title: Polarized 4-Manifolds, Extremal K\"ahler Metrics, and S-W Theory

abstract:
  Using Seiberg-Witten theory, it is shown that any Kaehler metric of constant
negative scalar curvature on a compact 4-manifold M minimizes the L^2-norm of
scalar curvature among Riemannian metrics compatible with a fixed decomposition
H^2(M)=(H^+) + (H^-). This implies, for example, that any such metric on a
minimal ruled surface must be locally symmetric.


title: The twistor transform of a Verlinde formula

abstract:
  The topology of the smooth moduli space of stable rank 2 bundles over a
Riemann surface of genus 3 is related to that of the real Grassmannian
Gr_4(R^8).


title: Integrable Gradient Flows and Morse Theory

abstract:
  Examples of Morse functions with integrable gradient flows on some classical
Riemannian manifolds are considered. In particular, we show that a generic
height function on the symmetric embeddings of classical Lie groups and certain
symmetric spaces is a perfect Morse function, i.e. has as many critical points
as the homology requires, and the corresponding gradient flow can be described
explicitly. This gives an explicit cell decomposition and geometric realization
of the homology for such a manifold. As another application of the integrable
Morse functions we give an elementary proof of Vassiljev's theorem on the flag
join of Grassmannians.


title: Adiabatic Limits and Spectral Geometry of Foliations

abstract:
  We study spectral asymptotics for the Laplace operator on differential forms
on a Riemannian foliated manifold equipped with a bundle-like metric in the
case when the metric is blown up in directions normal to the leaves of the
foliation. The asymptotical formula for the eigenvalue distribution function is
obtained. The relationships with the spectral theory of leafwise Laplacian and
with the noncommutative spectral geometry of foliations are discussed.


title: Extended moduli spaces and the Kan construction.II.Lattice gauge theory

abstract:
  Let $Y$ be a CW-complex with a single 0-cell, $K$ its Kan group, a model for
the loop space of $Y$, and let $G$ be a compact, connected Lie group. We give
an explicit finite dimensional construction of generators of the equivariant
cohomology of the geometric realization of the cosimplicial manifold
$\roman{Hom}(K,G)$ and hence of the space $\roman{Map}^o(Y,BG)$ of based maps
from $Y$ to the classifying space $BG$. For a smooth manifold $Y$, this may be
viewed as a rigorous approach to lattice gauge theory, and we show that it then
yields, (i) when {$\roman{dim}(Y)=2$,} equivariant de Rham representatives of
generators of the equivariant cohomology of twisted representation spaces of
the fundamental group of a closed surface including generators for moduli
spaces of semi stable holomorphic vector bundles on complex curves so that, in
particular, the known structure of a stratified symplectic space results; (ii)
when {$\roman{dim}(Y)=3$,} equivariant cohomology generators including the
Chern-Simons function; (iii) when {$\roman{dim}(Y) = 4$,} the generators of the
relevant equivariant cohomology from which for example Donaldson polynomials
are obtained by evaluation against suitable fundamental classes corresponding
to moduli spaces of ASD connections.


title:
Rokhlin Conjecture and Topology of Quotients of Complex Surfaces by
  Complex Conjugation

abstract:
  Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions
$conj\: X\to X$ tend to be completely decomposable when they are simply
connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if
$w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double
branched covering over $CP^2$, this phenomenon is related to unknottedness of
Arnold surfaces in $S^4=CP^2/conj$, which was conjectured by V.Rokhlin. The
paper contains proof of Rokhlin Conjecture and of decomposability of quotients
for plenty of double planes and in certain other cases. This results give, in
particular, an elementary proof of Donaldson's result on decomposability of $Y$
for K3 surfaces.


title:
The Centralizer of Invariant Functions and Division Properties of the
  Moment Map

abstract:
  Let Phi : M --> g^* be a proper moment map associated to an action of a
compact connected Lie group, G, on a connected symplectic manifold, (M,\omega).
A collective function is a pullback via \Phi of a smooth function on g^*. In
this paper we present four new results about the relationship between the
collective functions and the G-invariant functions in the Poisson algebra of
smooth functions on M.


title: Multiplicity-free Hamiltonian actions need not be K\"ahler

abstract:
  We show that Tolman's example (of a six dimensional Hamiltonian $T^2$-space
with isolated fixed points and no compatible K\"{a}hler structure) can be
constructed from the flag variety $U(3)/U(1)^3$ by $U(2)$-equivariant
symplectic surgery. This implies that Tolman's space has a ``transversal
multiplicity-free'' action of $U(2)$ and that Delzant's theorem ``every compact
multiplicity-free torus action is K\"{a}hler'' \cite{D1} does not generalize to
non-abelian actions.


title:
Volumes of Discrete Groups and Topological Complexity of Homology
  Spheres

abstract:
  We address two fundamental and well-known problems of Gromov and Lyndon:
  \demo{Problem A} (Gromov, see [5]). Consider a category $M_n$ of closed
manifolds of dimension $n$ with nonzero-degree ways as morphisms. Study a
partial order $M \ge N \Leftrightarrow Mor (M, N) \neq \phi$. For which $N$ the
degrees of maps $f: M \to N$ are bounded for all $M$?
  \demo{Problem B} (Lyndon, [12], problem 13). Extend and relate the theories
of deficiency, the rate of growth and the Euler-Poincar\'e characteristic. In
particular, what influence does the deficiency have on the structure of an
infinite group?


title:
Generalized Weierstrass formulae, soliton equations and Willmore
  surfaces. I. Tori of revolution and the mKdV equation

abstract:
  A new approach is proposed for study structure and properties of the total
squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the
generalized Weierstrass formulae for inducing surfaces. The quantity $W$
(Willmore functional) is shown to be invariant under the modified
Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case
and, in particular, Willmore tori of revolution, are studied in details. The
Willmore conjecture is proved for the mKDV--invariant Willmore tori.


title: Einstein Metrics on Complex Surfaces

abstract:
  We consider compact complex surfaces with Hermitian metrics which are
Einstein but not Kaehler. It is shown that the manifold must be CP2 blown up at
1,2, or 3 points, and the isometry group of the metric must contain a 2-torus.
Thus the Page metric on CP2#(-CP2) is almost the only metric of this type.


title: Curved flats in symmetric spaces

abstract:
  In this paper we study maps (curved flats) into symmetric spaces which are
tangent at each point to a flat of the symmetric space. Important examples of
such maps arise from isometric immersions of space forms into space forms via
their Gauss maps. Further examples are found in conformal geometry, e.g. the
curved flats obtained from isothermic surfaces and conformally flat 3-folds in
the 4-sphere. Curved flats admit a 1-parameter family of deformations (spectral
parameter) which enables us to make contact to integrable system theory. In
fact, we give a recipe to construct curved flats (and thus the above mentioned
geometric objects) from a hierarchy of finite dimensional algebraically
completely integrable flows.


title: $Pin$-structures on surfaces and quadratic forms

abstract:
  A correspondence between different $Pin$-type structures on a compact surface
and quadratic (linear) forms on its homology is constructed. Addition of
structures is defined and expressed in terms of these quadratic forms.


title: Cohomology of a Quaternionic Complex

abstract:
  We investigate the cohomology of a certain elliptic complex defined on a
compact quaternionic-K\"{a}hler manifold with negative scalar curvature. We
show that this particular complex is exact, with the possible exception of one
term.


title: Localisation of the Donaldson's invariants along Seiberg-Witten classes

abstract:
  This article is a first step in establishing a link between the Donaldson
polynomials and Seiberg-Witten invariants of a smooth 4-manifold.


title: On a Full Quantization of the Torus

abstract:
  I exhibit a prequantization of the torus which is actually a ``full''
quantization in the sense that a certain complete set of classical observables
is irreducibly represented. Thus in this instance there is no Groenewold-Van
Hove obstruction to quantization.


title: On the $L^{n\over 2}$-norm of Scalar Curvature

abstract:
  Comparisons on $L^{n\over 2}$-norms of scalar curvatures between Riemannian
metrics and standard metrics are obtained. The metrics are restricted to
conformal classes or under certain curvature conditions.


title: Quantization of K\"ahler manifolds admitting $H$-projective mappings

abstract:
  We discuss the quantization of mechanical systems for which the Hamiltonian
vector fields of observables form the deformation of $n$-dimensional oscilator
algebra. Because of this fact these systems can be considered as "deformations"
of the harmonic oscillator. The set of abovementioned mechanical systems are
realized at the classical level in the form of K\"ahler manifolds of constant
holomorphic curvature. Such mechanical systems are quantized later with the
help of the geometric quantization approach. We also discuss the quantization
of more general K\"ahler manifolds (not necessarily of constant holomorphic
curvature) admitting $H$-projective mappings.


title: Quantization and Coherent States over Lagrangian Submanifolds

abstract:
  A membrane technique, in which the symplectic and Ricci forms are integrated
over surfaces in a complexification of the phase space, as well a ``creation"
connection with zero curvature over lagrangian submanifolds, is used to obtain
a unified quantization including a noncommutative algebra of functions, its
representations, the Dirac axioms, coherent integral transformations for
solutions of spectral and Cauchy problems, and trace formulas. (To be published
in Russian Journal of Mathematical Physics,1995,v.3,n.3)


title: On isometric and minimal isometric embeddings

abstract:
  In this paper we study critial isometric and minimal isometric embeddings of
classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}.
Quasi-$k$-curved metrics generalize the metrics of space forms. We construct
explicit examples and prove results about existence and rigidity.


title: Introduction to Seiberg-Witten's Invariants, Part I: Theory of Spinors

abstract:
  In 1994, Witten has defined a monopole invariant and he has shown the
equivalence of this invariant with Donaldson's polynomial using his result in
\( \SS \)-duality. This new invariant is very powerful because the gauge group
is abelian. By using such an invariant, many new results are found in the
smooth, K\"{a}hler and even the symplectic categories. However, almost every
paper in this topic write the monopole equations in a different way. Therefore
is it necessary to clarify the basic idea behind the definition of such an
invariant.
  In this paper we investigate the algebraic structure (Clifford algebra and
\(\spinc\) representation ) underlying this invariant and explain the equations
explicitly, especially the K\"{a}hlerian case. Details of the computations are
shown explicitly, and some minute mistakes in the existing papers are
corrected.


title: Novikov type inequalities for differential forms with non-isolated zeros

abstract:
  We generalize the Novikov inequalities for 1-forms in two different
directions: first, we allow non-isolated critical points (assuming that they
are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the
inequalities by means of twisting by an arbitrary flat bundle.
  The proof uses Bismut's modification of the Witten deformation of the de Rham
complex; it is based on an explicit estimate on the lower part of the spectrum
of the corresponding Laplacian.
  In particular, we obtain a new analytic proof of the degenerate Morse
inequalities of Bott.


