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.
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
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.
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
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.
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.
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 and Thickened Surfaces, J. Comb. Theory. 1994).
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
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.
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 )
Departamento de Geometria y Topologia. Universidad de Santiago de Compostela.
trinipl@usc.es
TITLE... ``Harmonic-Killing, pluriharmonic
and
-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 -pluriharmonicity ( 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 -pluriharmonic maps; we call such vector fields pluriharmonic or -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. GARCf [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.
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
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
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 with free generators and ther bracket
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.
Bulgarian Academy of Sciences
mladenov@bgcict.acad.bg
TITLE... to be announced.
Abstract: to be announced.
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.
[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.
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 is a morphism.
Then
by [ML], and if it is equal to 4 then by [MP2]. We prove that if K2S=6 and 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 and it is known that the image of the
bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that
is birational if KS2=9 and that
the degree of 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
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 ", 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.
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.
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
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 (i.e. the integrability condition of the system , which characterizes a
parallel ( or locally symmetric, extrinsically) submanifold). Here is the curvature
operator of 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
. 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 (in particular, of totally geodesic M2);
(ii) surfaces with flat ; (iii) isotropic surface with nonflat and with
, 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.
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.
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.
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.
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
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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).
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.
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.
arXiv:math.DG/0005051.
9/25/2000