.uncor.edu


TITLE... ``On the dynamics of a rigid body in the hyperbolic space"

(accepted for publication in J. Geom. Physics)

Abstract: Let H be the three dimensional hyperbolic space and let G be the identity component of the isometry group of H. It is known that some aspects of the dynamics of a rigid body in H contrast strongly with the euclidean case, due to the lack of a subgroup of translations in G. We present the subject in the context of homogeneous Riemannian geometry, finding the metrics on G naturally associated with extended rigid bodies in H. We concentrate on the concept of dynamical center, characterizing it in various ways.

References.

- P. Dombrowski and J. Zitterbarth, On the planetary motion in the 3-dimensional standard spaces of constant curvature, Demonstratio Math. 24 Nr 3-4 (1991) 375-458.

- P. T. Nagy, Dynamical invariants of rigid motions on the hyperbolic plane, Geom. Dedicata 37 Nr 2 (1991) 125-139.

- J. Zitterbarth, Some remarks on the motion of a rigid body in a space of constant curvature without external forces, Demonstratio Math. 25 Nr 3-4 (1991) 465-494.





Justin SAWON

Mathematical Institute, Oxford University

sawon@maths.ox.ac.uk


TITLE... ``TQFTs and Hyperkahler Geometry"

Abstract: Murakami and Ohtsuki have constructed a "modified" TQFT based on the universal perturbative finite-type invariants of 3-folds known as the LMO invariant (q-alg/9512002). I will discuss how one hopes to obtain the TQFT conjectured to exist by Rozansky and Witten (hep-th/9612216) by applying a hyperkahler weight system (as in math/0002218) to Murakami and Ohtsuki's TQFT. The "Hilbert spaces" of this TQFT are cohomology groups of the hyperkahler manifold.





Jan SLOVAK

Masaryk University

slovak@math.muni.cz


TITLE... ``Bernstein-Gelfand-Gelfand sequences"

Abstract: During the last decades, the general theory of the so called parabolic geometries has been developed. These geometries are based on Cartan's concept of generalized spaces (modeled on G/P with G semisimple, P parabolic), a very general framework originally built in connection with the Cartan's equivalence problem (see e.g. [10], [11], [12] and the references therein). The name parabolic geometry, commonly adopted, originates in the closely related parabolic invariants program initiated by Fefferman. The relation to twistor theory renewed the interest in a good calculus for such geometries, with the aim to improve the techniques in conformal geometry and to extend them to other geometries, (see e.g. [1],[2] , [4], [9], and references therein for generalizations). One of the main objectives was the construction of invariant differential operators.

A new approach to this topic, combining Lie algebraic tools with the frame bundle approach was started in ([5]) and the first strong applications for all parabolic geometries were given in ([6]). The latter paper is the basic reference for this lecture. For further essential development of both the abstract calculus and the differential geometry in the general setting (see [3], [4]).

In the talk, I first wish to present the basic differential geometric and algebraic tools for the treatment of invariant operators in the realm of parabolic geometries. Then a general construction of a vast amount of invariant operators will be described and the basic results on the BGG-resolutions from representation theory will be recovered as the very special case of the homogeneous models. An important feature of our theory is the exclusive usage of the elementary (finite dimensional) representation theory. With a bit of exaggeration we could say that the representation theory enters rather as a language and the way of thinking. On the other hand, there are also purely representation theoretical aspects of interest as indicated in ([7]).


References


[1] R. J. Baston, M. G. Eastwood, The Penrose Transform. Its Interaction with Representation Theory, Oxford Science Publications, Clarendon Press, 1989

[2] T.N. Bailey, M.G. Eastwood, C.R. Graham, Invariant theory for conformal and CR geometry, Annals of Math. 139 (1994), 491-552

[3] A. Cap, A. R. Gover, Tractor calculi for parabolic geometries, ESI Preprint 792 (1999), electronically available on www.esi.ac.at

[4] A. Cap, J. Slovák, Weyl structures for parabolic geometries, ESI Preprint 801 (1999), electronically available at www.esi.ac.at

[5] A. Cap, J. Slovák, V. Soucek, Invariant operators on manifolds with almost Hermitian symmetric structures, I. Invariant differentiation, Acta Math. Univ. Commenianae, 66 (1997), 33-69, electronically available at www.emis.de; II. Normal Cartan connections, Acta Math. Univ. Commenianae, 66 (1997), 203-220, electronically available at www.emis.de; III. Standard operators, Diff. Geom. Appl., 12 (2000), 51-84.