title: Rank one symmetric spaces and Rigidity

abstract:
  In this paper we show that if the limit set is not small ,marked length
spectrum determines geometric structure of rank one locally symmetric
manifolds.


title: Equivalence of Geometric and Combinatorial Dehn Functions

abstract:
  In this paper it is proved that if a finitely presented group acts properly
discontinuously, cocompactly and by isometries on a simply connected Riemannian
manifold, then the two Dehn functions, of the group and the manifold,
respectively, are equivalent.


title: The Novikov-Bott inequalities

abstract:
  We generalize the Novikov inequalities for 1-forms in two different
directions: first, we allow non-isolated critical points (assuming that they
are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the
inequalities by means of twisting by an arbitrary flat bundle. We also obtain
an $L^2$ version of these inequalities with finite von Neumann algebras.
  The proof of the main theorem uses Bismut's modification of the Witten
deformation of the de Rham complex; it is based on an explicit estimate on the
lower part of the spectrum of the corresponding Laplacian.


title: Stiefel-Whitney Classes and the Conormal Cycle of a Singular Variety

abstract:
  A geometric construction of Sullivan's Stiefel-Whitney homology classes of a
real analytic variety $X$ is given by means of the conormal cycle of an
embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes
define additive natural transformations from certain constructible functions to
homology. We also show that, for a complex analytic variety, these classes are
the mod 2 reductions of the Chern-MacPherson classes.


title: Uniton Bundles

abstract:
  We show that a twistor construction of Hitchin and Ward can be adapted to
study unitons (harmonic spheres in a unitary group). Specifically, we show that
unitons are equivalent to holomorphic bundles with extra structure over a
rational ruled surface with energy given by Chern class. This equivalence
allows us to confirm the conjecture of Wood that unitons are rational. We also
construct an example of a two uniton in U(3) using this construction.


title: Controlled Geometry via Smoothing

abstract:
  We prove that Riemannian metrics with a uniform weak norm can be smoothed to
having arbitrarily high regularity. This generalizes all previous smoothing
results. As a consequence we obtain a generalization of Gromov's almost flat
manifold theorem. A uniform Betti number estimate is also obtained.


title: Manin Triples for Lie Bialgebroids

abstract:
  In his study of Dirac structures, a notion which includes both Poisson
structures and closed 2-forms, T. Courant introduced a bracket on the direct
sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi
identity except on certain subspaces. In this paper we systematize the
properties of this bracket in the definition of a Courant algebroid. This
structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric
bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit
expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric
bilinear forms on $E$. When $M$ is a point, the definition reduces to that of a
Lie algebra carrying an invariant nondegenerate symmetric bilinear form.
  For any Lie bialgebroid $(A,A^{*})$ over $M$ (a notion defined by Mackenzie
and Xu), there is a natural Courant algebroid structure on $A\oplus A^{*}$
which is the Drinfel'd double of a Lie bialgebra when $M$ is a point.
Conversely, if $A$ and $A^*$ are complementary isotropic subbundles of a
Courant algebroid $E$, closed under the bracket (such a bundle, with dimension
half that of $E$, is called a Dirac structure), there is a natural Lie
bialgebroid structure on $(A,A^{*})$ whose double is isomorphic to $E$. The
theory of Manin triples is thereby extended from Lie algebras to Lie
algebroids.
  Our work gives a new approach to bihamiltonian structures and a new way of
combining two Poisson structures to obtain a third one. We also take some
tentative steps toward generalizing Drinfel'd's theory of Poisson homogeneous
spaces from groups to groupoids.


title: On the Incompleteness of Berger's List of Holonomy Representations

abstract:
  In 1955, Berger \cite{Ber} gave a list of irreducible reductive
representations which can occur as the holonomy of a torsion-free affine
connection. This list was stated to be complete up to possibly a finite number
of missing entries. In this paper, we show that there is, in fact, an infinite
family of representations which are missing from this list, thereby showing the
incompleteness of Berger's classification. Moreover, we develop a method to
construct torsion-free connections with prescribed holonomy, and use it to give
a complete description of the torsion-free affine connections with these new
holonomies. We also deduce some striking facts about their global behaviour.


title: Affine connections on involutive G-structures

abstract:
  This paper is a review of the twistor theory of irreducible G-structures and
affine connections. Long ago, Berger presented a very restricted list of
possible irreducibly acting holonomies of torsion-free affine connections. His
list was complete in the part of metric connections, while the situation with
holonomies of non-metric torsion-free affine connections was and remains rather
unclear. One of the results discussed in this review asserts that any
torsion-free holomorphic affine connection with irreducibly acting holonomy
group can, in principle, be constructed by twistor methods. Another result
reveals a new natural subclass of affine connections with "very little torsion"
which shares with the class of torsion-free affine connections two basic
properties --- the list of irreducibly acting holonomy groups of affine
connections in this subclass is very restricted and the links with the twistor
theory are again very strong.


title:
On the Geometry of Complex Grassmann Manifold, Its Noncompact Dual and
  Coherent States

abstract:
  Different topics on the differential geometry of the complex Grassmann
manifold are surveyed in relation to the coherent states. A calculation of the
tangent conjugate locus and conjugate locus in the complex Grassmann manifold
is presented. The proofs use the Jordan's stationary angles. Also various
formulas for the distance on the complex Grassmann manifold are furnished.


title: Harmonic Maps with Prescribed Singularities on Unbounded Domains

abstract:
  The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and
axially symmetric case to a harmonic map with prescribed singularities
$\p\colon\R^3\sm\Sigma\to\H^{k+1}_\C$ into the $(k+1)$-dimensional complex
hyperbolic space. In this paper, we prove the existence and uniqueness of
harmonic maps with prescribed singularities $\p\colon\R^n\sm\Sigma\to\H$, where
$\Sigma$ is an unbounded smooth closed submanifold of $\R^n$ of codimension at
least $2$, and $\H$ is a real, complex, or quaternionic hyperbolic space. As a
corollary, we prove the existence of solutions to the reduced stationary and
axially symmetric Einstein/Abelian-Yang-Mills Equations.


title: On a class of K\"ahler manifolds whose geodesic flows are integrable

abstract:
  We study $n$-dimensional K\"ahler manifolds whose geodesic flows possess $n$
first integrals in involution that are fibrewise hermitian forms and
simultaneously normalizable. Under some mild assumption, one can associate with
such a manifold an $n$-dimensional commutative Lie algebra of infinitesimal
automorphisms. This, combined with the given $n$ first integrals, makes the
geodesic flow integrable. If the manifold is compact, then it becomes a toric
variety.


title: Notes on Seiberg-Witten Gauge Theory

abstract:
  These notes provide an introductory exposition of the Seiberg-Witten gauge
theory. They collect the material presented in a series of seminars given by
the author at the University of Milano.


title:
The Marked Length Spectrum vs. The Laplace Spectrum on Forms on
  Riemannian Nilmanifolds

abstract:
  The subject of this paper is the relationship among the marked length
spectrum, the length spectrum, the Laplace spectrum on functions, and the
Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show
that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in
this class has the same marked length spectrum, they necessarily share the same
Laplace spectrum on functions. In contrast, we present the first example of a
pair of isospectral Riemannian manifolds with the same marked length spectrum
but not the same spectrum on one-forms. Outside of the standard spheres vs. the
Zoll spheres, which are not even isospectral, this is the only example of a
pair of Riemannian manifolds with the same marked length spectrum, but not the
same spectrum on forms. This partially extends and partially contrasts the work
of Eberlein, who showed that on two-step nilmanifolds, the same marked length
spectrum implies the same Laplace spectrum both on functions and on forms.


title: Compact null hypersurfaces and collapsing Riemannian manifolds

abstract:
  Restrictions are obtained on the topology of a compact divergence-free null
hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is
zero or satisfies some weaker conditions. This is done by showing that each
null hypersurface of this type can be used to construct a family of
three-dimensional Riemannian metrics which collapses with bounded curvature and
applying known results on the topology of manifolds which collapse. The result
is then applied to general relativity, where it implies a restriction on the
topology of smooth compact Cauchy horizons in spacetimes with various types of
reasonable matter content.


title: On the space of harmonic $2$-spheres in ${\bf C}P^2$

abstract:
  Carrying further work of T.A. Crawford, we show that each component of the
space of harmonic maps from the $2$-sphere to complex projective $2$-space of
degree $d$ and energy $4 \pi E$ is a smooth closed submanifold of the space of
all $C^j$ maps $(j \geq 2)$. We achieve this by showing that the Gauss
transform which relates them to spaces of holomorphic maps of given degree and
ramification index is {\bf smooth} and has {\bf injective differential}.


title: Periodic Hamiltonian flows on four dimensional manifolds

abstract:
  We classify the periodic Hamiltonian flows on compact four dimensional
symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally,
we show that all these spaces are Kaehler, that every such space is obtained
from a simple model by a sequence of symplectic blowups, and that if the fixed
points are isolated then the space is a toric variety.


title: The Geometric Phase in the Three-Body Problem

abstract:
  Suppose that the initial triangle formed by the three moving masses of the
three-body problem is similar to the triangle formed at some later time. We
derive a simple integral formula for the overall rotation relating the two
triangles. The formula is based on the fact that the space of similarity
classes of triangles forms a two-sphere which we call the shape sphere. The
formula consists of a ``dynamic'' and ``geometric'' term. The geometric term is
the integral of a universal two-form on a``reduced configuration space''. This
space is a two-sphere bundle over the shape sphere. The fibering spheres are
instantaneous versions of the angular momentum sphere appearing in rigid body
motion. Our derivation of the formula is similar in spirit to our earlier
reconstruction formula for the rigid body motion.


title: Group systems, groupoids, and moduli spaces of parabolic bundles

abstract:
  Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear
form on its Lie algebra, let $\pi$ be the fundamental group of an orientable
(real) surface $M$ with a finite number of punctures, and let $\bold C$ be a
family of conjugacy classes in $G$, one for each puncture. A finite-dimensional
construction used earlier to obtain a symplectic structure on the moduli space
of flat $G$-bundles over compact $M$ is extended to the punctured case. It
yields a symplectic structure on a certain smooth manifold $\Cal M_{\bold C}$
containing the space $\roman{Hom}(\pi,G)_{\bold C}$ of homomorphisms mapping
the generators corresponding to the punctures into the corresponding conjugacy
classes. It also yields a Hamiltonian $G$-action on $\Cal M_{\bold C}$ such
that the reduced space equals the moduli space $\roman{Rep}(\pi,G)_{\bold C}$
of representations. For $G$ compact, each such space, obtained by
finite-dimensional symplectic reduction, is a {\it stratified symplectic
space\/}. For $G=U(n)$ one gets moduli spaces of semistable holomorphic
parabolic bundles or spaces closely related to them.


