title:
The universal cover of an affine three-manifold with holonomy of
  shrinkable dimension $\leq 2$

abstract:
  An affine manifold is a manifold with an affine structure, i.e. a
torsion-free flat affine connection. We show that the universal cover of a
closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or
discompacit\'e in French) less than or equal to two is diffeomorphic to
$\bR^3$. Hence, $M$ is irreducible. This follows from two results: (i) a simply
connected affine 3-manifold which is 2-convex is diffeomorphic to $\bR^3$,
whose proof using the Morse theory takes most of this paper; and (ii) a closed
affine manifold of holonomy of shrinkable dimension less or equal to $d$ is
$d$-convex. To prove (i), we show that 2-convexity is a geometric form of
topological incompressibility of level sets. As a consequence, we show that the
universal cover of a closed affine three-manifold with parallel volume form is
diffeomorphic to $\bR^3$, a part of the weak Markus conjecture. As
applications, we show that the universal cover of a hyperbolic 3-manifold with
cone-type singularity of arbitrarily assigned cone-angles along a link removed
with the singular locus is diffeomorphic to $\bR^3$. A fake cell has an affine
structure as shown by Gromov. Such a cell must have a concave point at the
boundary.


title: Circle-valued Morse theory and Reidemeister torsion

abstract:
  Let X be a closed manifold with zero Euler characteristic, and let f: X -->
S^1 be a circle-valued Morse function. We define an invariant I which counts
closed orbits of the gradient of f, together with flow lines between the
critical points. We show that our invariant equals a form of topological
Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679-695].
  We proved a similar result in our previous paper [Topology, 38 (1999)
861-888], but the present paper refines this by separating closed orbits and
flow lines according to their homology classes. (Previously we only considered
their intersection numbers with a fixed level set.) The proof here is
independent of the previous proof and also simpler.
  Aside from its Morse-theoretic interest, this work is motivated by the fact
that when X is three-dimensional and b_1(X)>0, the invariant I equals a
counting invariant I_3(X) which was conjectured in our previous paper to equal
the Seiberg-Witten invariant of X. Our result, together with this conjecture,
implies that the Seiberg-Witten invariant equals the Turaev torsion. This was
conjectured by Turaev [Math. Res. Lett. 4 (1997) 679-695] and refines the
theorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661-674].


title: Seiberg-Witten Equations on Three-Manifolds with Euclidean Ends

abstract:
  We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends
(connected sums of $\R^3$ and a compact manifold) with perturbations which
approximate $*dx_3$ at infinity, and describe the structure of the moduli
spaces. The setup is inspired by Taubes's program of relating the 4-dimensional
Seiberg-Witten invariant with `singular Gromov invariants' and has related
applications.


title:
Simple homotopy type of the Novikov complex and Lefschetz
  $\zeta$-function of the gradient flow

abstract:
  Let f be a Morse map from a closed manifold to a circle. S.P.Novikov
constructed an analog of the Morse complex for f. The Novikov complex is a
chain complex defined over the ring of Laurent power series with integral
coefficients and finite negative part. This complex depends on the choice of a
gradient-like vector field. The homotopy type of the Novikov complex is the
same as the homotopy type of the completed complex of the simplicial chains of
the cyclic covering associated to f. In the present paper we prove that for
every C^0-generic f-gradient there is a homotopy equivalence between these two
chain complexes, such that its torsion equals to the Lefschetz zeta-function of
the gradient flow. For these gradients the Novikov complex is defined over the
ring of rational functions and the Lefschetz zeta-function is also rational.
The paper contains also a survey of Morse-Novikov theory and of the previous
results of the author on the C^0-generic properties of the Novikov complex.


title: On the Flux Conjectures

abstract:
  The ``Flux conjecture'' for symplectic manifolds states that the group of
Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic
diffeomorphisms. We prove the conjecture for spherically rational manifolds and
for those whose minimal Chern number on 2-spheres either vanishes or is large
enough. We also confirm a natural version of the Flux conjecture for symplectic
torus actions. In some cases we can go further and prove that the group of
Hamiltonian diffeomorphisms is C^0-closed in the identity component of the
group of all symplectic diffeomorphisms.


title: Harmonic Spinors for Twisted Dirac Operators

abstract:
  We show that for a suitable class of ``Dirac-like'' operators there holds a
Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are
closed Riemannian manifolds of dimension $n\ge 3$ together with such operators,
then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that
the spectrum of its associated operator is close to the disjoint union of the
spectra of the two original operators. As an application, we show that in
dimension $n\equiv 3$ mod 4 harmonic spinors for the Dirac operator of a spin,
$\Spinc$, or $\Spinh$ manifold are not topologically obstructed.


title: Quaternionic Killing Spinors

abstract:
  In a previous article we proved a lower bound for the spectrum of the Dirac
operator on quaternionic Kaehler manifolds. In the present article we study the
limiting case, i. e. manifolds where the lower bound is attained as an
eigenvalue. We give an equivalent formulation in terms of a quaternionic
Killing equation and show that the only symmetric quaternionic Kaehler
manifolds with smallest possible eigenvalue are the quaternionic projective
spaces.


title:
Multivector Fields and Connections. Setting Lagrangian Equations in
  Field Theories

abstract:
  The integrability of multivector fields in a differentiable manifold is
studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable
multivector fields in $E$ are equivalent to integrable connections in the
bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is
applied to the particular case of multivector fields in the manifold $J^1E$ and
connections in the bundle $J^1E\to M$ (that is, jet fields in the repeated jet
bundle $J^1J^1E$), in order to characterize integrable multivector fields and
connections whose integral manifolds are canonical lifting of sections. These
results allow us to set the Lagrangian evolution equations for first-order
classical field theories in three equivalent geometrical ways (in a form
similar to that in which the Lagrangian dynamical equations of non-autonomous
mechanical systems are usually given). Then, using multivector fields; we
discuss several aspects of these evolution equations (both for the regular and
singular cases); namely: the existence and non-uniqueness of solutions, the
integrability problem and Noether's theorem; giving insights into the
differences between mechanics and field theories.


title: Systolic freedom of orientable manifolds

abstract:
  In 1972, Marcel Berger defined a metric invariant that captures the `size' of
k-dimensional homology of a Riemannian manifold. This invariant came to be
called the k-dimensional SYSTOLE. He asked if the systoles can be constrained
by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct
metrics, inspired by M. Gromov's 1993 example, which give a negative answer for
large classes of manifolds, for the product of systoles in a pair of
complementary dimensions. An obstruction (restriction on k modulo 4) to
constructing further examples by our methods seems to reside in the free part
of real Bott periodicity. The construction takes place in a split neighborhood
of a suitable k-dimensional submanifold whose connected components (rationally)
generate the k-dimensional homology group of the manifold. Bounded geometry
(combined with the coarea inequality) implies a lower bound for the k-systole,
while calibration with support in this neighborhood provides a lower bound for
the systole of the complementary dimension. In dimension 4 everything reduces
to the case of S^2 x S^2.


title: Almost Diameter Rigidity for the Cayley Plane

abstract:
  In this paper we give a generalisation of the Radius Rigidity theorem of
F.Wilhelm. This is done by showing that if a Riemannian submersion of $S^{15}$
with 7-dimensional fibres has at least one fibre which is a great sphere then
all the fibres are so. Some weaker than known conditions which force the
existence of such a fibre are also discussed.


title: Poisson Actions and Scattering Theory for Integrable Systems

abstract:
  Conservation laws, heirarchies, scattering theory and B\"acklund
transformations are known to be the building blocks of integrable partial
differential equations. We identify these as facets of a theory of Poisson
group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which
includes the non-linear Schr\"odinger equation, modified KdV, and the n-wave
equation). We also discuss a number of applications in geometry, including the
sine-Gordon equation, harmonic maps, Schr\"odinger flows on Hermitian symmetric
spaces, Darboux orthogonal coordinates, and isometric immerisons of one space
form in another.


title: Equivariant Higher Analytic Torsion and Equivariant Euler Characteristic

abstract:
  We show that J. Lott's equivariant higher analytic torsion for compact group
actions depends only on the equivariant Euler characteristic.


title: Linearization of Nambu Structures

abstract:
  We classify linear Nambu structures (which are generalized Poisson structures
in Hamiltonian dynamics and which give rise to integrable differential forms
and singular foliations), then give a linearization for Nambu structures anf
integrable differential forms near a nondegenerate singular point.


title: Zeta Functions, Determinants and Torsion for Open Manifolds

abstract:
  On an open manifold, the spaces of metrics or connections of bounded
geometry, respectively, split into an uncountable number of components. We show
that for a pair of metrics or connections, belonging to the same component,
relative $\zeta$-functions, determinants, torsion for pairs of generalized
Dirac operators are well defined.


title: On nodal sets for Dirac and Laplace operators

abstract:
  We prove that the nodal set (zero set) of a solution of a generalized Dirac
equation on a Riemannian manifold has codimension 2 at least. If the underlying
manifold is a surface, then the nodal set is discrete. We obtain a quick proof
of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami
operator on a Riemannian manifold consists of a smooth hypersurface and a
singular set of lower dimension. We also see that the nodal set of a
$\Delta$-harmonic differential form on a closed manifold has codimension 2 at
least; a fact which is not true if the manifold is not closed. Examples show
that all bounds are optimal.