[6] A. Cap, J. Slovák, V. Soucek, Bernstein-Gelfand-Gelfand sequences, ESI Preprint 722 (1999), electronically available at www.esi.ac.at

[7] M.G. Eastwood, J. Slovák, Semi-holonomic Verma modules, J. of Algebra, 197 (1997), 424-448

[8] C. Fefferman, Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), 131-262

[9] A.R. Gover, Aspects of parabolic invariant theory, in: Proc. Winter School Geometry and Physics, Srni 1998, Supp. Rend. Circ. Mat. Palermo, 59 (1999), 25-47

[10] T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22, (1993), 263-347

[11] N. Tanaka, On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J., 8 (1979), 23-84

[12] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Mathematics 22 (1993), 413-494





Mauro SPERA

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Universita' di Padova

spera@ulam.dmsa.unipd.it


TITLE... ``Higher order linking numbers and curvature"

(joint work with Vittorio Penna, in final preparation)

Abstract: A differential geometric approach to the higher order linking numbers of Massey and Milnor is devised, via Chen's iterated path integrals and connections; their equality is proved by parallel transport evaluation, in two different ways, of suitable Tavares' nilpotent connections manifactured from the topology of the link (as in the Penna-Rasetti-Spera approach).





Vladimir TKACHEV

Volgograd State University

ares_vg@zmail.ru


TITLE... ``Elliptic functions and zero mean curvature surfaces in Minkowski and Euclidean spaces."

Abstract: The main area of the talk are the recent results of the author and Sergienko V.V. concerning the examples of two-dimensional zero-mean curvature (ZMC) surfaces with symmetries in the Minkowski space. If we are working in the space with nontrivial signature (Lorentz spaces) we need to consider the a priori restrictions on the surfaces such as space-likeness and others. This lead us to the corresponding PDE's which have a mixed type (elliptic and hyperbolic). Such a situation yields that the special language for the surfaces of the mixed nature and the special tools for their study need to be organized.

1. We give a review of our recent examples of maximal surfaces with mixed type singularities. This theory closely concerns the elliptic functions and so-called generative matrixes (which parametrize the surfaces of the mean curvature in the Minkowski space with nontrivial group of spatial symmetry). The main results concern maximal surfaces which have analitic structure near the isolated mixed type singularity.

2. The corresponding results for the minimal and higherdimensional minimal surfaces in the Euclidean space will be disscused.

3. We consider also the theory of minimal tubes via elliptic functions representations.





Finlay THOMPSON

Victoria University of Wellington, New Zealand

finlay.thompson@vuw.ac.nz


TITLE... ``Quaternionic Gerbes on Conformal Four Manifolds."

Abstract: The algebra of quaternions combines naturally with its group of automorphisms to form a groupoid. I will demonstrate that this groupoid is equivalent to the groupoid of quaternionic bimodules (of real dimension four) and their isomorpisms. The Euclidean conformal group in four dimensions appears as a tensor product on this groupoid.

As principal bundles are related to groups, so gerbes are related to groupoids. In the same way that the transition functions for a principle bundle can be organised into a cocycle, we demonstrate the construction of a ``cocycle'' for quaternionic gerbes. We give two presentations of this cocycle, first explicitly in terms of quaternionic valued functions, and then as quaternionic bi-torsors and their maps. Cocycle and coboundary conditions are presented and we show that equivalence classes of cocycles classify quaternionic gerbes.

A conformal four manifold carries a cannonical quaternionic gerbe related to the tangent bundle. We present this ``tangent gerbe'' explicitly in the form of a cocycle.


9/25/2000