title:
Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and
  flattening of Legendrian curves

abstract:
  We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its
relation with flattening points of Legendrian curves and Sturm theory.


title: Theorem on six vertices of a plane curve via the Sturm theory

abstract:
  We discuss the theorem on the existence of six points on a convex closed
plane curve in which the curve has a contact of order six with the osculating
conic. (This is the ``projective version'' of the well known four vertices
theorem for a curve in the Euclidean plane.) We obtain this classical fact as a
corollary of some general Sturm-type theorems.


title: Substantial Riemannian Submersions of S(15) with 7- dimensional fibres

abstract:
  In this paper we show that a substantial Riemannian submersion of S(15) with
7- dimensional fibres is congruent to the standard Hopf fibration. As a
consequence we prove a slightly weak form of of the Diameter Rigidity theorem
for the Cayley plane which is considerably stronger than the very recent Radius
Rigidity Theorem of Wilhelm.


title: Torsions for manifolds with boundary and glueing formulas

abstract:
  We extend the definition of analytic and Reidemeister torsion from closed
compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M,
\partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of
finite type and a decomposition of the boundary $\partial M =\partial_- M \cup
\partial_+ M$ into disjoint components. In particular we extend the $L-2$
analytic and Reidemeister torsions to compact manifolds with boundary. If the
system $(M,\partial_-M, \partial_+M, \Cal F)$ is of determinant class we
compute the quotient of the analytic and the Reidemeister torsion and prove
glueing formulas for both of them. In particular we answer positively
Conjecture 7.6 in [LL]


title: Quadratic harmonic morphisms and O-systems

abstract:
  We introduce O-systems (Definition \ref{DO}) of orthogonal transformations of
${\Bbb R}^{m}$, and establish $1-1$ correspondences both between equivalence
classes of Clifford systems and that of O-systems, and between O-systems and
orthogonal multiplications of the form $\mu :{\Bbb R}^{n} \times {\Bbb R}^{m}
\longrightarrow {\Bbb R}^{m} $, which allow us to solve the existence problems
both for O-systems and for umbilical quadratic harmonic morphisms (Theorems
\ref{ES} and \ref{EU}) simultaneously. The existence problem for general
quadratic harmonic morphisms is then solved (Theorem \ref{EG}) by the Splitting
Lemma (Lemma \ref{Split}). We also study properties (see, e.g., Theorems
\ref{single} and \ref{TL}) possessed by all quadratic harmonic morphisms for
fixed pairs of domain and range spaces (\S5).


title:
On the classification of quadratic harmonic morphisms between Euclidean
  spaces

abstract:
  We give a classification of quadratic harmonic morphisms between Euclidean
spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence
between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford
systems. In the case $ {\Bbb R}^{4}\longrightarrow {\Bbb R}^{3} $, we determine
all quadratic harmonic morphisms and show that, up to a constant factor, they
are all bi-equivalent (Definition 3.2) to the well-known Hopf construction map
and induce harmonic morphisms bi-equivalent to the Hopf fibration ${\Bbb S}^{3}
\longrightarrow {\Bbb S}^{2}$.


title: Complete lifts of harmonic maps and morphisms between Euclidean spaces

abstract:
  We introduce the complete lifts of maps between (real and complex) Euclidean
spaces and study their properties concerning holomorphicity, harmonicity and
horizontal weakly conformality. As applications, we are able to use this
concept to characterize holomorphic maps $\phi:{\Bbb C}^{m}\supset
U\longrightarrow {\Bbb C}^{n}$ (Proposition 2.3) and to construct many new
examples of harmonic morphisms (Theorem 3.3). Finally we show that the complete
lift of the quaternion product followed by the complex product is a simple and
explicit example of a harmonic morphism which does not arise (see Definition
4.8 in \cite{BaiWoo95}) from any K{\"a}hler structure.


title:
On one family of 13-dimensional closed Riemannian positively curved
  manifolds

abstract:
  An infinite family of pairwise nonhomeomorphic 13-dimensional positively
curved manifolds is constructed


title: Modified Novikov--Veselov equation and differential geometry of surfaces

abstract:
  Global deformations of surfaces, immersed into the Euclidean 3-space, by
using the modified Novikov--Veselov equation are investigated. relation to the
theory of the Willmore functional is discussed


title: Geometric zeta-functions of locally symmetric spaces

abstract:
  The theory of geometric zeta functions for locally symmetric spaces as
initialized by Selberg and continued by numerous mathematicians is generalized
to the case of higher rank spaces. We show analytic continuation, describe the
divisor in terms of tangential cohomology and in terms of group cohomology
which generalizes the Patterson conjecture. We also extend the range of zeta
functions in considering higher dimensional flats.


title: Examples of non-Kaehler Hamiltonian torus actions

abstract:
  An important question with a rich history is the extent to which the
symplectic category is larger than the Kaehler category. Many interesting
examples of non-Kaehler symplectic manifolds have been constructed. However,
sufficiently large symmetries can force a symplectic manifold to be Kaehler. In
this paper, we solve several outstanding problems by constructing the first
symplectic manifold with large non-trivial symmetries which does not admit an
invariant Kaehler structure. The proof that it is not Kaehler is based on the
Atiyah-Guillemin-Sternberg convexity theorem. Using the ideas of this paper, C.
Woodward shows that even the symplectic analogue of spherical varieties need
not be Kaehler.


title: Hamiltonian torus actions on symplectic orbifolds and toric varieties

abstract:
  In the first part of the paper, we build a foundation for further work on
Hamiltonian actions on symplectic orbifolds. Most importantly we prove the
orbifold versions of the abelian connectedness and convexity theorems.
  In the second half, we prove that compact symplectic orbifolds with
completely integrable torus actions are classified by convex simple rational
polytopes with a positive integer attached to each facet and that all such
orbifolds are algebraic toric varieties.


title: Regularized and $L^2$-Determinants

abstract:
  It is shown that in a tower of coverings the regularized determinant of a
generalized Laplacian converges to the $L^2$-determinant. This shows generic
nontriviality of analytic torsion or regularized determinants since the
$L^2$-counterparts are easier to compute. We further have an "Euler product
expansion" for regularized determinants in terms of equivariant
$L^2$-determinants.


title: A Generalisation of Obata's theorem

abstract:
  In a complete Riemannian manifold $(M, g)$ if the hessian of a real valued
function satisfies some suitable conditions then it restricts the geometry of
$(M, g)$. In this paper we characterize all compact rank-1 symmetric spaces, as
those Riemannian manifolds $(M, g)$ admitting a real valued function $u$ such
that the hessian of $u$ has atmost two eigenvalues $-u$ and $-{{u+1}\over 2}$,
under some mild hypothesis on $(M, g)$. This generalises a well known result of
Obata which characterizes all round spheres.


title:
$\pi_1$ of symplectic automorphism groups and invertibles in quantum
  homology rings

abstract:
  We define a homomorphism from (a certain extension of) the fundamental group
of the Hamiltonian automorphism group of a symplectic manifold to the group of
invertibles in its quantum cohomology ring. The manifold must satify a
technical condition similar to weak monotonicity.
  This revised version has been entirely rewritten. It is more general and
contains new examples and applications.


title: Constraints for Seiberg-Witten basic classes of glued manifolds

abstract:
  We use rudiments of the Seiberg-Witten gluing theory for trivial circle
bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two
$4$-manifolds containing Riemann surfaces of the same genus and
self-intersection zero with those of the $4$-manifold resulting as a connected
sum along the surface. We study examples in which this is enough to describe
completely the basic classes.


title: Donaldson invariants for some glued manifolds

abstract:
  We prove that every suitable $4$-manifold with $b_1=0$ and with an embedded
Riemann surface of genus $2$ is of simple type. We find a relationship between
the basic classes of two of these $4$-manifolds and those of the connected sum
along the Riemann surface.


title: The First Eigenvalue of $P$-manifolds

abstract:
  Antonio Ros gave a lower bound for the first eigenvalue $\lambda_1$ of
$\Delta$ of a $P$-manifold $(M, g)$ in terms of the lower bound on the Ricci
curvature $Ric_M$ and asked what happened when this lower bound was achieved.
In this paper we look in to this question and show that there are strong
implications on the geometry and topology of the underlying manifold. In
particular we show that in case of spheres or real projective spaces we have
isometry with the standard metric. In other cases, with some additional
hypothesis, we again show isometry with standard models.


title: 4-Manifolds without Einstein Metrics

abstract:
  It is shown that there are infinitely many compact orientable smooth
4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the
strict Hitchin-Thorpe inequality 2 chi > 3 |tau|. The examples in question
arise as non-minimal complex algebraic surfaces of general type, and the method
of proof stems from Seiberg-Witten theory.


title:
Periodic orbits in magnetic fields and Ricci curvature of Lagrangian
  systems

abstract:
  We consider a periodic problem for the motion of a charged particle in a
magnetic field. Introducing a notion of Ricci curvature for such Lagrangian
systems and using the methods of the calculus of variations in the large, we
prove the existence of periodic motions for such particles under a condition of
positivity of the Ricci curvature.


title: A remark on positively curved manifolds of dimensions 7 and 13

abstract:
  Totally geodesically embeddings of infinitely many closed 7-manifolds into
13-dimensional positively curved closed Riemannian manifolds are constructed.
The problems of computing pinching constants and existence of other totally
geodesical embeddings are discussed.


title: Connected sum constructions for constant scalar curvature metrics

abstract:
  We give a general procedure for gluing together possibly noncompact manifolds
of constant scalar curvature which satisfy an extra nondegeneracy hypothesis.
Our aim is to provide a simple paradigm for making `analytic' connected sums.
In particular, we can easily construct complete metrics of constant positive
scalar curvature on the complement of certain configurations of an even number
of points on the sphere, which is a special case of Schoen's \cite{S1}
well-known, difficult construction. Applications of this construction produces
metrics with prescribed asymptotics. In particular, we produce metrics with
cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the
complement of an infinite number of points are also constructed by an iteration
of our construction.


title: On the index of Dirac operators on arithmetic quotients

abstract:
  Using the Arthur-Selberg trace formula we express the index of a Dirac
operator on an arithmetic quotient over a totally real field with at least two
real embeddings as the integral over the index form plus a sum of orbital
integrals. For the Euler operator these orbital integrals are shown to vanish
for products of rank one spaces. In this case the index theorem looks exactly
as in the compact case.