title: L^2-torsion of hyperbolic manifolds of finite volume

abstract:
  Suppose $\bar{M}$ is a compact connected odd-dimensional manifold with
boundary, whose interior $M$ comes with a complete hyperbolic metric of finite
volume. We will show that the $L^2$-topological torsion of $\bar{M}$ and the
$L^2$-analytic torsion of the Riemannian manifold $M$ are equal. In particular,
the $L^2$-topological torsion of $\bar{M}$ is proportional to the hyperbolic
volume of $M$, with a constant of proportionality which depends only on the
dimension and which is known to be nonzero in dimension 3, 5 and 7. In
dimension 3 this proves the conjecture Of Lott and Lueck which gives a complete
calculation of the $L^2$-topological torsion of compact $L^2$-acyclic
3-manifolds which admit a geometric torus-decomposition. In an appendix we give
a counterexample to an extension of the Cheeger-Mueller theorem to manifolds
with boundary: if the metric is not a product near the boundary, in general
analytic and topological torsion are not equal, even if the Euler
characteristic of the boundary vanishes.
  Keywords: L^2-torsion, hyperbolic manifolds, 3-manifolds


title: Torsion and fibrations

abstract:
  We study the behaviour of analytic torsion under smooth fibrations. Namely,
let F \to E \to^{f} B be a smooth fiber bundle of connected closed oriented
smooth manifolds and let $V$ be a flat vector bundle over $E$. Assume that $E$
and $B$ come with Riemannian metrics and $V$ comes with a unimodular (not
necessarily flat) Riemannian metric. Let $\rho_{an}(E;V)$ be the analytic
torsion of $E$ with coefficients in $V$ and let $\Pf_B$ be the Pfaffian
$\dim(B)$-form. Let $H^q_{dR}(F;V)$ be the flat vector bundle over $B$ whose
fiber over $b \in B$ is $H^q_{dR}(F_b;V)$ with the Riemannian metric which
comes from the Hilbert space structure on the space of harmonic forms induced
by the Riemannian metrics. Let $\rho_{an}(B;H^q_{dR}(F;V))$ be the analytic
torsion of $B$ with coefficients in this bundle. The Leray-Serre spectral
sequence for deRham cohomology determines a certain correction term
$\rho^{Serre}_{dR}(f)$. We prove $\rho_{an}(E;V) = \int_B \rho_{an}(F_b;V)
\cdot \Pf_B + \sum_{q} (-1)^q \cdot \rho_{an}(B;H^q_{dR}(F;V)) +
\rho^{Serre}_{dR}(f)$.


title:
Dimension theory of arbitrary modules over finite von Neumann algebras
  and applications to $L^2$-Betti numbers

abstract:
  We define for arbitrary modules over a finite von Neumann algebra $\cala$ a
dimension taking values in $[0,\infty]$ which extends the classical notion of
von Neumann dimension for finitely generated projective $\cala$-modules and
inherits all its useful properties such as additivity, cofinality and
continuity. This allows to define $L^2$-Betti numbers for arbitrary topological
spaces with an action of a discrete group $\Gamma$ extending the well-known
definition for regular coverings of compact manifolds. We show for an amenable
group $\Gamma$ that the $p$-th $L^2$-Betti number depends only on the
$\cc\Gamma$-module given by the $p$-th singular homology. Using the generalized
dimension function we detect elements in $G_0(\cc\Gamma)$, provided that
$\Gamma$ is amenable. We investigate the class of groups for which the zero-th
and first $L^2$-Betti numbers resp. all $L^2$-Betti numbers vanish. We study
$L^2$-Euler characteristics and introduce for a discrete group $\Gamma$ its
Burnside group extending the classical notions of Burnside ring and Burnside
ring congruences for finite $\Gamma$.
  Keywords: Dimension functions for finite von Neumann algebras, $L^2$-Betti
numbers, amenable groups, Grothendieck groups, Burnside groups


title: Remark About Heat Diffusion on Periodic Spaces

abstract:
  Let M be a complete Riemannian manifold with a free cocompact Z^k-action. Let
k(t,x,y) be the heat kernel on M. We compute the asymptotics of k(t,x,y) in the
limit in which t goes to infinity and d(x,y) is comparable to sqrt{t}. We show
that in this limit, the heat diffusion is governed by an effective Euclidean
metric on R^k coming from the Hodge inner product on H^1(M/Z^k; R).


title:
Sequences of Levy Transformations and Multi-Wro\'nski Determinant
  Solutions of the Darboux System

abstract:
  Sequences of Levy transformations for the Darboux system of conjugates nets
in multidimensions are studied. We show that after a suitable number of Levy
transformations, with at least a Levy transformation in each direction, we get
closed formulae in terms of multi-Wro\'nski determinants. These formulae are
for the tangent vectors, Lam\`e coefficients, rotation coefficients and points
of the surface.


title: Nonlocal invariants in index theory

abstract:
  This article surveys the relations among local and nonlocal invariants in
Atiyah-Singer index theory. We discuss the local invariants that arise from the
heat equation approach to the index theorem for geometric operators, as well as
the nonlocal invariants (the eta invariant, the determinant of the
Laplacian/analytic torsion) that occur in more refined index theorems, such as
the determinant line bundle setting and the index theorem for families of
manifolds with boundary. We also discuss the higher torsion forms of Bismut and
Lott and their conjectured relation to the rational homotopy of the
diffeomorphism group of aspherical manifolds.


title:
A Strong Maximum Principle for Weak Solutions of Quasi-Linear Elliptic
  Equations with Applications to Lorentzian and Riemannian Geometry

abstract:
  The strong maximum principle is proved to hold for weak (in the sense of
support functions) sub- and super-solutions to a class of quasi-linear elliptic
equations that includes the mean curvature equation for $C^0$ spacelike
hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped
product splitting theorem is given.


title: Volumes, middle-dimensional systoles, and Whitehead products

abstract:
  Let X be a closed manifold of dimension 2m >= 6 with torsion-free
middle-dimensional homology. We construct metrics on X of arbitrarily small
volume, such that every middle-dimensional submanifold of less than unit volume
necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.


title:
Scalar Curvature Rigidity for asymptotically locally hyperbolic
  manifolds

abstract:
  Rigidity results for asymptotically locally hyperbolic manifolds with lower
bounds on scalar curvature are proved using spinor methods related to the
Witten proof of the positive mass theorem. The argument is based on a study of
the Dirac operator defined with respect to the Killing connection. The
existence of asymptotic Killing spinors is related to the spin structure on the
end. The expression for the mass is calculated and proven to vanish for
conformally compact Einstein manifolds with conformal boundary a spherical
space form, giving rigidity. In the 4-dimensional case, the signature of the
manifold is related to the spin structure on the end and explicit formulas for
the relevant invariants are given.


title:
Moduli spaces of flat connections on 2-manifolds, cobordism, and
  Witten's volume formulas

abstract:
  We discuss Witten's formulas for the symplectic volumes of moduli spaces of
flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as
introduced by Ginzburg-Guillemin-Karshon.


title: Fusion of Hamiltonian loop group manifolds and cobordism

abstract:
  We construct an oriented cobordism between moduli spaces of flat connections
on the three holed sphere and disjoint unions of toric varieties, together with
a closed two-form which restricts to the symplectic forms on the ends. As
applications, we obtain formulas for mixed Pontrjagin numbers and Witten's
formulas for symplectic volumes.


title: Comparison and Rigidity Theorems in Semi-Riemannian Geometry

abstract:
  The comparison theory for the Riccati equation satisfied by the shape
operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds
of arbitrary index, using one-sided bounds on the Riemann tensor which in the
Riemannian case correspond to one-sided bounds on the sectional curvatures.
Starting from 2-dimensional rigidity results and using an inductive technique,
a new class of gap-type rigidity theorems is proved for semi-Riemannian
manifolds of arbitrary index, generalizing those first given by Gromov and
Greene-Wu. As applications we prove rigidity results for semi-Riemannian
manifolds with simply connected ends of constant curvature.


title: Lie Group Valued Moment Maps

abstract:
  We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map
takes values in the group G itself rather than in the dual of the Lie algebra.
The theory includes counterparts of Hamiltonian reductions, the
Guillemin-Sternberg symplectic cross-section theorem and of convexity
properties of the moment map. As an application we obtain moduli spaces of flat
connections on an oriented compact 2-manifold with boundary as
quasi-Hamiltonian quotients of the space G^2 x ... x G^2.


title: On the Determinant of One-Dimensional Elliptic Boundary Value Problems

abstract:
  We discuss the $\zeta-$regularized determinant of elliptic boundary value
problems on a line segment. Our framework is applicable for separated and
non-separated boundary conditions.


title: Singular Reduction and Quantization

abstract:
  Consider a compact prequantizable symplectic manifold M on which a compact
Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with
reduction'' theorem asserts that the G-invariant part of the equivariant index
of M is equal to the Riemann-Roch number of the symplectic quotient of M,
provided the quotient is nonsingular. We extend this result to singular
symplectic quotients, using partial desingularizations of the symplectic
quotient to define its Riemann-Roch number. By similar methods we also compute
multiplicities for the equivariant index of the dual of a prequantum bundle,
and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is
invariant under symplectic reduction.


title: Fedosov Manifolds

abstract:
  In this paper we study geometry of symmetric torsion-free connections which
preserve a given symplectic form


title: A remark on Berezin's quantization and cut locus

abstract:
  The consequences for Berezin's quantization on symmetric spaces of the
identity of the set of coherent vectors orthogonal to a fixed one with the cut
locus are stated precisely. It is shown that functions expressing the coherent
states, the covariant symbols of operators, the diastasis function, the
characteristic and two-point functions are defined when one variable does not
belong to the cut locus of the other one.


title: Coherent states, transition amplitudes and embeddings

abstract:
  The transition amplitudes between coherent states on a coherent state
manifold are expressed in terms of the embedding of the coherent state manifold
into a projective Hilbert space. Consequences for the dimension of projective
Hilbert space and a simple geometric interpretation of Calabi's diastasis
follows.


title: Geometry via coherent states

abstract:
  It is shown how the coherent states permit to find different geometrical
objects as the geodesics, the conjugate locus, the cut locus, the Calabi's
diastasis and its domain of definition, the Euler-Poincar\'e characteristic,
the number of Borel-Morse cells, the Kodaira embedding theorem.


title: Yamabe Invariants and Spin^c Structures

abstract:
  The Yamabe Invariant of a smooth compact manifold is by definition the
supremum of the scalar curvatures of unit-volume Yamabe metrics on the
manifold. For an explicit infinite class of 4-manifolds, we show that this
invariant is positive but strictly less than that of the 4-sphere. This is done
by using spin^c Dirac operators to control the lowest eigenvalue of a
perturbation of the Yamabe Laplacian. These results dovetail perfectly with
those derived from the perturbed Seiberg-Witten equations, but the present
method is much more elementary in spirit.


title:
The surfaces capable of division into Infinitesimal Squares by their
  Curves of Curvature

abstract:
  Classically, isothermic surfaces are characterized as those surfaces which
are "divisible into infinitesimal squares by their curvature lines". This
characterization is the direct analogue to the definition of discrete
isothermic nets. In order to understand the relations between the discrete and
the smooth theory better, it is described how to give the classical
characterization a rigorous meaning in the sense of modern differential
geometry.