title: A Lefschetz formula for flows

abstract:
  For the geodesic flow of an odd dimensional hyperbolic manifold we prove a
Lefschetz type formula. The local terms are Fuller indices of the closed
orbits. The global "Frobenius operator" is the generator of the flow and its
action on tangential cohomology.


title: The Spinor Representation of Minimal Surfaces

abstract:
  The spinor representation is developed and used to investigate minimal
surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of
planar-ended minimal spheres and real projective planes are determined, and new
families of minimal tori and Klein bottles are given. These surfaces compactify
in $S^3$ to yield surfaces critical for the M\"obius invariant squared mean
curvature functional $W$. On the other hand, all $W\!$-critical spheres and
real projective planes arise this way. Thus we determine at the same time the
moduli spaces of $W\!$-critical spheres and real projective planes via the
spinor representation.


title:
The Space of Harmonic Maps from the 2-sphere to the Complex Projective
  Plane

abstract:
  We study the topology of the space of harmonic maps from $S^2$ to \CP 2$. We
prove that the subspaces consisting of maps of a fixed degree and energy are
path connected. By a result of Guest and Ohnita it follows that the same is
true for the space of harmonic maps to $\CP n$ for $n\geq 2$. We show that the
components of maps to $\CP 2$ are complex manifolds.


title: Quotients by complex conjugation for real complete intersection surfaces

abstract:
  Quotients $Y=X/conj$ by the complex conjugation $conj\: X\to X$ for complex
surfaces $X$ defined over $\R$ tend to be completely decomposable when they are
simply connected, i.e., split into connected sums $\#_n CP^2\#_m\barCP^2$ if
$w_2(Y)\ne0$, or into $\#_n(S^2\times S^2)$ if $w_2(Y)=0$. The author proves
this property for complete intersections which are constructed by method of a
small perturbation.


title:
Energy of embedded surfaces invariant under Moebius tranformations,
  addendum

abstract:
  In a previous preprint we defined an energy associated to every embedding of
a surface into $R^n$ or $S^n$. This energy is invariant under Moebius
tranformations and the "round" sphere is its only absolute minimum. Here we
sketch a proof of the compactness property for a variant of it. The details
will appear elsewhere.


title: Exotic deformation quantization

abstract:
  We consider formal deformations of the Poisson algebra of functions (with
singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case:
$\dim M=1$ ($M=S^1, {\bf R}$), there exists a non-trivial $\star$-product on
this algebra non-equivalent to the standard Moyal product.


title: Harmonic morphisms between almost Hermitian manifolds

abstract:
  We obtain conditions on the Lee form under which a holomorphic map between
almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under
what conditions (i) the image of a holomorphic map from a cosymplectic manifold
is also cosymplectic, (ii) a holomophic map with Hermitian image defines a
Hermitian structure on its domain.


title: A closed form for unitons

abstract:
  Unitons, i.e.\ harmonic spheres in a unitary group, correspond to \lq uniton
bundles\rq, i.e.\ holomorphic bundles over the compactified tangent space to
the complex line with certain triviality and other properties. In this paper,
we use a monad representation similar to Donaldson's representation of
instanton bundles to obtain a simple formula for the unitons. Using the monads,
we show that real triviality for uniton bundles is automatic. We interpret the
uniton number as the `length' of a jumping line of the bundle, and identify the
uniton bundles which correspond to based maps into Grassmannians. We also show
that energy-$3$ unitons are $1$-unitons, and give some examples.


title:
Weierstrass representations for harmonic morphisms on Euclidean spaces
  and spheres

abstract:
  We construct large families of harmonic morphisms which are holomorphic with
respect to Hermitian structures by finding heierarchies of Weierstrass-type
representations. This enables us to find new examples of complex-valued
harmonic morphisms from Euclidean spaces and spheres.


title: $L^2$-Cohomology of Geometrically Infinite Hyperbolic 3-Manifolds

abstract:
  We give results on the following questions about a topologically tame
hyperbolic 3-manifold M :
  1. Does M have nonzero square-integrable harmonic 1-forms?
  2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on
M)/Ker(d)?


title:
Regularisable and minimal orbits for group actions in infinite
  dimensions

abstract:
  We introduce a class of regularisable infinite dimensional principal fibre
bundles which includes fibre bundles arising in gauge field theories like
Yang-Mills and string theory and which generalise finite dimensional Riemannian
principal fibre bundles induced by an isometric action. We show that the orbits
of regularisable bundles have well defined, both heat-kernel and zeta function
regularised volumes. We introduce two notions of minimality (which extend the
finite dimensional one) for these orbits, using both heat-kernel and zeta
function regularisation methods and show they coincide. For each of these
notions, we give an infinite dimensional version of Hsiang's theorem which
extends the finite dimensional case, interpreting minimal orbits as orbits with
extremal (regularised) volume.


title:
Some curvature estimates for Riemannian manifolds equipped with
  foliations of rank 2

abstract:
  Some curvature estimates are derived from geometrical data concerning
quasi-conformality properties of some commuting linearly independent vector
fields on a compact Riemannian manifold.


title: Hamilton dynamics and $H$-planar curves

abstract:
  Hamilton flows on K\"ahler manifold for which all trajectories are $H$-planar
curves (complex analog of geodesics) are considered. These flows are called
$H$-planar. The equation which has to obey the Hamiltonian of $H$-planar
Hamilton flow is received and the method of finding general solution of this
equation is proposed.
  Trajectories of charged particles in magnetic fields of special form on
K\"ahler manifolds of constant holomorphic sectional curvature are studied.
Using the fact that K\"ahler manifolds of constant holomorphic sectional
curvature admit $H$-projective mapping on flat space the equation of particle
motion is reduced to an ordinary differential equation of second order.


title: Cobordism theory and localization formulas for Hamiltonian group actions

abstract:
  We announce the following result and give several applications: A Hamiltonian
$T$-space (for $T$ a torus) with isolated fixed points is cobordant to a
disjoint union of weighted projective spaces which are constructed from its
fixed point data. The applications concern the Duistermaat-Heckman formula, the
topological Jeffrey-Kirwan localization theorem, and geometric quantization.


title: Supersymmetry and the generalized Lichnerowicz formula

abstract:
  A classical result in differential geometry due to Lichnerowicz [8] is
concerned with the decomposition of the square of Dirac operators defined by
Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian
manifold $M$. Recently, this formula has been generalized to arbitrary Dirac
operators [2]. In this paper we prove a supersymmetric version of the
generalized Lichnerowicz formula, motivated by the fact that there is a
one-to-one correspondence between Clifford superconnections and Dirac
operators. We extend this result to obtain a simple formula for the
supercurvature of a generalized Bismut superconnection. This might be seen as a
first step to prove the local index theorem also for families of arbitrary
Dirac operators.


title: Two nontrivial index theorems in odd dimensions

abstract:
  A recent anomaly computation of Horava and Witten is proved and generalized
in the form of two index theorems in odd dimensions. Theorem A is a fixed point
formula for orientation-reversing involutions. Theorem B is an index theorem
for manifolds with boundary using local boundary conditions. Both hold for
families of operators.


title: On the Negative Case of the Singular Yamabe Problem

abstract:
  The negative case of the Singular Yamabe Problem concerns the existence and
behavior of complete metrics with constant negative scalar curvature on the
complement of a closed set in a compact Riemannian manifold which are
conformally equivalent to a smooth metric on this compact manifold. When the
closed set is a smooth submanifold, it is known by the results of
Loewner-Nirenberg and Aviles-McOwen that there exists such a complete metric if
and only if $d > (n-2)/2$, and in general the Hausdorff dimension of the set
must be at least $(n-2)/2$. In this paper, we show that the existence of such a
complete conformal metric with constant negative scalar curvature depends on
the tangent structure of the closed set. Specifically, provided the set has a
nice tangent cone at a point, we show that when the dimension of this tangent
cone is less than $(n-2)/2$ there can not exist such a negative Singular Yamabe
metric.


title: Adiabatic limits of Seiberg-Witten equations on Seifert manifolds

abstract:
  We present a simpler proof for the existence of adiabatic limits. Moreover,
we added a new section where the adiabatic process is reversed and in some
nondegenerate cases we deform the adiabatic limits to genuine irreducible
solutions of the SW equations.


title: Gluing and moduli for noncompact geometric problems

abstract:
  In this paper we survey a number of recent results concerning the existence
and moduli spaces of solutions of various geometric problems on noncompact
manifolds. The three problems which we discuss in detail are:
  I. Complete properly immersed minimal surfaces in $\RR^3$ with finite total
curvature.
  II. Complete embedded surfaces of constant mean curvature in $\RR^3$ with
finite topology.
  III. Complete conformal metrics of constant positive scalar curvature on $M^n
\setminus \Lambda$, where $M^n$ is a compact Riemannian manifold, $n\geq3$ and
$\Lam \subset M$ is closed.
  The existence results we discuss for each of these problems are ones whereby
known solutions (sometimes satisfying certain nondegeneracy hypotheses) are
glued together to produce new solutions. Although this sort of procedure is
quite well-known, there have been some recent advances on which we wish to
report here. We also discuss what has been established about the moduli spaces
of all solutions to these problems, and report on some work in progress
concerning global aspects of these moduli spaces. In the final section we
present a new compactness result for the `unmarked moduli spaces' for problem
III.


title:
The quantization of constrained systems: from symplectic reduction to
  Rieffel induction

abstract:
  This is an introduction to the author's recent work on constrained systems.
Firstly, a generalization of the Marsden-Weinstein reduction procedure in
symplectic geometry is presented - this is a reformulation of ideas of
Mikami-Weinstein and Xu. Secondly, it is shown how this procedure is quantized
by Rieffel induction, a technique in operator algebra theory. The essential
point is that a symplectic space with generalized moment map is quantized by a
pre-(Hilbert) C^*-module. The connection with Dirac's constrained quantization
method is explained. Three examples with a single constraint are discussed in
some detail: the reduced space is either singular, or defined by a constraint
with incomplete flow, or unproblematic but still interesting. In all cases, our
quantization procedure may be carried out. Finally, we re-interpret and
generalize Mackey's quantization on homogeneous spaces. This provides a double
illustration of the connection between C^*-modules and the moment map.


title: Indefinite Kaehler-Einstein metrics on Compact Complex Surfaces

abstract:
  Indefinite Kaehler solutions of the Einstein equations are studied, and it is
almost completely determined which compact complex surfaces admit such metrics.


title: Seiberg-Witten Floer Homology and Heegaard Splittings

abstract:
  This paper presents the construction of the Seiberg-Witten-Floer homology of
three-manifolds with non-trivial rational homology, and some properties of the
invariant of three-manifolds obtained by computing the Euler characteristic.
This new version corrects a mistake and some misprints that appear in the
published version (Internat. J. Math., Vol.7, N.5 (1996) 671-696). Various
sections are rewritten in more detail.


title: A compact symmetric symplectic non-Kaehler manifold

abstract:
  In this paper I construct, using off the shelf components, a compact
symplectic manifold with a non-trivial Hamiltonian circle action that admits no
Kaehler structure. The non-triviality of the action is guaranteed by the
existence of an isolated fixed point.
  The motivation for this work comes from the program of classification of
Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of
Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of
such manifolds are Kaehler. Delzant's classification of $2n$-dimensional
symplectic manifolds with Hamiltonian action of $n$-dimensional tori showed
that all such manifolds are projective toric varieties, hence Kaehler. An
example in this paper show that not all compact symplectic manifolds that admit
Hamiltonian torus actions are Kaehler. Similar technique allows us to construct
a compact symplectic manifold with a Hamiltonian circle action that admits no
invariant complex structures, no invariant polarizations, etc.


title: On Poisson actions of compact Lie groups on symplectic manifolds

abstract:
  Let $G_{\P}$ be a compact simple Poisson-Lie group equipped with a Poisson
structure $\P$ and $(M, \o)$ be a symplectic manifold. Assume that $M$ carries
a Poisson action of $G_{\P}$ and there is an equivariant moment map in the
sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G^*_{\P}$,
$\m: M\rightarrow G^*_{\P}$. We prove that $M$ always possesses another
symplectic form $\to$ so that the $G$-action preserves $\tilde{\o}$ and there
is a new moment map $\mu= e^{-1} \circ \m: M\rightarrow \g^*$. Here $e$ is a
universal (independent of $M$) invertible equivariant map $e: \g^*\rightarrow
G^*_{\P}$. We suggest new short proves of the convexity theorem for the
Poisson-Lie moment map, Poisson reduction theorem and the Ginzburg-Weinstein
theorem on the isomorphism of $\g^*$ and $G^*_{\P}$ as Poisson spaces.


title: Associativity properties of the symplectic sum

abstract:
  In this note we apply a 4-fold sum operation to develop an associativity rule
for the pairwise symplectic sum. This allows us to show that certain
diffeomorphic symplectic $4$-manifolds made out of elliptic surfaces are in
fact symplectically deformation equivalent. We also show that blow-up points
can be traded from one side of a symplectic sum to another without changing the
symplectic deformation class of the resulting manifold.


title: Seiberg-Witten-Floer Theory for Homology 3-Spheres

abstract:
  We give the definition of the Seiberg-Witten-Floer homology group for a
homology 3-sphere. Its Euler characteristic number is a Casson-type invariant.
For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten
invariant is defined taking values in the Seiberg-Witten-Floer homology group,
these relative Seiberg-Witten invariants are applied to certain homology
spheres bounding Stein surfaces.


title:
Space of linear differential operators on the real line as a module over
  the Lie algebra of vector fields

abstract:
  Let ${\cal D}^k$ be the space of $k$-th order linear differential operators
on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural
1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal
D}^k$. (To define this family, one considers arguments of differential
operators as tensor-densities of degree $\lambda$.) In this paper we solve the
problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$
corresponding to different values of $\lambda$. The result is as follows: for
$k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each
other for every values of $\lambda\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm
\frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant)
intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case
$(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to
two different values of the degree: $\lambda$ and $\lambda^{\prime}$, are
isomorphic if and only if $\lambda+\lambda^{\prime}=1$.


title: Conformal Invariants of Manifolds of Non-positive Scalar Curvature

abstract:
  Conformal invariants of manifolds of non-positive scalar curvature are
studied in association with growth in volume and fundamental group.


title:
Three-manifolds class field theory (Homology of coverings for a
  non-virtually Haken manifold)

abstract:
  This is a first in a series of papers, devoted to the relation betwwen
three-manifolds and number fields. The present paper studies first homology of
finite coverings of a three-manifold with primary interest in the Thurston
$b_1$ conjecture.The main result reads: if $M$ does not yield the Thurston
conjecture, then the pro-p completion of its fundamental group is a Poincar\'e
duality pro-p group. Conceptually, it means that we have a ``p-adic''
three-manifold. We develop several algebraic techniques, including a new
powerful specral seguence, to actually compute homology of coverings, assumong
only information on homology of $M$, a thing never done before.A number of
applications to the structure of finite group cohomology rings is also given.


title: Equivariant Holomorphic Morse Inequalities I: A Heat Kernel Proof

abstract:
  Assume that the circle group acts holomorphically on a compact K\"ahler
manifold with isolated fixed points and that the action can be lifted
holomorphically to a holomorphic Hermitian vector bundle. We give a heat kernel
proof of the equivariant holomorphic Morse inequalities. We use some techniques
developed by Bismut and Lebeau. These inequalities, first obtained by Witten
using a different argument, produce bounds on the multiplicities of weights
occurring in the twisted Dolbeault cohomologies in terms of the data of the
fixed points.


title:
Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian
  Group Actions

abstract:
  We extend the equivariant holomorphic Morse inequalities of circle actions to
cases with torus and non-Abelian group actions on holomorphic vector bundles
over Kahler manifolds and show the necessity of the Kahler condition. For torus
actions, there is a set of inequalities for each choice of action chambers
specifying directions in the Lie algebra of the torus. We apply the results to
invariant line bundles over toric manifolds. If the group is non-Abelian, there
is in addition an action of the Weyl group on the fixed-point set of its
maximal torus. The sum over the fixed points can be rearranged into sums over
the Weyl group (having incorporated the character of the isotropy
representation on the fiber) and over its orbits.


title:
Lagrangian embedding, Maslov indexes and Integer graded symplectic Floer
  cohomology

abstract:
  We define an integer graded symplectic Floer cohomology and a spectral
sequence which are new invariants for monotone Lagrangian sub-manifolds and
exact isotopies. Such an integer graded Floer cohomology is an integral lifting
of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of
applications of the spectral sequence, we offer an affirmative answer to an
Audin's question for oriented, embedded, monotone Lagrangian tori, i.e. $\Si
(L) = 2$.


title: Polyhedral representations of discrete differential manifolds

abstract:
  Any discrete differential manifold $M$ (finite set endowed with an algebraic
differential calculus) can be represented by appropriate polyhedron ${\cal
P}(M)$. This representation demonstrates the adequacy of the calculus of
discrete differential manifolds and links this approach with that based on
finitary substitutes of continuous spaces introduced by R.D.Sorkin.


title: On the 8-dimensional hyperKaehler manifolds

abstract:
  6 pages, LaTeX, to appear in Proc. of International Conf. "Geometrization of
Physics - II", Kazan, Russia, Oct 28 - Nov 2, 1995. Withdrawn


title: Polygon spaces and Grassmannians

abstract:
  We study the moduli spaces of polygons in R^2 and R^3, identifying them with
subquotients of 2-Grassmannians using a symplectic version of the
Gel'fand-MacPherson correspondence. We show that the bending flows defined by
Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the
Grassmannian, and with these determine the pentagon and hexagon spaces up to
equivariant symplectomorphism. Other than invocation of Delzant's theorem, our
proofs are purely polygon-theoretic in nature.


title: Holomorphic torsion for Hermitian locally symmetric spaces

abstract:
  The holomorphic torsion of a compact locally symmetric manifold is expressed
as a special value of a zeta function built out of geometric data (closed
geodesics) of the manifold.


title: Equivariant torsion of locally symmetric spaces

abstract:
  The equivariant holomorphic torsion of a compact locally symmetric manifold
and an automorphism is expressed as a special value of a zeta function built
out of geometric data (closed geodesics) of the manifold.


title:
Group cohomology and the singularities of the Selberg zeta function
  associated to a Kleinian group

abstract:
  We prove Patterson's conjecture about the singularities of the Selberg zeta
function associated to a convex-cocompact, torsion free group acting on a
hyperbolic space.


title:
Decomposability of quotients by complex conjugation for rational and
  Enriques surfaces

abstract:
  The quotients $Y=X/conj$ by the complex conjugation $conj\: X\to X$ for
complex rational and Enriques surfaces $X$ defined over $\R$ are shown to be
diffeomorphic to connected sums of $\barCP2$, whenever $Y$ are simply
connected.


title: Harmonic morphisms, conformal foliations and shear-free ray congruences

abstract:
  Equivalences between conformal foliations on Euclidean $3$-space, Hermitian
structures on Euclidean $4$-space, shear-free ray congruences on Minkowski
$4$-space, and holomorphic foliations on complex $4$-space are explained
geometrically and twistorially; these are used to show that 1) any
real-analytic complex-valued harmonic morphism without critical points defined
on an open subset of Minkowski space is conformally equivalent to the direction
vector field of a shear-free ray congruence, 2) the boundary values at infinity
of a complex-valued harmonic morphism on hyperbolic $4$-space define a
real-analytic conformal foliation by curves of an open subset of Euclidean
$3$-space and all such foliations arise this way. This gives an explicit method
of finding such foliations; some examples are given.


title:
The Incidence Coefficients in the Novikov Complex are generically
  rational functions

abstract:
  For a Morse map $f:M\to S^1$ Novikov [11] has introduced an analog of Morse
complex, defined over the ring $\ZZZ[[t]][t^{-1}]$ of integer Laurent power
series. Novikov conjectured, that generically the matrix entries of the
differentials in this complex are of the form $\sum_ia_it^i$, where $a_i$ grow
at most exponentially in $i$. We prove that for any given $f$ for a $C^0$
generic gradient-like vector field all the incidence coefficients above are
rational functions in $t$ (which implies obviously the exponential growth rate
estimate).


title: On symmetries of constant mean curvature surfaces

abstract:
  We start the investigation of immersions $\Psi$ of a simply connected domain
$D$ into three dimensional Euclidean space $R^3$, which have constant mean
curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which
leave the image $\Psi(D)$ invariant. On one hand, this leads to a detailed
description of symmetric CMC-surfaces and the associated symmetry groups. On
the other hand, it allows us to start the classification of CMC-immersions of
an arbitrary, compact or noncompact Riemann surface $M$ into $R^3$ in terms of
Weierstrass-type data, as introduced by Pedit, Wu, and one of the authors [D].
We use our general results to prove, that there are no CMC-tori or Delaunay
surfaces in the dressing orbit of the cylinder. As an example, we apply the
discussion to Smyth surfaces and to a CMC-surface with a branchpoint.


title: Connectedness of spaces of symplectic embeddings

abstract:
  We prove that the space of symplectic packings of ${\Bbb C}P^2$ by $k$ equal
balls is connected for $3\leq k\leq 6$. The proof is based on Gromov-Witten
invariants and on the inflation technique due to Lalonde and McDuff.