title: The Symmetries of Solitons

abstract:
  This paper is an expository account of the development of soliton
mathematics, from its inception in famous numerical experiments of
Fermi-Pasta-Ulam and Zabusky-Kruskal to the recent synthesis of Terng-Uhlenbeck
(dg-ga/9707004) that explains hidden symmetries of soliton equations in terms
of loop-groups acting by dressing transformations. The inverse scattering
method is explained in detail, first using inverse scattering for the
Schroedinger equation to solve the IVP for the KdV equation (the original
application) and then using inverse scattering for the Zero Curvature Lax
Equation to solve the IVP for the Nonlinear Schroedinger equation, and more
generally other integrable PDE arising from the ZS-AKNS scheme devised by
Zakharov and Shabat and by Ablowitz, Kaup, Newell and Segur. The paper is a
revised version of notes from a series of Rudolf Lipschitz Lectures delivered
by the author at Bonn University in January and February of 1997.


title:
Exterior differential systems: a geometric approach to PDE. Lecture
  notes from the 1997 Daewoo workshop

abstract:
  This is an elementary introduction to exterior differential systems motivated
by two examples: minimal submanifolds and the isometric embedding problem. The
two main goals of the lectures are: 1. To explain how to find an appropriate
geometric setting for studying a given system of pde. 2. To explain the Cartan
algorithm to determine the moduli space of local solutions to any given
exterior differential system and an appropriate initial value problem for the
system.


title: Pseudo-holomorphic curves and the Weinstein conjecture

abstract:
  We established existence of periodic Reeb orbits for a large class of tight
contact structures on closed 3-manifolds, notably the Stein fillable
structures, based on a fundamental theorem of Cliff Taubes on symplectic
4-manifolds.


title:
On the Lusternik-Schnirelmann category of symplectic manifolds and the
  Arnold conjecture

abstract:
  We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed
symplectic manifold $(M, \omega)$ equals the dimension $dim(M)$ provided that
the symplectic cohomology class vanishes on the image of the Hurewicz
homomorphism. This holds, in particular, when $\pi_2(M)=0$. The Arnold
conjecture asserts that the number of fixed points of a Hamiltonian
symplectomorphism of $M$ is greater than or equal to the number of critical
points of some function on $M$. A modified form of the conjecture, replacing
the latter quantity (via Lusternik-Schnirelmann theory) by $cup(M) + 1$, has
been proved recently by various authors using techniques of Floer. The first
author has also recently shown that the original form of the conjecture holds
when $cat(M) =dim(M)$. Thus, this paper completes the proof of the original
Arnold conjecture for closed symplectic manifolds with, for example,
$\pi_2(M)=0$.


title:
On analytical applications of stable homotopy (the Arnold conjecture,
  critical points)

abstract:
  We prove the Arnold conjecture for closed symplectic manifolds with
$\pi_2(M)=0$ and $\cat M=\dim M$. Furthermore, we prove an analog of the
Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity''
property.


title: Tree-like curves and their inflection points

abstract:
  We give a criterion when a planar tree-like curve, i.e. a generic immersed
plane curve each double point of which cuts it into two disjoint parts, can be
send by a diffeomorphism of the plane onto a curve with no inflection points.
We also present some upper and lower bounds for the minimal number of
inflection points on such curves unremovable by diffeomorphisms of the plane in
terms of their combinatorics.


title:
On the noncommutative residue for pseudodifferential operators with
  log-polyhomogeneous symbols

abstract:
  We study various aspects of the noncommutative residue for an algebra of
pseudodifferential operators whose symbols have an expansion $a\sim
\sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,\xi)=\sum_{l=0}^k a_{m-j,l}(x,\xi)
\log^l|\xi|,$ where $a_{m-j,l}$ is homogeneous in $\xi$ of degree $m-j$. We
will explain why this algebra of pseudodifferential operators is natural.
  For a pseudodifferential operator in this class, $A$, and a classical
elliptic pseudodifferential operator, $P$, we show that the generalized
zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole
complex plane, however possibly with higher order poles.
  Our algebra of operators has a bigrading given by the order and the highest
log-power occuring in the symbol expansion. We construct "higher"
noncommutative residue functionals on the subspaces given by the log-grading.
However, in contrast to the classical case we prove that the whole algebra does
not admit any nontrivial traces.
  Finally we show that the analogue of the Kontsevich-Vishik trace also exists
on our algebra. Our method also provides an alternative approach to the
Kontsevich-Vishik trace.


title: Contact Topology and Hydrodynamics

abstract:
  We draw connections between the field of contact topology and the study of
Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We
demonstrate an equivalence between Reeb fields (vector fields which preserve a
transverse nowhere-integrable plane field) up to scaling and rotational
Beltrami fields on three-manifolds. Thus, we characterise Beltrami fields in a
metric-independant manner.
  This correspondence yields a hydrodynamical reformulation of the Weinstein
Conjecture, whose recent solution by Hofer (in several cases) implies the
existence of closed orbits for all $C^\infty$ rotational Beltrami flows on
$S^3$. This is the key step for a positive solution to the hydrodynamical
Seifert Conjecture: all $C^\omega$ steady state flows of a perfect
incompressible fluid on $S^3$ possess closed flowlines. In the case of Euler
flows on $T^3$, we give general conditions for closed flowlines derived from
the homotopy data of the normal bundle to the flow.


title: On the Kobayashi-Royden pseudonorm for almost complex manifolds

abstract:
  In this paper we define Kobayashi-Royden pseudonorm for almost complex
manifolds. Its basic properties known from the complex analysis are preserved
in the nonintegrable case as well. We prove that the pseudodistance induced by
this pseudonorm coincides with the Kobayashi pseudodistance defined for the
almost complex case earlier. We also consider a geometric application for
moduli spaces of pseudoholomorphic curves.


title: Invariants of Velocities, and Higher Order Grassmann Bundles

abstract:
  An $(r,n)$-velocity is an $r$-jet with source at $0 \in \R^n$, and target in
a manifold $Y$. An $(r,n)$-velocity is said to be regular, if it has a
representative which is an immersion at $0 \in \R^{n}$. The manifold
$T^{r}_{n}Y$ of $(r,n)$-velocities as well as its open, $L^{r}_{n}$-invariant,
dense submanifold $\Imm T^{r}_{n}Y$ of regular $(r,n)$-velocities, are endowed
with a natural action of the differential group $L^{r}_{n}$ of invertible
$r$-jets with source and target $0 \in \R^{n}$. In this paper, we describe all
continuous, $L^{r}_{n}$-invariant, real-valued functions on $T^{r}_{n}Y$ and
$\Imm T^{r}_{n}Y$. We find local bases of $L^{r}_{n}$-invariants on $\Imm
T^{r}_{n}Y$ in an explicit, recurrent form. To this purpose, higher order
Grassmann bundles are considered as the corresponding quotients $P^{r}_{n}Y =
\Imm T^{r}_{n}Y/L^{r}_{n}$, and their basic properties are studied. We show
that nontrivial $L^{r}_{n}$-invariants on $\Imm T^{r}_{n}Y$ cannot be
continuously extended onto $T^{r}_{n}Y$.


title: On the geometry and topology of manifolds of positive bi-Ricci curvature

abstract:
  We introduce some new curvature quantities such as conformal Ricci curvature
and bi-Ricci curvature and extend the classical Myers theorem under these new
curvature conditions. Moreover, we are able to obtain the Myers type theorem
for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature.
Some topological applications are discussed. We also give examples of manifolds
of positive bi-Ricci curvature and prove that the connect sum of manifolds of
positive bi-Ricci curvature admits metrics of positive bi-Ricci curvature.


title: The structure of stable minimal hypersurfaces in R^n

abstract:
  We provide a new topological obstruction for complete stable minimal
hypersurfaces in R^n. For $n\geq 4$, we prove that any complete orientable
stable hypersurfaces in R^n has only one end. This follows from a more general
analytic theorem we prove in the paper.


title: Ricci curvature, minimal surfaces and sphere theorems

abstract:
  Using an analogue of Myers' theorem for minimal surfaces and three
dimensional topology, we prove the diameter sphere theorem for Ricci curvature
in dimension three and a corresponding eigenvalue pinching theorem. This
settles these two open problems for closed 3 manifolds with positive Ricci
curvature since they are both false in dimensions greater than three.


title: Kahler manifolds with large isometry group

abstract:
  We investigate compact Kahler manifolds, which are acted on by a semisimple
compact Lie group G of isometries with one hypersurface orbit. In case of
ordinary action and projectable complex structure, we set up a one to one
correspondence between such manifolds and abstract models. The Ricci tensor is
then computed and we fully characterize the Kahler-Einstein manifolds with an
ordinary cohomogeneity one action and projectable complex structure.


title:
Symplectic and contact Lie algebras with an application to Monge-Ampere
  equations

abstract:
  In this paper we consider symplectic and contact Lie algebras. We define
contactization and symplectization procedures and describe its main properties.
We also give classification of such algebras in dimensions 3 and 4. The
classification in dimension~4 is closely connected with normal forms of
nondegenerate elliptic equations of the second order on two-dimensional
surfaces with transitive symmetry group in first jets. We point out this
connection and discuss normal forms.


title: Higher-Order Lagrangian Formalism on Grassmann Manifolds

abstract:
  The Lagrangian formalism on a arbitrary non-fibrating manifold is considered.
The kinematical description of this generic situation is based on the concept
of (higher-order) Grassmann manifolds which is the factorization of the regular
velocity manifold to the action of the differential group. Here we introduce in
this context the basic concepts of the Lagrangian formalism as Lagrange,
Euler-Lagrange and Helmholtz-Sonin forms. These objects come in pairs, namely
we have homogeneous objects (defined on the regular velocity manifold) and
non-homogeneous objects (defined on the Grassmann manifold). We will establish
the connection between the homogeneous objects and their non-homogeneous
counterparts. As a result we will conclude that the generic expressions for a
variationally trivial Lagrangian and for a locally variational differential
equation remain the same as in the fibrating case.


title: An inverse problem of the flux for minimal surfaces

abstract:
  For a complete minimal surface in the Euclidean 3-space, the so-called flux
vector corresponds to each end. The flux vectors are balanced, i.e., the sum of
those over all ends are zero. Consider the following inverse problem: For each
balanced n vectors, find an n-end catenoid which attains given vectors as flux.
Here, an n-end catenoid is a complete minimal surface of genus 0 with ends
asymptotic to the catenoids. In this paper, the problem is reduced to solving
algebraic equation. Using this reduction, it is shown that, when n=4, the
inverse problem for 4-end catenoid has solutions for almost all balanced 4
vectors. Further obstructions for n-end catenoids with parallel flux vectors
are also discussed.