title: Supermanifold Forms and Integration. A Dual Theory

abstract:
  We investigate forms on supermanifolds defined as Lagrangians of ``copaths''
(that is, systems of equations, which may or may not specify submanifolds). For
this, we consider direct products $M^{n|m}\times\Bbb R^{r|s}$ and study
isomorphisms corresponding to simultaneously advancing the number of additional
parameters $r|s$ and the number of equations. We define an exteriour
differential in terms of variational derivatives w.r.t. a copath and establish
its main properties. In the resulting stable picture we obtain infinite
complexes $\D:\Om{r}{s}\to\Om{r+1}{s}$ for $M^{n|m}$, where $0 \le s \le m$ and
$r$ can be any integer. For $r\ge 0$ a canonical isomorphism with forms
constructed as Lagrangians of $r|s$-paths is established. We discover the
``lacking half'' of forms on supermanifolds: $r|s$-forms with $r<0$, previously
unknown except for $s=m$. (They have been partly replaced earlier by an
augmentation of the ``non-negative'' part of the complexes.) All these results
are new. The study of these questions is in progress now.


title: Quasi-rigidity of hyperbolic 3-manifolds and scattering theory

abstract:
  Take two isomorphic convex co-compact co-infinite volume Kleinian groups,
whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by
these groups gives two hyperbolic 3-manifolds whose scattering operators may be
compared. We prove that the operator norm of the difference between the
scattering operators is small, then the groups are related by a coorespondingly
small quasi-conformal deformation. This in turn implies that the two hyperbolic
3-manifolds are quasi-isometric.


title: Minimal Geodesics and Nilpotent Fundamental Groups

abstract:
  Hedlund constructed Riemannian metrics on n-tori, $n \geq 3$ for which
minimal geodesics are very rare. In this paper we construct similar examples
for every nilpotent fundamental group. These examples show that Bangert's
existence results of minimal geodesics are optimal for nilpotent fundamental
groups.


title: Finite dimensional imbeddings of harmonic spaces

abstract:
  In a noncompact harmonic manifold $M$ we establish finite dimensionality of
the eigenspaces $V_{\lambda}$ generated by radial eigenfunctions of the form
$\cosh r + c$. As a consequence, for such harmonic manifolds, we give an
isometric imbedding of $M$ into $(V_{\lambda},B)$, where $B$ is a nondegenerate
symmetric bilinear indefinite form on $V_{\lambda}$ (analogous to the imbedding
of the real hyperbolic space $I\!\!\!H^{n}$ into $I\!\!\!R^{n+1}$ with the
indefinite form $Q(x,x) = -x_0^2 + \sum x_i^2$). Finally we give certain
conditions under which $M$ is symmetric.


title: Harmonic manifolds with some specific volume densities

abstract:
  We show that noncompact simply connected harmonic manifolds with volume
density $\Theta_{p}(r) =\sinh ^{n-1} r$ is isometric to the real hyperbolic
space and noncompact simply connected K\"{a}hler harmonic manifold with volume
density $\Theta_{p}(r) =\sinh ^{2n-1} r \cosh r$ is isometric to the complex
hyperbolic space. A similar result is also proved for Quaternionic K\"{a}hler
manifolds. Using our methods we get an alternative proof, without appealing to
the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci
flat harmonic manifold is isometric to the euclidean space. Finally a rigidity
result for real hyperbolic space is presented.


title: A Pinching constant for harmonic manifolds

abstract:
  In this note we shall show that the sectional curvature of a harmonic
manifold is bounded on both sides. In fact we shall give a pinching constant
for all harmonic manifolds. We shall use the imbedding theorem for harmonic
manifolds proved by Z.I.Szabo and the description of screw lines in hilbert
spaces to prove the result.


title: Non-abelian convexity by symplectic cuts

abstract:
  In this paper we extend the results of Kirwan et alii on convexity properties
of the moment map for Hamiltonian group actions, and on the connectedness of
the fibers of the moment map, to the case of non-compact orbifolds.
  Our motivation is twofold. First, the category of orbifolds is important in
symplectic geometry because, generically, the symplectic quotient of a
symplectic manifold is an orbifold. Second, our proof is conceptually very
simple since it reduces the non-abelian case to the abelian case.


title: Donaldson Wall-Crossing Invariants Via Topology

abstract:
  The wall-crossing formula for Donaldson invariants of smooth, simply
connected four manifolds with $b^+=1$ is shown to be a topological invariant of
the manifold for reducible connections with two or fewer singular points. The
explicit formulas derived agree with those of Ellingsrud and Gottische and
Friedman and Qin for algebraic manifolds.


title:
The hyperbolic moduli space of flat connections and the isomorphism of
  symplectic multiplicity spaces

abstract:
  Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a
maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by
$X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out
the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued
connections on a Riemann sphere with three holes which satisfy the additional
condition $\overline{A(z)}=-A(\overline{z})$. We call the quotient of this
space over the action of the gauge group $\overline{g(z)}=g^{-1}(\overline{z})$
a \emph{hyperbolic} moduli space of flat connections. We prove that the
following three symplectic spaces are isomorphic: 1. The hyperbolic moduli
space of flat connections. 2. The symplectic multiplicity space obtained as
symplectic quotient of the triple product of co-adjoint orbits of $K$. 3. The
Poisson-Lie multiplicity space equal to the Poisson quotient of the triple
product of dressing orbits of $K$.


title:
Conformal Deformation of Warped Products and Scalar Curvature Functions
  on Open Manifolds

abstract:
  We discuss conformal deformation and warped products on some open manifolds.
We discuss how these can be applied to construct Riemannian metrics with
specific scalar curvature functions.


title:
Self-duality in Dimensions $2n>4$: Equivalence of Various Definitions,
  and an Upper Bound for $p_2$

abstract:
  We show that the self-duality defined in [Trautman, Int.J.Theor.Phys.,{\bf
16},561 (1977)] is equivalent to strong self-duality defined in [Bilge, Dereli
and Kocak, Lett.Math.Phys., {\bf 36}, 301-309, (1996)] and we obtain an upper
bound on $\int (p_2)^{n}$, where $p_2$ is the second Pontrjagin class of an
$SO(N)$ bundle over a $8n$ dimensional manifold.


title:
Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of
  Revolution

abstract:
  We find sharp upper bounds for the multiplicities and the numerical values of
all the distinct eigenvalues on a surface of revolution diffeomorphic to the
sphere.


title:
Incidence coefficients in the Novikov complex for Morse forms:
  rationality and exponential growth properties

abstract:
  In this paper we continue the study of generic properties of the Novikov
complex, began in the work "The incidence coefficients in the Novikov complex
are generically rational functions" ( dg-ga/9603006). For a Morse map $f:M\to
S^1$ there is a refined version of Novikov complex, defined over the Novikov
completion of the fundamental group ring. We prove that for a $C^0$ generic
$f$-gradient the corresponding incidence coefficients belong to the image in
the Novikov ring of a (non commutative) localization of the fundamental group
ring. The Novikov construction generalizes also to the case of Morse 1-forms.
In this case the corresponding incidence coefficients belong to the suitable
completion of the ring of integral Laurent polynomials of several variables. We
prove that for a given Morse form $\omega$ and a $C^0$ generic
$\omega$-gradient these incidence coefficients are rational functions. The
incidence coefficients in the Novikov complex are obtained by counting the
algebraic number of the trajectories of the gradient, joining the zeros of the
Morse form. There is V.I.Arnold's version of the exponential growth conjecture,
which concerns the total number of trajectories. We confirm it for any given
Morse form and a $C^0$ dense set of its gradients. We also give an example of
explicit computation of the Novikov complex.


title: The Zero-in-the-Spectrum Question

abstract:
  This is an expository article on the question of whether zero lies in the
spectrum of the Laplace-Beltrami operator acting on differential forms on a
manifold.


title: Rotationally Symmetric $p$-harmonic maps

abstract:
  We study a second order ordinary differential equation corresponding to
rotationally symmetric $p$-harmonic maps. We show unique continuation and
Liouville's type theorems for positive solutions. We discuss the existence of
bounded positive entire solutions. Asymptotic properties of the positive
solutions are investigated.


title: Smooth structures on collarable ends of 4-manifolds

abstract:
  We use Furuta's result, usually referred to as ``10/8-conjecture'', to show
that for any compact 3-manifold $M$ the open manifold $M\times\r$ has
infinitely many different smooth structures. Another consequence of Furuta's
result is existence of infinitely many smooth structures on open topological
4-manifolds with a topologically collarable end, provided there are only
finitely many ends homeomorphic to it. We also show that for each closed spin
4-manifold there are exotic \rf's that can not be smoothly embedded into it.


title: Torus actions on compact quotients

abstract:
  We consider the action of a noncompact torus H on the compact quotient G/L,
where G is a Lie group containing H and L is a uniform lattice in G. Using
harmonic analysis on G we prove a formula relating the compact orbits of H to
the action of H on the (infinite dimensional) tangential cohomology. The
formula may be viewed as a Lefschetz formula or an equivariant index theorem
for noncompact group actions.


title: Obstruction Results in Quantization Theory

abstract:
  We define the quantization structures for Poisson algebras necessary to
generalise Groenewold and Van Hove's result that there is no consistent
quantization for the Poisson algebra of Euclidean phase space. Recently a
similar obstruction was obtained for the sphere, though surprising enough there
is no obstruction to the quantization of the torus. In this paper we want to
analyze the circumstances under which such obstructions appear. In this context
we review the known results for the Poisson algebras of Euclidean space, the
sphere and the torus.


title: The Fermi Flow and its Application to Geometry

abstract:
  We introduce the notion of Fermi flow for hypersurfaces in Riemannian
manifolds. It turns out that this is a powerful tool to study the geometry of
distance surfaces about a given initial hypersurface. Some of the results in
this paper are known in one form or another, however the aim is to demonstrate
how they can all be derived by the same method and proved in a very simple
manner. In addition we obtain some new results and results that are stronger
than those stated in the literature.


title: Flux homomorphism on symplectic groupoids

abstract:
  An anologue of the Calabi invariant for Poisson manifolds is considered. For
any Poisson manifold $P$, the Poisson bracket on $C^{\infty}(P)$ extends to a
Lie bracket on the space $\Omega^{1}(P)$ of all differential one-forms, under
which the space $Z^{1}(P)$ of closed one-forms and the space $B^{1}(P)$ of
exact one-forms are Lie subalgebras. These Lie algebras are related by the
exact sequence: $$0\lon \reals \lon C^{\infty}(P)\stackrel{d}{\lon}
Z^{1}(P)\stackrel{f}{\lon} H^{1}(P, \reals)\lon 0, $$ where $H^{1}(P,\reals)$
is considered as a trivial Lie algebra, and $f$ is the map sending each closed
one-form to its cohomology class. The goal of the present paper is to lift this
exact sequence to the group level for compact Poisson manifolds under certain
integrability condition. In particular, we will give a geometric description of
a Lie group integrating the underlying Poisson algebra $C^{\infty}(P) $.


title: Constant scalar curvature metrics with isolated singularities

abstract:
  We extend the results and methods of \cite{MP} to prove the existence of
constant positive scalar curvature metrics $g$ which are complete and conformal
to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a
disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. The
existence of solutions with isolated singularities occupies the majority of the
paper; their existence was previously established by Schoen \cite{S}, but the
proof we give here, based on the techniques of \cite{MP}, is more direct, and
provides more information about their geometry. When $\Lambda$ is discrete we
also establish that these solutions are smooth points in the moduli spaces of
all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}


title: A canonical way to deform a Lagrangian submanifold

abstract:
  We derive some important geometric identities for Lagrangian submanifolds
immersed in a K\"ahler manifold and prove that there exists a canonical way to
deform a Lagrangian submanifold by a parabolic flow through a family of
Lagrangian submanifolds if the ambient space is a Ricci-flat Calabi-Yau
manifold.


title: Theory of connections on graded principal bundles

abstract:
  The geometry of graded principal bundles is discussed in the framework of
graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that
a graded principal bundle is globally trivial if and only if it admits a global
graded section and, further, that the sheaf of vertical derivations on such a
bundle coincides with the graded distribution induced by the action of the
structure graded Lie group. This result leads to a natural definition of the
graded connection in terms of graded distributions; its relation with Lie
superalgebra-valued graded differential forms is also exhibited. Finally, we
define the curvature for the graded connection; we prove that the curvature
controls the involutivity of the horizontal graded distribution corresponding
to the graded connection.


title: Structure "Hyper-Lie Poisson"

abstract:
  The main purpose of the paper is to study hyperkahler structures from the
viewpoint of symplectic geometry. We introduce a notion of hypersymplectic
structures which encompasses that of hyperkahler structures. Motivated by the
work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras, we define
hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra
and give criterion which implies their existence. We study an explicit example
of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to
Nahm's equations assocaited to Lie algebra $\frak{su}(2)$ are realized as
hypersymplectic leaves and are related to the (co)adjoint orbits of
$\frak{sl}(2, \complex)$.


title: Rotationally Symmetric F-harmonic Maps Equations

abstract:
  We study a second order differential equation corresponding to rotationally
symmetric $F$-harmonic maps between certain noncompact manifolds. We show
unique continuation and Liouville's type theorems for positive solutions.
Asymptotic properties and the existence of bounded positive solutions are
investigated.


title: Yamabe Constants and the Perturbed Seiberg-Witten Equations

abstract:
  Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of
the Fubini-Study metric is shown to have the largest Yamabe constant. The
proof, which involves perturbations of the Seiberg-Witten equations, also
yields new results on the total scalar curvature of almost-K\"ahler
4-manifolds.


title: Symplectic packing in dimension 4

abstract:
  We discuss closed symplectic 4-manifolds which admit full symplectic packings
by $N$ equal balls for large $N$'s. We give a homological criterion for
recognizing such manifolds. As a corollary we prove that ${\Bbb C}P^2$ can be
fully packed by $N$ equal balls for every $N\geq 9$.


title: Harmonic two-spheres in compact symmetric spaces, revisited

abstract:
  Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic
2-spheres in a Lie group G and proved that when G=SU(n) the uniton number
cannot exceed n-1. In this paper, using new methods inspired by Morse Theory,
we explain this result and extend it to an arbitrary compact group G. The same
methods also yield Weierstrass formulae for these harmonic maps and simple
proofs of most of the known classification theorems for harmonic 2-spheres in
symmetric spaces.


title: Equivariant Seiberg-Witten Floer Homology

abstract:
  This paper circulated previously in a draft version. Now, upon general
request, it is about time to distribute the more detailed (and much longer)
version. The main technical issues revolve around the fine structure of the
compactification of the moduli spaces of flow lines and the obstruction bundle
technique, with related gluing theorems, needed in the proof of the topological
invariance of the equivariant version of the Floer homology.


title: From symplectic deformation to isotopy

abstract:
  Let $X$ be an oriented 4-manifold which does not have simple SW-type, for
example a blow-up of a rational or ruled surface. We show that any two
cohomologous and deformation equivalent symplectic forms on $X$ are isotopic.
This implies that blow-ups of these manifolds are unique, thus extending work
of Biran. We also establish uniqueness of structure for certain fibered
4-manifolds.


title: Lectures on Gromov invariants for symplectic 4-manifolds

abstract:
  These are notes of lectures given at the NATO Summer School, Montreal 1995.
Taubes's recent spectacular work setting up a correspondence between
$J$-holomorphic curves in symplectic 4-manifolds and solutions of the
Seiberg-Witten equations counts $J$-holomorphic curves in a somewhat new way.
The "standard" theory concerns itself with moduli spaces of connected curves,
and gives rise to Gromov-Witten invariants. However, Taubes's curves arise as
zero sets of sections and so need not be connected. These notes are in the main
expository. We first discuss the invariants as Taubes defined them, and then
discuss some alternatives, showing, for example, a way of dealing with
multiply-covered exceptional spheres. We also calculate some examples, in
particular finding the Gromov invariant of the fiber class of an elliptic
surface by counting $J$-holomorphic curves, rather than going via
Seiberg--Witten theory.


title: Positive paths in the linear symplectic group

abstract:
  A positive path in the linear symplectic group $\Sp(2n)$ is a smooth path
which is everywhere tangent to the positive cone. These paths are generated by
negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean
space. A special case are autonomous positive paths, which are generated by
time-independent Hamiltonians, and which all lie in the set $\Uu$ of
diagonalizable matrices with eigenvalues on the unit circle. However, as was
shown by Krein, the eigenvalues of a general positive path can move off the
unit circle. In this paper, we extend Krein's theory: we investigate the
general behavior of positive paths which do not encounter the eigenvalue 1,
showing, for example, that any such path can be extended to have endpoint with
all eigenvalues on the circle. We also show that in the case $2n=4$ there is a
close relation between the index of a positive path and the regions of the
symplectic group that such a path can cross. Our motivation for studying these
paths came from a geometric squeezing problem in symplectic topology. However,
they are also of interest in relation to the stability of periodic Hamiltonian
systems and in the theory of geodesics in Riemannian geometry.


title: Example of a non-log-concave Duistermaat-Heckman measure

abstract:
  We construct a compact symplectic manifold with a Hamiltonian circle action
for which the Duistermaat-Heckman function is not log-concave.


title: Novikov inequalities with symmetry

abstract:
  We establish an equivariant generalization of the Novikov inequalities which
allow to estimate the topology of the set of critical points of a closed basic
invariant 1-form by means of twisted equivariant cohomology of the manifold. We
test and apply our inequalities in the case of a finite group.
  As an application we obtain Novikov type inequalities for a manifold with
boundary.


title:
On Livsic's Theorem, Superrigidity, and Anosov Actions of Semisimple Lie
  Groups

abstract:
  We prove a generalization of Livsic's Theorem on the vanishing of the
cohomology of certain types of dynamical systems. As a consequence, we
strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions
of semisimple Lie groups. Combining this with Topological Superrigidity, we
find a Holder geometric structure for multiplicity free Anosov actions.


title:
Smooth classification of Cartan actions of higher rank semisimple Lie
  groups and their lattices

abstract:
  Let G be a connected semisimple Lie group without compact factors whose real
rank is at least 2, and let \Gamma \subset G be an irreducible lattice. We
provide a C^\infty classification for volume-preserving Cartan actions of
\Gamma and G. Also, if G has real rank at least 3, we provide a C^\infty
classification for volume-preserving, multiplicity free, trellised, Anosov
actions on compact manifolds.


title: Analytic fields on compact balanced Hermitian manifolds

abstract:
  On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using
the torsion and the curvature of the Chern connection. On a compact balanced
Hermitian manifold we find necessary and sufficient conditions in terms of the
tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form
to be harmonic. We prove that if $H$ is positive definite then the first Betti
number $b_1 = 0$ and the Hodge number $h^{1,0} = 0$. We obtain an obstruction
to the existence of Killing vector fields in terms of the Ricci tensor of the
Chern connection: if the Chern form of the Chern connection on a compact
balanced Hermitian manifold is non- positive definite then every Killing vector
field is analytic; if moreover the Chern form is negative definite then there
are no Killing vector fields. It is proved that on a compact balanced Hermitian
manifold every affine with respect to the Chern connection vector field is an
analytic vector field.


title: Hinges and geometric constructions of boundaries of symmetric spaces

abstract:
  We give elementary constructions for Satake-Furstenberg, Martin and
Karpelevich boundaries of symmetric spaces. We also consruct some "new"
boundaries


title: Homological algebra of Novikov-Shubin invariants and Morse inequalities

abstract:
  It is shown that the topological phenomenon "zero in the continuous
spectrum", discovered by S.P.Novikov and M.A.Shubin, can be explained in terms
of a homology theory on the category of finite polyhedra with values in certain
abelian category. This approach implies homotopy invariance of the
Novikov-Shubin invariants. Its main advantage is that it allows to use the
standard homological techniques, such as spectral sequences, derived functors,
universal coefficients etc., while studying the Novikov-Shubin invariants. It
also leads to some new quantitative invariants, measuring the Novikov-Shubin
phenomenon in a different way, which are used in order to strengthen the Morse
type inequalities of Novikov and Shubin.


title: Combinatorial invariants computing the Ray-Singer analytic torsion

abstract:
  It is shown that for any piecewise-linear closed orientable manifold of odd
dimension there exists an invariantly defined metric on the determinant line of
cohomology with coefficients in an arbitrary flat bundle E over the manifold (E
is not required to be unimodular). The construction of this metric (called
Poincare - Reidemeister metric) is purely combinatorial; it combines the
standard Reidemeister type construction with Poincare duality. The main result
of the paper states that the Poincare-Reidemeister metric computes
combinatorially the Ray-Singer metric. It is shown also that the Ray-Singer
metrics on some relative determinant lines can be computed combinatorially
(including the even-dimensional case) in terms of metrics determined by
correspondences.