title:
General existence of minimal surfaces of genus zero with catenoidal ends
  and prescribed flux

abstract:
  For each end of complete minimal surface in the Euclidean 3-space, the flux
vector is defined. It is well-known that the sum of the flux vector over all
ends are zero. Consider the following inverse problem: For each balanced
n-vectors, find an n-end catenoid which realizes these vectors as flux. Here,
an n-end catenoid is a complete minimal surface of genus zero with ends
asymptotic to the catenoids. In this paper, we show that the inverse problem
can be solved for almost all balanced n vectors for arbitrary n, which is
grater than 4. The assumption "almost all" is needed because nonexistence is
known for special balanced vectors. We remark that in the case of n=4, the same
result has been obtained by the authors (dg-ga/9709006). And the case n=3 is
treated by Lopez and Berbanel.


title:
Irreducible constant mean curvature 1 surfaces in hyperbolic space with
  positive genus

abstract:
  In this work we give a method for constructing a one-parameter family of
complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space
that correspond to a given complete minimal surface with finite total curvature
in Euclidean 3-space. We show that this one-parameter family of surfaces with
the same symmetry properties exists for all given minimal surfaces satisfying
certain conditions. The surfaces we construct in this paper are irreducible,
and in the process of showing this, we also prove some results about the
reducibility of surfaces.
  Furthermore, in the case that the surfaces are of genus 0, we are able to
make some estimates on the range of the parameter for the one-parameter family.


title: Euler class and free generation

abstract:
  the main theorem gives a sufficient condition for a n elements of SL(2,R) to
generate a free group.The idea behind it is to use a nonorientable version of
the Dehn-Wolpert-Goldman twist and to sew it with the original representation
of a free group to get representation of the closed surfase group and then to
apply Goldman' theorem.


title:
continious cohomology of groups of volume-preserving and symplectic
  diffepmorphisms, measurable transfer and higher asymtotic cycles

abstract:
  I construct the real counterparts (which I call Borel-Bott classes) of the
R/Z classes constructed in "Characteristic classes in symplectic topology", to
appear, in the cohomology of volume-preserving and symplectomorhisms of a
compact (symplectic) manifold.I show that, for the symplectic action of the
mapping class group in the moduli space of stable vector bundles over a Riemann
surface, the restriction of the first constructed class from the
symplectomorphism group gives a generator for the second (bounded) cohomology
of the mapping class group.


title:
A Bernstein theorem for complete spacelike constant mean curvature
  hypersurfaces in Minkowski space

abstract:
  We obtain a gradient estimate for the Gauss maps from complete spacelike
constant mean curvature hypersurfaces in Minkowski space into the hyperbolic
space. As applications, we prove a Bernstein theorem which says that if the
image of the Gauss map is bounded from one side, then the spacelike constant
mean curvature hypersurface must be linear. This result extends the previous
theorems obtained by B. Palmer and Y.L. Xin where they assume that the image of
the Gauss map is bounded. We also proved a Bernstein theorem for spacelike
complete surfaces with parallel mean curvature vector in four-dimensional
spaces.


title:
From a Relativistic Phenomenology of Anyons to a Model of Unification of
  Forces via the Spencer Theory of Lie Structures

abstract:
  Starting from a relativistic phenomenology of anyons in crystals, we discuss
the concept of relativistic interaction and the need to unify electromagnetism
and gravitation within the Spencer cohomology of Lie equations. Then, from the
sophisticated non-linear Spencer complex of the Poincar\'e and conformal Lie
pseudogroups, we build up a non-linear relative complex assigned to a gauge
sequence for electromagnetic and gravitational potentials and fields. Then,
using a conformally equivariant Lagrangian density, we deduce, first, the two
first steps of its corresponding Janet complex and second, the dual relative
linear complex. We conclude by giving suggestions for higher unification with
the weak and strong interactions and interpretations of the Lagrangian density
as a thermodynamical function and quantum wave-function.


title:
Deformations of representations of fundamental groups of open Kaehler
  manifolds

abstract:
  Given a compact Kaehler manifold, we consider the complement U of a divisor
with normal crossings. We study the variety of unitary representations of the
fundamental group of U with certain restrictions related to the divisor. We
show that the possible singularities of this variety as well as of the
corresponding moduli space of irreducible representations are quadratic. In the
course of our proof we exhibit a differential graded Lie algebra of which
reflects our deformation problem.


title:
The First Eigenvalue of the Dirac Operator on Quaternionic Kaehler
  Manifolds

abstract:
  In a previous paper we proved a lower bound for the spectrum of the Dirac
operator on quaternionic Kaehler manifolds. In the present article we show that
the only manifolds in the limit case, i.e. the only manifolds where the lower
bound is attained as an eigenvalue, are the quaternionic projective spaces. We
use the equivalent formulation in terms of the quaternionic Killing equation
and show that a nontrivial solution defines a parallel spinor on the associated
hyperkaehler manifold.


title: Pseudo-Kaehler Quantization on Flag Manifolds

abstract:
  A unified approach to geometric, symbol and deformation quantizations on a
generalized flag manifold endowed with an invariant pseudo-Kaehler structure is
proposed. The Hilbert space of states is realized via the Bott-Borel-Weil
theorem in the sheaf cohomology of the geometric quantization line bundle. The
corresponding deformation quantization is a quantization with separation of
variables. In particular cases we arrive at Berezin's quantization via
covariant and contravariant symbols.


title: Transports along maps in fibre bundles

abstract:
  Generalized are the investigated in other works of the author transports
along paths in fibre bundles to transports along arbitrary maps in them. Their
structure and some properties are studied. Special attention is paid to the
linear case and the case when the map's domain is a Cartesian product of two
sets. Also considered are the consistency with bundle morphisms and a number of
special cases.


title:
Linear transports along paths in vector bundles V. Properties of
  curvature and torsion

abstract:
  A geometric interpretation of curvature and torsion of linear transports
along paths is presented. A number of (Bianchi type) identities satisfied by
these quantities are derived. The obtained results contain as special cases the
corresponding classical ones concerning curvature and torsion of linear
connections.


title:
Investigation and application of the dressing action on surfaces of
  constant mean curvature

abstract:
  We investigate the dressing action on surfaces of constant mean curvature
(CMC surfaces) in Euclidean space. In particluar, we show that for CMC surfaces
with umbilics the isotropy group under dressing is always trivial. This result
is applied to the investigation of CMC surfaces whose metric is invariant under
a group of automorphisms of the parameter domain.


title:
Supplement on Curved flats in the space of point pairs and Isothermic
  surfaces: A Quaternionic Calculus

abstract:
  A quaternionic calculus for surface pairs in the conformal 4-sphere is
elaborated. This calculus is then used to discuss the relation between curved
flats in the symmetric space of point pairs and Darboux and Christoffel pairs
of isothermic surfaces. A new viewpoint on relations between surfaces of
constant mean curvature in certain space forms is presented --- in particular,
a new form of Bryant's Weierstrass type representation for surfaces of constant
mean curvature 1 in hyperbolic 3-space is given.


title:
Foliation by Constant Mean Curvature Spheres on Asymptotically Flat
  Manifolds

abstract:
  In this paper, the existence and uniqueness of foliations by constant mean
curvature spheres on asymptotically flat manifolds of nonzero ADM mass in all
dimensions were established. (A similar result in the case of positive mass was
obtained independently by G. Huisken and S. T. Yau, see the introduction of
this paper and their paper in Inv. Math.)


title: Corrigenda to: Introduction to Symplectic Topology

abstract:
  This contains a list of (mostly very minor) corrections to the book
Introduction to Symplectic Topology, Clarendon Press, Oxford, (1995), together
with rewritten versions of two lemmas and some additional comments.


title: PU(2) monopoles and relations between four-manifold invariants

abstract:
  Using quantum field-theoretic arguments, Witten has established a relation
between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds.
In this survey article, we describe the program to prove this relation using a
moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson
moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1)
monopoles. We provide an overview of some of our transversality and Uhlenbeck
compactness results for PU(2) monopoles, along with some of our calculations of
Donaldson invariants in terms of Seiberg-Witten invariants. We give a brief
outline of issues concerning the gluing theory, focussing on some of the
analytical difficulties that are particular to PU(2) monopoles, and its
application to the PU(2) monopole program to prove the relation between
Donaldson and Seiberg-Witten invariants.


title: Integral Invariants of 3-Manifolds

abstract:
  This note describes an invariant of rational homology 3-spheres in terms of
configuration space integrals which in some sense lies between the invariants
of Axelrod and Singer and those of Kontsevich.


title: Residual Amenability and the Approximation of L^2-invariants

abstract:
  We generalize Luck's Theorem to show that the L^2-Betti numbers of a
residually amenable covering space are the limit of the L^2-Betti numbers of a
sequence of amenable covering spaces. We show that any residually amenable
covering space of a finite simplicial complex is of determinant class, and that
the L^2 torsion is a homotopy invariant for such spaces. We give examples of
residually amenable groups, including the Baumslag-Solitar groups.


title: Non-Symplectic Geometry of First Order Time-Dependent Mechanics

abstract:
  The usual formulation of time-dependent mechanics implies a given splitting
$Y=R\times M$ of an event space $Y$. This splitting, however, is broken by any
time-dependent transformation, including transformations between inertial
frames. The goal is the frame-covariant formulation of time-dependent mechanics
on a bundle $Y\to R$ whose fibration $Y\to M$ is not fixed. Its phase space is
the vertical cotangent bundle $V^*Y$ provided with the canonical 3-form and the
corresponding canonical Poisson structure. An event space of relativistic
mechanics is a manifold $Y$ whose fibration $Y\to R$ is not fixed.


title:
Isospectral deformations of closed Riemannian manifolds with different
  scalar curvature

abstract:
  We construct the first examples of continuous families of isospectral
Riemannian metrics that are not locally isometric on closed manifolds, more
precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and
$S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally
homogeneous; in particular, the scalar curvature of each metric is nonconstant.
For some of the deformations, the maximum scalar curvature changes during the
deformation.


title:
The differential equation $\Delta u = 8\pi - 8\pi h\exp {u}$ on a
  compact Riemann surface

abstract:
  Let $M$ be a compact Riemann surface, $h(x)$ a positive smooth function on
$M$. In this paper, we consider the functional $$J(u)={1/2}\int
\sb{M}|\bigtriangledown u|\sp 2 + 8\pi \int\sb{M}u -8\pi \log\int\sb{M}h\exp
{u}$$. We give a sufficient condition under which $J$ achieves its minimum.


title: An analysis of the two-vortex case in the Chern-Simons Higgs model

abstract:
  Extending work of Caffarelli-Yang and Tarantello, we present a variational
existence proof for two-vortex solutions of the periodic Chern-Simons Higgs
model and analyze the asymptotic behavior of these solutions as the parameter
coupling the gauge field with the scalar field tends to 0.


title: Lie algebroids and Poisson-Nijenhuis structures

abstract:
  Poisson-NIjenhuis structures for an arbitrary Lie agebroid are defined and
studied by means of tangent lifts of tensor fields.


title: Deficiencies of Lattice Subgroups of Lie Groups

abstract:
  Let L be a lattice in a connected Lie group. We show that besides a few
exceptional cases, the deficiency of L is nonpositive.


title: Cocycles, symplectic structures and intersection

abstract:
  We investigate the cross ratio for closed negatively curved manifolds. As one
of several applications, we obtain that for two such homotopy equivalent
manifolds M and N, the following is true : If M and N have the same marked
length spectrum and if the Anosov splitting for M is C^1 then M and N have the
same volume.


title: Mathematical Background of Formalism of Operator Manifold

abstract:
  The analysis of mathematical structure of the method of operator manifold
guides our discussion. The latter is a still wider generalization of the method
of secondary quantization with appropriate expansion over the geometric
objects. The nature of operator manifold provides its elements with both
quantum field and geometry aspects, a detailed study of which is a subject of
present paper. It yields a quantization of geometry differing in principle from
all earlier suggested schemes.


title: The beta function of a knot

abstract:
  We introduce the beta function of a knot in euclidean three-space. This is a
meromorphic function of a complex variable which we prove admits a Bernstein
type functional equation. We determine the first residues.


title:
Uhlenbeck compactness and transversality for the moduli space of PU(2)
  monopoles

abstract:
  This research announcement gives a brief report of the main results in our
paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and
transversality" (Journal of Differential Geometry, to appear). We describe the
existence of perturbations for the PU(2) monopole equations, yielding both
useful transversality properties and an Uhlenbeck compactification for this
perturbed moduli space.


title:
Tangent and cotangent lifts and graded Lie algebras associated with Lie
  algebroids

abstract:
  Generalized Schouten, Froelicher-Nijenhuis and Froelicher-Richardson brackets
are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie
algebroids are introduced and discussed and the behaviour of the related graded
Lie brackets under these lifts is studied. In the case of the canonical Lie
algebroid on the tangent bundle, a new common generalization of the
Froelicher-Nijenhuis and the symmetric Schouten brackets, as well as embeddings
of the Nijenhuis-Richardson and the Froelicher-Nijenhuis bracket into the
Schouten bracket, are obtained.


title: Double vector bundles and duality

abstract:
  The most important examples of a double vector bundle are provided by
iterated tangent and cotangent functors: TTM, TT^*M, T^*TM, and T^*T^*M. We
introduce the notions of the dual double vector bundle and the dual double
vector bundle morphism. Theorems on canonical isomorphisms are formulated and
proved. Several examples are given.


title:
A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and
  applications

abstract:
  Using the Bryant representation, we define a new flux on homology classes of
CMC-1 surfaces in hyperbolic 3-space, satisfying a balancing formula which is
useful to show nonexistencd of certain kinds of complete CMC-1 surfaces.


title: Intersection Cohomology of Symplectic Quotients

abstract:
  Paper withdrawn - lemma 4.1 was false. Quite a lot of changes need to be
made!


title: Topological rigidity of Hamiltonian loops and quantum homology

abstract:
  This paper studies the question of when a loop $\phi$ in the group
Symp$(M,\omega)$ of symplectomorphisms of a symplectic manifold $(M,\omega)$ is
isotopic to a loop that is generated by a time-dependent Hamiltonian function.
(Loops with this property are said to be Hamiltonian.) Our main result is that
Hamiltonian loops are rigid in the following sense: if $\phi$ is Hamiltonian
with respect to $\omega$, and if $\phi'$ is a small perturbation of $\phi$ that
preserves another symplectic form $\omega'$, then $\phi'$ is Hamiltonian with
respect to $\omega'$. This allows us to get some new information on the
structure of the flux group, i.e. the image of $\pi_1(Symp(M,\omega))$ under
the flux homomorphism. We give a complete proof of our result for some
manifolds, and sketch the proof in general. The argument uses methods developed
by Seidel for studying properties of Hamiltonian loops via the quantum homology
of $M$.


title: Three Cocycles on $\Diff(S^1)$ Generalizing the Schwarzian Derivative

abstract:
  The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on
the M\"obius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules
of linear differential operators on $S^1$ is calculated. We introduce three
non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the
Schwarzian derivative.


title: Cap-prodcut structures on the Fintushel-Stern spectral sequence

abstract:
  We show that there is a well-defined cap-product structure on the
Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product
structure on the ${\BZ}_8$-graded instanton Floer homology. The cap-product
structure provides an essentially new property of the instanton Floer homology,
from a topological point of view, which multiplies a finite dimensional
cohomology class by an infinite dimensional homology class (Floer cycles) to
get another infinite dimensional homology class.


title: The Weierstrass representation of closed surfaces in $R^3$

abstract:
  In the present paper which a sequel to dg-ga/9511005 and dg-ga//9610013 a
global Weierstrass representation of an arbitrary closed oriented surface of
genus $\geq 1$ in the the three-space is constructed. The Weierstrass spectrum
of a torus immersed into $R^3$ is introduced and finite-zone planes as well as
finite-zone solutions to the modified Novikov-Veselov equations are
constructed.


title: Topology of the set of smooth solutions to the Liouville equation

abstract:
  We prove that the space of smooth initial data and the set of smooth
solutions of the Liouville equation are homeomorphic.


title: Analytic Torsion for Quaternionic manifolds and related topics

abstract:
  In this paper we show that the Ray-Singer complex analytic torsion is trivial
for even dimensional Calabi-Yau manifolds.
  Then we define the quaternionic analytic torsion for quaternionic manifolds
and prove that they are metric independent.
  In dimension four, the quaternionic analytic torsion equals to the self-dual
analytic torsion. For higher dimensional manifolds, the self-dual analytic
torsion is a conformal invariant.


title: Existence results for mean field equations

abstract:
  Let $\Omega$ be an annulus. We prove that the mean field equation
$-\Delta\psi=\frac{e\sp{-\beta\psi}}{\int\sb{\Omega}e\sp{-\beta\psi}} $ admits
a solution with zero boundary for $\beta\in (-16\pi,-8\pi)$. This is a
supercritical case for the Moser-Trudinger inequality.


title:
Nonexistence of Finite-dimensional Quantizations of a Noncompact
  Symplectic Manifold

abstract:
  We prove that there is no faithful finite-dimensional representation by
skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact
symplectic manifold M. Consequently there exists no finite-dimensional
quantization of any Lie subalgebra of the Poisson algebra C^\infty(M)
containing B.


title: Topology of solutions of the Liouville equation

abstract:
  Suggestions concerning the generalization of the geometric quantization to
the case of nonlinear field theories are given. Results for the Liouville field
theory are presented.


title: Introduction to the Tangent Grupoid

abstract:
  We present some plausible definitions for the tangent grupoid of a manifold
M, as well as some of the known applications of the structure. This is a kind
of introductory note.


title: Generalized Symplectic Manifolds

abstract:
  This paper uses a generalization of symplectic geometry, known as
$n$-symplectic geometry and developed by Norris, to find observables on
three-dimensional manifolds. It will be seen that for the cases considered, the
$n$-symplectic observables are derivable from the symplectic observables of
$C^2$. The quantization of these observables, as well as those on the frame
bundle of $R^n$, is also examined.


title:
Surfaces in Lie sphere geometry and the stationary Davey-Stewartson
  hierarchy

abstract:
  We introduce two basic invariant forms which define generic surface in
3-space uniquely up to Lie sphere equivalence. Two particularly interesting
classes of surfaces associated with these invariants are considered, namely,
the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For
diagonally-cyclidic surfaces we derive the stationary modified Veselov-Novikov
equation, whose role in the theory of these surfaces is similar to that of
Calapso's equation in the theory of isothermic surfaces. Since Calapso's
equation itself turns out to be related to the stationary Davey-Stewartson
equation, these results shed some new light on differential geometry of the
stationary Davey-Stewartson hierarchy. Diagonally-cyclidic surfaces are the
natural Lie sphere analogs of the isothermally-asymptotic surfaces in
projective differential geometry for which we also derive the stationary
modified Veselov-Novikov equation with the different real reduction. Parallels
between invariants of surfaces in Lie sphere geometry and reciprocal invariants
of hydrodynamic type systems are drawn in the conclusion.


title: Ring structure of the Floer Cohomology of $\Sigma \times S^1$

abstract:
  We give a presentation for the Floer cohomology ring $HF^*(\Sigma \times
S^1)$, where $\Sigma$ is a Riemann surface of genus bigger than one, which
coincides with the conjectural presentation for the quantum cohomology ring of
the moduli space of flat SO(3)-connections of odd degree over $\Sigma$. We
study the spectrum of the action of the homology of $\Sigma$ on the Floer
cohomology and prove a physical assumption made by Vafa et al.


title: Classical lifting processes and multiplicative vector fields

abstract:
  We extend the calculus of multiplicative vector fields and differential forms
and their intrinsic derivatives from Lie groups to Lie groupoids; this
generalization turns out to include also the classical process of complete
lifting from arbitrary manifolds to tangent and cotangent bundles. Using this
calculus we give a new description of the Lie bialgebroid structure associated
with a Poisson groupoid.


title: Frobenius Manifolds from Yang-Mills Instantons

abstract:
  We present an elementary self-contained account of semisimple Frobenius
manifolds in three dimensions, and exhibit a new family of explicit examples.
These examples are constructed from Yang-Mills instantons with a certain
symmetry.


title:
PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and
  Transversality

abstract:
  We prove the existence of perturbations for the PU(2) monopole equations,
yielding transversality on the complement of the anti-self-dual or reducible
solutions, and the existence of an Uhlenbeck compactification for the moduli
space of solutions to these perturbed PU(2) monopole equations. In December
1994, V. Pidstrigach and A. Tyurin and then others proposed a method to prove
Witten's conjecture concerning the relation between the Donaldson and
Seiberg-Witten invariants of smooth four-manifolds. Their proposal uses a
moduli space of solutions to the PU(2) monopole equations, which are a natural
generalization of the U(1) monopole equations of Seiberg and Witten and the
equation for anti-self-dual SO(3) connections, to construct a cobordism between
links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the
moduli space of anti-self-dual connections, which appear as singularities in
this larger moduli space. A basic requirement of this cobordism technique is
the existence of an Uhlenbeck compactification for the moduli space of PU(2)
monopoles and of generic-parameter transversality results for all the moduli
spaces of PU(2) monopoles which appear in this compactification, on the
complement of the anti-self-dual and U(1) solutions.