title: Twistor spaces of non-flat Bochner-K\"ahler manifolds

abstract:
  For the twistor spaces of the Bochner-K\"ahler manifold $M = H^l \times P^n$,
systems of holomorphic coordinates are constructed. As an application of them,
an explicit description of the moduli space of relative deformations of fibers
of $M$'s twistor space is given.


title: The index of operators on foliated bundles

abstract:
  We compute the equivariant cohomology Chern character of the index of
elliptic operators along the leaves of the foliation of a flat bundle. The
proof is based on the study of certain algebras of pseudodifferential operators
and uses techniques for analizing noncommutative algebras similar to those
developed in Algebraic Topology, but in the framework of cyclic cohomology and
noncommutative geometry.


title: An Intrinsic Approach to Lichnerowicz Conjecture

abstract:
  In this paper we give a proof of Lichnerowicz Conjecture for compact simply
connected manifolds which is intrinsic in the sense that it avoids the {\it
Nice Embeddings} into eigen spaces of the Laplacian. Even if one wants to use
these embeddings this paper gives a more streamlined proof.


title: Exotic holonomies $\E_7^{(a)}$

abstract:
  It is proved that the Lie groups $\E_7^{(5)}$ and $\E^{(7)}_7$ represented in
$\R^{56}$ and the Lie group $\E_7^{\C}$ represented in $\R^{112}$ occur as
holonomies of torsion-free affine connections. It is also shown that the moduli
spaces of torsion-free affine connnections with these holonomies are finite
dimensional, and that every such connection has a local symmetry group of
positive dimension.


title:
Differential operators of Fuchs type, conical singularities, and
  asymptotic methods

abstract:
  This text is a revised version of the authors Habilitationsschrift which was
submitted to the University of Augsburg, 1993.
  Fuchs type differential operators are used to model the analysis on manifolds
with cone--like singularities, or more general, stratified spaces. This book
provides a self--contained treatment of the analysis of the heat equation and
index theory for these operators.
  Major topics are short--time asymptotics, $\eta$-- and $\zeta$--functions,
(relative) index theorems. Another chapter is devoted to the discussion of
deficiency indices and Dirac Schr\"odinger operators.


title: Diffeomorphisms, Analytic Torsion and Noncommutative Geometry

abstract:
  We prove an index theorem concerning the pushforward of flat B-vector
bundles, where B is an appropriate algebra. We construct the associated
analytic torsion form T. If Z is a smooth closed aspherical manifold, we show
that T gives invariants of the homotopy groups of Diff(Z).


title: Backlund transformations and knots of constant torsion

abstract:
  The Backlund transformation for pseudospherical surfaces, which is equivalent
to that of the sine-Gordon equation, can be restricted to give a transformation
on space curves that preserves constant torsion. We study its effects on closed
curves (in particular, elastic rods) that generate multiphase solutions for the
vortex filament flow (also known as the Localized Induction Equation). In doing
so, we obtain analytic constant-torsion representatives for a large number of
knot types.


title: Higher spectral flow

abstract:
  For a continuous curve of families of Dirac type operators we define a higher
spectral flow as a $K$-group element. We show that this higher spectral flow
can be computed analytically by $\heta$-forms, and is related to the family
index in the same way as the spectral flow is related to the index. We
introduce a notion of Toeplitz family and relate its index to the higher
spectral flow. Applications to family indices for manifolds with boundary are
also given.


title: Pseudo Harmonic Morphisms

abstract:
  We study a geometrical condition (PHWC) which is weaker than horizontal weak
conformality. In particular, we show that harmonic maps satisfying this
condition, which will be called {\em pseudoharmonic morphisms}, include
harmonic morphisms and can be described as pulling back certain germs to
certain other germs. Finally, we construct a canonical f-structure associated
to every map satisfying (PHWC) and find conditions on this f-structure to
ensure the harmonicity of the map.


title: Pluriharmonic Morphisms

abstract:
  Pluriharmonic maps form an important class of harmonic maps which includes
holomorphic maps. We study their morphisms, in particular the
inter-relationships between $(1,1)$-geodesic, pluriharmonic and
$\pm$holomorphic maps. Then we characterise pluriharmonic morphisms between
Hermitian manifolds. We make a special study of the situation where the target
is K{\"a}hler, pluriharmonic morphisms being particularly well understood for
this case.


title: Gromov-Witten invariants of general symplectic manifolds

abstract:
  We present an approach to Gromov-Witten invariants that works on arbitrary
(closed) symplectic manifolds. We avoid genericity arguments and take into
account singular curves in the very formulation. The method is by first
endowing mapping spaces from (prestable) algebraic curves into the symplectic
manifold with the structure of a Banach orbifold and then exhibiting the space
of stable $J$-curves (``stable maps'') as zero set of a Fredholm section of a
Banach orbibundle over this space. The invariants are constructed by pairing
with a homology class on the locally compact topological space of stable
$J$-curves that is generated as localized Euler class of the section.


title: Almost complex structures and geometric quantization

abstract:
  We study two quantization schemes for compact symplectic manifolds with
almost complex structures. The first of these is the Spin-c quantization. We
prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which
shows that the index space of this operator provides an honest (not virtual)
vector space semiclassically. We also introduce a new quantization scheme,
based on a rescaled Laplacian, for which we are able to prove strong
semiclassical properties. The two quantizations are shown to be close
semiclassically.


title: Discriminants of convex curves are homeomorphic

abstract:
  For a given real generic curve $\ga: S^1\to \Bbb {RP}^n$ let $D_\ga$ denote
the ruled hypersurface in $\Bbb {RP}^n$ consisting of all osculating subspaces
to $\ga$ of codimension 2. A curve $\ga: S^1\to \Bbb {RP}^n$ is called convex
if the total number of its intersection points (counted with multiplicities)
with any hyperplane in $\Bbb {RP}^n$ does not exceed $n$. In this short note we
show that for any two convex real projective curves $\ga_1:S^1\to\Bbb {RP}^n$
and $\ga_2:S^1\to\Bbb {RP}^n$ the pairs $(\Bbb {RP}^n,D_{\ga_1})$ and $(\Bbb
{RP}^n,D_{\ga_2})$ are homeomorphic answering a question posed by V.Arnold.


title: Kaehler structures on Kc/(P,P)

abstract:
  Let K be a compact semi-simple Lie group. We classify K-invariant Kaehler
structures on the space Kc/(P,P), where Kc is the complexification of K, P is a
parabolic subgroup of Kc, and (P,P) the commutator subgroup. For each Kaehler
structure, we study its moment map and associated pre-quantum line bundle for
geometric quantization. Some holomorphic sections of the line bundle form a
unitary K-representation space, and we study the multiplicity of its
irreducible subrepresentations.


title:
A universal lower bound for the first eigenvalue of the Dirac operator
  on quaternionic Kaehler manifolds

abstract:
  A universal lower bound for the first positive eigenvalue of the Dirac
operator on a compact quaternionic Kaehler manifold M of positive scalar
curvature is calculated. It is shown that it is equal to the first positive
eigenvalue on the quaternionic projective space. For this, the horizontal
tangent bundle on the canonical SO(3)-bundle over M is equipped with a
hyperkaehlerian structure and the corresponding splitting of the horizontal
spinor bundle is considered. The desired estimate is obtained by looking at
hyperkaehlerian twistor operators on horizontal spinors.


title: Stability of symmetric tops via one variable calculus

abstract:
  We study the stability of symmetric trajectories of a particle on the Lie
group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant
metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce
the number of degrees of freedom at {\em singular} values of the $SO(2)\times
SO(2)$ momentum map and study the stability of the equilibria of the reduced
systems as a function of spin. The result is an elementary analysis of the
fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since
an $SO(2)\times SO(2)$ invariant potential on $SO(3)$ can be thought of as
${\bf Z}_2$ invariant function on a circle, we get a condition on the second
and fourth derivatives of the potential at the symmetric points that guarantees
that the corresponding system gains stability as the spin increases.


title: On the eta-invariant of certain nonlocal boundary value problems

abstract:
  Motivated by the work of Vishik on the analytic torsion we introduce a new
class of generalized Atiyah-Patodi-Singer boundary value problems. We are able
to derive a full heat expansion for this class of operators generalizing
earlier work of Grubb and Seeley. As an application we give another proof of
the gluing formula for the eta invariant. Our class of boundary conditions
contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary
value problems as well as the (local) relative and absolute boundary conditions
for the Gauss-Bonnet operator.


title:
Approximating L^2 invariants of amenable covering spaces: A heat kernel
  approach

abstract:
  In this paper, we prove that the L^2 Betti numbers of an amenable covering
space can be approximated by the average Betti numbers of a regular exhaustion,
under some hypotheses. We also prove that some L^2 spectral invariants can be
approximated by the corresponding average spectral invariants of a regular
exhaustion. The main tool which is used is a generalisation of the "principle
of not feeling the boundary" (due to M. Kac), for heat kernels associated to
boundary value problems.


title:
Approximating L^2 invariants of amenable covering spaces: A
  combinatorial approach

abstract:
  In this paper, we prove that the $L^2$ Betti numbers of an amenable covering
space can be approximated by the average Betti numbers of a regular exhaustion,
proving a conjecture that we made in an earlier paper. We also prove that an
arbitrary amenable covering space of a finite simplicial complex is of
determinant class.


title:
Spinors, Nonlinear Connections, and Nearly Autoparallel Maps of
  Generalized Finsler Spaces

abstract:
  We review the geometric setting of the field theory with locally anisotropic
interactions. The concept of locally anisotropic space is introduced as a
general one for various type of extensions of Lagrange and Finsler geometry and
higher dimension (Kaluza--Klein type) spaces. The problem of definition of
spinors on generalized Finsler spaces is solved in the framework of the
geometry of Clifford bundles provided with compatible nonlinear and
distinguished connections and metric structures. We construct the spinor
differential geometry of locally anisotropic spaces and discuss some related
issues connected with the geometric aspects of locally anisotropc interactions
for gravitational, gauge, spinor, Dirac spinor and Proca fields. Motion
equations in generalized Finsler spaces, of the mentioned type of fields, are
defined in a geometric manner by using bundles of linear and affine frames
locally adapted to the nonlinear connection structure. The nearly autoparallel
maps are introduced as maps with deformation of connections extending the class
of geodesic and conformal transforms. Using this approach we propose two
variants of solution of the problem of definition of conservation laws on
locally anisotropic spaces.
  1991 Mathematics Subject Classification: 53A50, 53B15, 53B40, 53B50, 53C05,
70D10, 81D27, 81E13, 81E20, 83C40, 83C47, 83D05, 83E15 Key words and phrases:
Spinors, nonlinear connections, generalized Lagrange and Finsler spaces,
locally anisotropic fields, nearly autoparallel maps, conservation laws.