title: On the variation of the Poisson structures of certain moduli spaces

abstract:
  Given a Lie group G whose Lie algebra is endowed with a nondegenerate
invariant symmetric bilinear form, we construct a Poisson algebra of continuous
functions on a certain open subspace R of the space of representations in G of
the fundamental group of a compact connected orientable topological surface
with finitely many boundary circles; when G is compact and connected, R may be
taken dense in the space of all representations. The space R contains spaces of
representations where the values of those generators of the fundamental group
which correspond to the boundary circles are constrained to lie in fixed
conjugacy classes and, on these representation spaces, the Poisson algebra
restricts to stratified symplectic Poisson algebras constructed elsewhere
earlier. Hence the Poisson algebra on R gives a description of the variation of
the stratified symplectic Poisson structures on the smaller representation
spaces as the chosen conjugacy classes move.


title: Mutation and Gauge Theory I: Yang-Mills Invariants

abstract:
  Mutation is an operation on 3-manifolds containing an embedded surface of
genus 2. It is defined by cutting along the surface and regluing using the
`hyperelliptic' involution, and is known to preserve many 3-manifold
invariants. I show that mutation of a homology 3-sphere preserves its
(instanton) Floer homology, and that a related operation on 4-manifolds
preserves the Donaldson invariants. A companion article (in preparation) will
treat invariants based on the Seiberg-Witten equations.


title: Equivariant K-Theory of Simply Connected Lie Groups

abstract:
  We compute the equivariant $K$-theory $K_G^*(G)$ for a simply connected Lie
group $G$ (acting on itself by conjugation). We prove that $K_G^*(G)$ is
isomorphic to the algebra of Grothendieck differentials on the representation
ring. We also study a special example of a non-simply connected Lie group $G$,
namely PSU(3), and compute the corresponding equivariant $K$-theory.


title: Upper bounds for regularized determinants

abstract:
  Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If
we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon
smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one
defines its regularized determinant $det'(\Delta)$. We conjectured elsewhere
that, when $h$ varies, this determinant $det'(\Delta)$ remains bounded from
above.
  In this paper we prove this in two special cases. The first case is when $X$
is a Riemann surface, $E$ is a line bundle and $dim(H^0 (X,E)) + dim(H^1 (X,E))
\leq 2$, and the second case is when $X$ is the projective line, $E$ is a line
bundle, and all metrics under consideration are invariant under rotation around
a fixed axis.


title: The moduli space of special Lagrangian submanifolds

abstract:
  This paper considers the natural geometric structure on the moduli space of
deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau
manifold. From the work of McLean this is a smooth manifold with a natural
$L^2$ metric. It is shown that the metric is induced from a local
  Lagrangian immersion into the product of cohomology groups $H^1(L)\times
H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry
discussed by Strominger, Yau and Zaslow is given in terms of the classical
Legendre transform.


title: On the localization theorem in equivariant cohomology

abstract:
  We present a simple proof of a precise version of the localization theorem in
equivariant cohomology. As an application, we describe the cohomology algebra
of any compact symplectic variety with a multiplicity-free action of a compact
Lie group. This applies in particular to smooth, projective spherical
varieties.


title:
Critical-exponent Sobolev norms and the slice theorem for the quotient
  space of connections

abstract:
  The use of certain critical-exponent Sobolev norms is an important feature of
methods employed by Taubes to solve the anti-self-dual and similar non-linear
elliptic partial differential equations. Indeed, the estimates one can obtain
using these critical-exponent norms appear to be the best possible when one
needs to bound the norm of a Green's operator for a Laplacian, depending on a
connection varying in a non-compact family, in terms of minimal data such as
the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature
of the connection. Following Taubes, we describe a collection of
critical-exponent Sobolev norms and general Green's operator estimates
depending only on first positive eigenvalues or the L^2 norm of the
connection's curvature. Such estimates are particularly useful in the gluing
construction of solutions to non-linear partial differential equations
depending on a degenerating parameter, such as the approximate, reference
solution in the anti-self-dual or PU(2) monopole equations. We apply them here
to prove an optimal slice theorem for the quotient space of connections. The
result is optimal in the sense that if a point [A] in the quotient space is
known to be just L^2_1-close enough to a reference point [A_0], then the
connection A can be placed in Coulomb gauge relative to the connection A_0,
with all constants depending at most on the first positive eigenvalue of the
covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In
this paper we shall for simplicity only consider connections over
four-dimensional manifolds, but the methods and results can adapted to
manifolds of arbitrary dimension to prove slice theorems which apply when the
reference connection is allowed to degenerate.


title:
New examples of conservative systems on S^2 possessing an integral cubic
  in momenta

abstract:
  It has been proved that on 2-dimensional orientable compact manifolds of
genus $g>1$ there is no integrable geodesic flow with an integral polynomial in
momenta. There is a conjecture that all integrable geodesic flows on $T^2$
possess an integral quadratic in momenta. All geodesic flows on $S^2$ and $T^2$
possessing integrals linear and quadratic in momenta have been described by
Kolokol'tsov, Babenko and Nekhoroshev.
  So far there has been known only one example of conservative system on $S^2$
possessing an integral cubic in momenta: the case of Goryachev-Chaplygin in the
dynamics of a rigid body. The aim of this paper is to propose a new
one-parameter family of examples of complete integrable conservative systems on
$S^2$ possessing an integral cubic in momenta. We show that our family does not
include the case of Goryachev-Chaplygin.


title:
Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding
  Seifert fibrations

abstract:
  We corrected a few errors in the previous submission. These do not affect any
of the topological conclusions of the earlier version. We have also included a
few observations about the Casson invariant of the Brieskorn homology spheres.


title: Morse theory of harmonic forms

abstract:
  We consider the problem of whether it is possible to improve the Novikov
inequalities for closed 1-forms, or any other inequalities of a similar nature,
if we assume, additionally, that the given 1-form is harmonic with respect to
some Riemannian metric. We show that, under suitable assumptions, it is
impossible. We use, in an essential way, a theorem of E.Calabi characterizing
1-forms which are harmonic with respect to some metric. We also study some
interesting examples illustrating our results.


title:
Classification of stationary compact homogeneous special pseudo K\"ahler
  manifolds of semisimple groups

abstract:
  The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical
K\"ahler metric on its Kuranishi moduli space, known as the Weil-Petersson
metric. Similarly, special pseudo K\"ahler manifolds correspond to certain
(abstract) variations of Hodge structure which generalize the above example. We
give the classification of homogeneous special pseudo K\"ahler manifolds of
semisimple groups with compact stabilizer.


title: Prescribing Mean Curvature: Existence and Uniqueness Problems

abstract:
  This paper presents results on the extent to which mean curvature data can be
used to determine a surface in space or its shape. The emphasis is on Bonnet's
problem: classify and study the surface immersions in $\R^3$ whose shape is not
uniquely determined by the first fundamental form and the mean curvature
function. The properties of immersions with umbilics and global rigidity
results for closed surfaces are presented in the first part of this paper. The
second part of the paper outlines an existence theory for conformal immersions
based on Dirac spinors along with its immediate applications to Bonnet's
problem. The presented existence paradigm provides insight into the topology of
the moduli space of Bonnet immersions of a closed surface, and reveals a
parallel between Bonnet's problem and Pauli's exclusion principle.


title: Continuous families of isospectral metrics on simply connected manifolds

abstract:
  We construct continuous families of Riemannian metrics on certain simply
connected manifolds with the property that the resulting Riemannian manifolds
are pairwise isospectral for the Laplace operator acting on functions. These
are the first examples of simply connected Riemannian manifolds without
boundary which are isospectral, but not isometric. For example, we construct
continuous isospectral families of metrics on the product of spheres S^4\times
S^3\times S^3. The metrics considered are not locally homogeneous. For a big
class of such families, the set of critical values of the scalar curvature
function changes during the deformation. Moreover, the manifolds are in general
not isospectral for the Laplace operator acting on 1-forms.


title: Equivariant torsion and G-CW-complexes

abstract:
  For any compact Lie group G we discuss the relation of the equivariant
Reidemeister and analytic torsion of G-manifolds with their G-CW structures.


title: Totally geodesic orbits of isometries

abstract:
  We study cohomogeneity one Riemannian manifolds and we establish some simple
criterium to test when a singular orbit is totally geodesic. As an application,
we classify compact, positively curved Riemannian manifolds which are acted on
isometrically by a non semisimple Lie group with an hypersurface orbit.


title:
Complex Generated by Variational Derivatives. Lagrangian Formalism of
  Infinite Order and a Generalized Stokes' Formula

abstract:
  We prove that an analog of the exterior differential acts on the space of
arbitrary Lagrangians of multidimensional paths on any manifold or
supermanifold, thus making this space into a cochain complex. An analog of the
Stokes' formula holds. The construction and the proofs are purely geometrical,
in terms of the variation of corresponding actions.


title: Kahler geometry of toric varieties and extremal metrics

abstract:
  Recently Guillemin gave an explicit combinatorial way of constructing "toric"
Kahler metrics on (symplectic) toric varieties, using only data on the moment
polytope. In this paper, differential geometric properties of these metrics are
investigated using Guillemin's construction. In particular, a nice
combinatorial formula for the scalar curvature is given, and the Euler-Lagrange
condition for such "toric" metrics being extremal (in the sense of Calabi) is
derived. A construction, due to Calabi, of a 1-parameter family of extremal
metrics of non-constant scalar curvature is recast very simply and explicitly.
Finally, a curious combinatorial formula for convex polytopes, that follows
from the relation between the total integral of the scalar curvature and the
wedge product of the first Chern class with a suitable power of the Kahler
class, is presented.


title:
Isometry groups and geodesic foliations of Lorentz manifolds. Part I:
  Foundations of Lorentz dynamics

abstract:
  This is the first part of a series on non-compact groups acting isometrically
on compact Lorentz manifolds. This subject was recently investigated by many
authors. In the present part we investigate the dynamics of affine, and
especially Lorentz transformations. In particular we show how this is related
to geodesic foliations. The existence of geodesic foliations was (very
succinctly) mentioned for the first time by D'Ambra and Gromov, who suggested
that this may help in the classification of compact Lorentz manifolds with
non-compact isometry groups.
  In the Part II of the series, a partial classification of compact Lorentz
manifolds with non-compact isometry group will be achieved with the aid of
geometrical tools along with the dynamical ones presented here.


title: Scattering theory and deformations of asymptotically hyperbolic metrics

abstract:
  For an asymptotically hyperbolic metric on the interior of a compact manifold
with boundary, we prove that the resolvent and scattering operators are
continuous functions of the metric in the appropriate topologies.


title: Discrete surfaces of constant mean curvature via dressing

abstract:
  We translate a classification scheme for periodic CMC surfaces developed by
J. Dorfmeister and the author to discrete CMC surfaces in the sense of A.
Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC
surfaces to arrive at a classification for periodic discrete CMC surfaces.


title: Relative torsion for representations in finite type Hilbert modules

abstract:
  For a closed manifold equipped with a Riemannian metric, a triangulation, a
representation of its fundamental group on an Hilbert module of finite type
(over of finite von Neumann algebra), and a Hermitian structure on the flat
bundle associated to the representation, one defines a numerical invariant, the
relative torsion. The relative torsion is a positive real number and unlike the
analytic torsion or the Reidemeister torsion, which are defined only when the
pair manifold- representation is of determinant class, is always defined. When
the pair is of determinant class the relative torsionis equal to the quotient
of the analytic and the Reidemeister torsion.We calculate the relative torsion.


title:
Poisson harmonic forms, Kostant harmonic forms, and the
  $S^1$-equivariant cohomology of $K/T$

abstract:
  We characterize the harmonic forms on a flag manifold $K/T$ defined by
Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson
harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This
enables us to give Poisson geometrical proofs of many of the special properties
of these harmonic forms. In particular, we construct explicit representatives
for the Schubert basis of the $S^1$-equivariant cohomology of $K/T$, where the
$S^1$-action is defined by $\rho$. Using a simple argument in equivariant
cohomology, we recover the connection between the Kostant harmonic forms and
the Schubert calculus on $K/T$ that was found by Kostant and Kumar in 1986. We
also show that the Kostant harmonic forms are limits of the more familiar Hodge
harmonic forms with respect to a family of Hermitian metrics.


title:
Isometry groups and geodesic foliations of Lorentz manifolds. Part II:
  Geometry of analytic Lorentz manifolds with large isometry groups

abstract:
  This is Part II of a series on noncompact isometry groups of
  Lorentz manifolds. We have introduced in Part I, a compactification of these
isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a
``trivial '' compactification. Here we show a geometric rigidity of
non-bi-polarized Lorentz manifolds; that is, they are (at least locally) warped
products of constant curvature Lorentz manifolds by Riemannian manifolds.


title: Higher analytic torsion and cohomology of diffeomorphism groups

abstract:
  We consider a closed odd-dimensional oriented manifold $M$ together with an
acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle
with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a
natural flat connection and a vertical Riemannian metric. The higher analytic
torsion form of Bismut/Lott associated to the situation is invariant with
respect to the connected component of the identity of the diffeomorphism group
of $M$. Using that the space of Riemannian metrics is contractible we define
continuous cohomology classes of the diffeomorphism group and its Lie algebra.
For the circle we compute this classes in degree 2 and show that the group
cohomology class is non-trivial, while the Lie algebra cohomology class
vanishes.


title: Positively curved 7-dimensional manifolds

abstract:
  We deal with seven dimensional compact Riemannian manifolds of positive
curvature which admit a cohomogeneity one action by a compact Lie group G. We
prove that the manifold is diffeomorphic to a sphere if the dimension of the
semisimple part of G is bigger than 6.


title: The Positive Fundamental Group of Sp(2) and Sp(4)

abstract:
  In this paper, we examine the homotopy classes of positive loops in Sp(2) and
Sp(4). We show that two positive loops are homotopic if and only if they are
homotopic through positive loops.


title:
Generalized Symmetries of Partial Differential Equations and Quasiexact
  Solvability

abstract:
  Using the adjoint action of the infinitesimal translations (with respect to
some (in)dependant variables) on specific finite-dimensional subspaces of the
space of generalized symmetries of some system of partial differential
equations, we explicitly determine the dependance of coefficients of
generalized symmetries from these subspaces on the above-mentioned variables.
We establish the connection of our results with the theory of quasiexactly
solvable models. Some generalizations of the approach proposed also are
discussed.

categories: dg-ga math-ph math.DG math.MP nlin.SI solv-int


title:
PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's
  conjecture in low degrees

abstract:
  In this article we complete the proof---for a broad class of
four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten
series coincide, at least through terms of degree less than or equal to c-2,
where c is a linear combination of the Euler characteristic and signature of
the four-manifold. This article is a revision of sections 4--7 of an earlier
version, while a revision of sections 1--3 of that earlier version now appear
in a separate companion article (math.DG/0007190). Here, we use our
computations of Chern classes for the virtual normal bundles for the
Seiberg-Witten strata from the companion article (math.DG/0007190), a
comparison of all the orientations, and the PU(2) monopole cobordism to compute
pairings with the links of level-zero Seiberg-Witten moduli subspaces of the
moduli space of PU(2) monopoles. These calculations then allow us to compute
low-degree Donaldson invariants in terms of Seiberg-Witten invariants and
provide a partial verification of Witten's conjecture.


title: The decomposition and classification of radiant affine 3-manifolds

abstract:
  An affine manifold is a manifold with torsion-free flat affine connection. A
geometric topologist's definition of an affine manifold is a manifold with an
atlas of charts to the affine space with affine transition functions; a radiant
affine manifold is an affine manifold with holonomy consisting of affine
transformations fixing a common fixed point. We decompose an orientable closed
radiant affine 3-manifold into radiant 2-convex affine manifolds and radiant
concave affine 3-manifolds along mutually disjoint totally geodesic tori or
Klein bottles using the convex and concave decomposition of real projective
$n$-manifolds developed earlier. Then we decompose a 2-convex radiant affine
manifold into convex radiant affine manifolds and concave-cone affine
manifolds. To do this, we will obtain certain nice geometric objects in the
Kuiper completion of holonomy cover. The equivariance and local finiteness
property of the collection of such objects will show that their union covers a
compact submanifold of codimension zero, the complement of which is convex.
Finally, using the results of Barbot, we will show that a closed radiant affine
3-manifold admits a total cross section, confirming a conjecture of Carri\`ere,
and hence every radiant affine 3-manifold is homeomorphic to a Seifert fibered
space with trivial Euler number, or a virtual bundle over a circle with fiber
homeomorphic to a torus.


title: The functional definition of geodesics

abstract:
  In this paper a functional definition of geodesics is introduced which allows
to generalize the notion of a geodesic from smooth to topological manifolds. It
is shown that in the smooth case the new definition coincides with the
classical definition of geodesics of a linear connection. If the smoothness is
not required, it is shown by an example that the new definition includes
geodesics of non-linear, homogeneous connections. Moreover, an example of
generalized geodesics which does not arise from any connection is presented
here.


title: Projectivity of moment map quotients

abstract:
  Let G be a complex reductive group and K a maximal compact subgroup. If X is
a smooth projective G-variety, with a fixed (not necessarily integral)
K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero
fiber of the corresponding moment map. It is well known that the quotient M/K
is a complex space in a natural way. We prove that M/K is a projective variety.
In particular, it follows that semistability with respect to a moment map is
equivalent to semistability in the sense of Mumford.


title: Notes on affine isometric actions of discrete groups

abstract:
  Consider a lattice $\Gamma$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$,
$SL_2(\Q_p)$. We discuss actions of $\Gamma$ by affine isometric
transformations of Hilbert spaces. We show that for irreducible affine
isometric action of $G$ its restriction to $\Gamma$ is irreducible. We prove
the existence of canonical irreducible affine isometric actions of $\Gamma$
associated to actions of $\Gamma$ on $\R$- trees. Using such actions we
construct irreducible representations of semigroup of probabilistic measures on
$\Gamma$ and construct the series of representations of the groups of
diffeomorphisms of Riemann surfaces enumerated by the points of Thurston
compactification of Teichm\"uller (Teichmuller) space.


title: Weyl groups of Hamiltonian manifolds, I

abstract:
  We consider a connected compact Lie group K acting on a symplectic manifold M
such that a moment map m exists. A pull-back function via m Poisson commutes
with all K-invariants. Guillemin-Sternberg raised the problem to find a
converse. In this paper, we solve this problem by determining the Poisson
commutant of the algebra of K-invariants. It is completely controlled by the
image of m and a certain subquotient W_M of the Weyl group of K. The group W_M
is also a reflection group and forms a symplectic analogue of the little Weyl
group of a symmetric space. The proof rests ultimately on techniques from
algebraic geometry. In fact, a major part of the paper is of independent
interest: it establishes connectivity and reducedness properties of the fibers
of the (complex algebraic) moment map of a complex cotangent bundle.


title: Quadratic Differentials, Quaternionic Forms, and Surfaces

abstract:
  Global isothermic immersions are defined and studied with the aid of a
connection between quadratic differentials and immersions. The applications are
two problems stemming from the fundamental question: how much data is needed to
identify a surface immersion (Christoffel's problem) or its shape (Bonnet's
problem). A short complete solution of Christoffel's problem, including closed
surfaces, is given. It is shown that every immersion of an oriented closed
surface (genus $\neq$ 1) is uniquely determined up to similitude by its
conformal class and the tangent planes map. A classification of all generic
Bonnet surfaces follows from a series of papers by Bonnet, Cartan, and Chern.
The existence of a new class of Bonnet surfaces is shown here. The
understanding of this class is necessary in order to study the rigidity of
closed surfaces.


title: Integration of Lie bialgebroids

abstract:
  We prove that under certain mild assumptions a Lie bialgebroid integrates to
a Poisson groupoid. This includes, in particular, a new proof of the existence
of local symplectic groupoids for any Poisson manifold, a theorem of Karasev
and of Weinstein.


title: Double Lie algebroids and second-order geometry, II

abstract:
  We complete the construction of the double Lie algebroid of a double Lie
groupoid begun in the first paper of this title. We show that the Lie algebroid
structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie
groupoid structure; in the case of a double groupoid this prolonged structure
for either LA--groupoid is canonically isomorphic to the Lie algebroid
structure associated with the other; this extends many canonical isomorphisms
associated with iterated tangent and cotangent structures. We also show that
the cotangent of a double Lie groupoid is a symplectic double groupoid, and
that the side groupoids of any symplectic double groupoid are Poisson groupoids
in duality. Thus any double Lie groupoid gives rise to a dual pair of Poisson
groupoids.


title: Parallel Connections Over Symmetric Spaces

abstract:
  Let M be a simply connected Riemannian symmetric space, with at most one flat
direction. We show that every Riemannian (or unitary) vector bundle with
parallel curvature over M is an associated vector bundle of a canonical
principal bundle, with the connection inherited from the principal bundle. The
problem of finding Riemannian (or unitary) vector bundles with parallel
curvature then reduces to finding representations of the structure group of the
canonical principal bundle.


title: Simple Type is Not a Boundary Phenomenon

abstract:
  This is an expository article, explaining recent work by D. Groisser and
myself [GS] on the extent to which the boundary region of moduli space
contributes to the ``simple type'' condition of Donaldson theory. The
presentation is intended to complement [GS], presenting the essential ideas
rather than the analytical details. It is shown that the boundary region of
moduli space contributes 6/64 of the homology required for simple type,
regardless of the topology or geometry of the underlying 4-manifold. The simple
type condition thus reduces to a statement about the interior of moduli space,
namely that the interior of the (k+1)st ASD moduli space, intersected with two
representatives of (4 times) the point class, be homologous to 58 copies of the
(k)-th moduli space. This is peculiar, since the only known embeddings of the
(k)-th moduli space into the (k+1)st involve Taubes patching, and the image of
such an embedding lies entirely in the boundary region.


title: Simple Type and the Boundary of Moduli Space

abstract:
  We measure, in two distinct ways, the extent to which the boundary region of
moduli space contributes to the ``simple type'' condition of Donaldson theory.
Using a geometric representative of \mu(pt), the boundary region of moduli
space contributes 6/64 of the homology required for simple type, regardless of
the topology or geometry of the underlying 4-manifold. The simple type
condition thus reduces to the interior of the k+1st ASD moduli space,
intersected with two representatives of (4 times) the point class, being
homologous to 58 copies of the k-th moduli space. This is peculiar, since the
only known embeddings of the k-th moduli space into the k+1st involve Taubes
gluing, and the images of such embeddings lie entirely in the boundary region.
  When using de Rham representatives of mu(pt), the boundary region contributes
1/8 of what is needed for simple type, again regardless of the topology or
geometry of the underlying 4-manifold. The difference between this and the
geometric representative answer is surprising but not contradictory, as the
contribution of a fixed region to the Donaldson invariants is geometric, not
topological.


title: Jacobi vector fields of integrable geodesic flows

abstract:
  We show that an invariant surface allows to construct the Jacobi vector field
along a geodesic and construct the formula for the normal component of the
Jacobi field. If a geodesic is the transversal intersection of two invariant
surfaces (such situation we have, for example, if the geodesic is hyperbolic),
then we can construct a fundamental solution of the the Jacobi-Hill equation.
This is done for quadratically integrable geodesic flows.


title:
New families of conservative systems on $S^2$ possessing an integral of
  fourth degree in momenta

abstract:
  There is a well-known example of integrable conservative system on $S^2$, the
case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of
fourth degree in momenta. Goryachev proposed a one-parameter family of examples
of conservative systems on $S^2$ possessing an integral of fourth degree in
momenta which includes the case of Kovalevskaya. In this paper we proposed new
examples of conservative systems on $S^2$ possessing an integral of fourth
degree in momenta.


title: The minimum principle from a Hamiltonian point of view

abstract:
  Let G be a complex Lie group, G_R a real form of G and X a G_R-stable domain
of holomorphy in a complex G-manifold. If there is a G_R-invariant strictly
plurisubharmonic function on X which has certain exhaustion properties, then we
show that the extended domain G.X is also a domain of holomorphy. As an
application we give a proof of the extended future tube conjecture. This is the
assertion that G.X is a domain of holomorphy in the case where X is the N-fold
product of the tube domain in C^4 over the positive light cone in R^4 and G is
the connected complex Lorentz group acting diagonally.


title: Weinstein Conjecture and GW Invariants

abstract:
  In this paper, we establish a general relationship between the nonvanishing
of GW invariants with the existence of the closed orbits of a Hamiltonian
system. As an application, we completely solved the stabilized Weinstein
conjecture.


title: On the Spinor Representation of Surfaces in Euclidean 3-Space

abstract:
  The aim of the present paper is to clarify the relationship between
immersions of surfaces and solutions of the inhomogeneous Dirac equation. The
main idea leading to the description of a surface M^2 by a spinor field is the
observation that the restriction to M^2 of any parallel spinor phi on R^3 is
(with respect to the inner geometry of M^2) a non-trivial spinor field on M^2
of constant length which is a solution of the inhomogeneous Dirac equation and
vice versa.


title: A Baire Category Approach to the Bang-Bang Property

abstract:
  Aim of this paper is to develop a new technique, based on the Baire category
theorem, in order to establish the closure of reachable sets and the existence
of optimal trajectories for control systems, without the usual convexity
assumptions. The bang-bang property is proved for a new class of ``concave"
multifunctions, characterized by the existence of suitable linear selections.
The proofs rely on Lyapunov's theorem in connection with a Baire category
argument.


title: Integral representation for a class of $C^1$-convex functionals

abstract:
  In view of the applications to the asymptotic analysis of a family of
obstacle problems, we consider a class of convex local functionals $F(u,A)$,
defined for all functions $u$ in a suitable vector valued Sobolev space and for
all open sets $A$ in ${\bf R}^n$. Sufficient conditions are given in order to
obtain an integral representation of the form $F(u,A)=\int_A f(x,u(x))\,d\mu +
\nu(A)$, where $\mu$ and $\nu$ are Borel measures and $f$ is convex in the
second variable.


title: Nonstandard hulls of Banach-Lie groups and algebras

abstract:
  We propose a new construction of Banach-Lie groups and algebras relying on
nonstandard analysis. A major standard application is the Local Theorem which
to certain extent reduces the problem of associating a Lie group to a given
banach-Lie algebra to a similar problem for finitely generated Lie subalgebras.
We discuss possible applications, e.g., to gauge theories.


title:
Universal arrows to forgetful functors from categories of topological
  algebra

abstract:
  We survey the present trends in theory of universal arrows to forgetful
functors from various categories of topological algebra and functional analysis
to categories of topology and topological algebra. Among them are free
topological groups, free locally convex spaces, free Banach-Lie algebras, and
much more. An accent is put on relationship of those constructions with other
areas of mathematics and their possible applications. A number of open problems
is discussed; some of them belong to universal arrow theory, and others may
hopefully become amenable to methods of this theory.


title: Extremal Selections of Multifunctions Generating a Continuous Flow

abstract:
  Let $F:[0,T]\times\R^n\mapsto 2^{\R^n}$ be a continuous multifunction with
compact, not necessarily convex values. In this paper, we prove that, if $F$
satisfies the following Lipschitz Selection Property: \begin{itemize}
\item[{(LSP)}] {\sl For every $t,x$, every $y\in \overline{co} F(t,x)$ and
$\varepsilon>0$, there exists a Lipschitz selection $\phi$ of $\overline{co}F$,
defined on a neighborhood of $(t,x)$, with $|\phi(t,x)-y|<\varepsilon$.}
\end{itemize} then there exists a measurable selection $f$ of $ext F$\ such
that, for every $x_0$, the Cauchy problem $$ \dot x(t)=f(t,x(t)),\qquad\qquad
x(0)=x_0 $$ has a unique Caratheodory solution, depending continuously on
$x_0$. We remark that every Lipschitz multifunction with compact values
satisfies (LSP). Another interesting class, for which (LSP) holds, consists of
those continuous multifunctions $F$ whose values are compact and have convex
closure with nonempty interior.


title:
Circle Actions on C*-Algebras, Partial Automorphisms and a Generalized
  Pimsner-Voiculescu Exact Sequence

abstract:
  We introduce a method to study C*-algebras possessing an action of the circle
group, from the point of view of its internal structure and its K-theory. Under
relatively mild conditions our structure Theorem shows that any C*-algebra,
where an action of the circle is given, arises as the result of a construction
that generalizes crossed products by the group of integers.
  Such a generalized crossed product construction is carried out for any
partial automorphism of a C*-algebra, where by a partial automorphism we mean
an isomorphism between two ideals of the given algebra.
  Our second main result is an extension to crossed products by partial
automorphisms, of the celebrated Pimsner-Voiculescu exact sequence for
K-groups.
  The representation theory of the algebra arising from our construction is
shown to parallel the representation theory for C*-dynamical systems. In
particular, we generalize several of the main results relating to regular and
covariant representations of crossed products.


title: C*-algebras and numerical linear algebra

abstract:
  We consider problems associated with the computation of spectra of
self-adjoint operators in terms of the eigenvalue distributions of their n x n
sections. Under rather general circumstances, we show how these eigenvalues
accumulate near points of the essential spectrum of the given operator, and we
prove that their averages converge to a measure concentrated precisely on the
essential spectrum.
  In the primary cases of interest, namely the discretized Hamiltonians of
one-dimensional quantum systems, this limiting measure is associated with a
tracial state on a certain simple C*-algebra. These results have led us to
conclude that one must view this kind of numerical analysis in the context of
C*-algebras.


title:
Improper filtrations for C*-algebras: spectra of unilateral tridiagonal
  operators

abstract:
  We extend the results of our previous paper "C*-algebras and numerical linear
algebra" to cover the case of "unilateral" sections. This situation bears a
close resemblance to the case of Toeplitz operators on Hardy spaces, in spite
of the fact that the operators here are far from Toeplitz operators. In
particular, there is a short exact sequence 0 --> K --> A --> B --> 0 whose
properties are essential to the problem of computing the spectra of self
adjoint operators.


title: Approximately Finite C*-Algebras and Partial Automorphisms

abstract:
  We prove that every AF-algebra is isomorphic to a crossed product of a
commutative AF-algebra by a partial automorphism. The case of UHF-algebras is
treated in detail.


title: A Short Proof that $M_{n}(A)$ is local if $A$ is local and Fr\'echet

abstract:
  We give a short and very general proof of the fact that the property of a
dense Fr\'echet subalgebra of a Banach algebra being local, or closed under the
holomorphic functional calculus in the Banach algebra, is preserved by
tensoring with the $n\times n$ matrix algebra of the complex numbers.


title:
A Non-Spectral Dense Banach Subalgebra of the Irrational Rotation
  Algebra

abstract:
  We give an example of a dense, simple, unital Banach subalgebra $A$ of the
irrational rotation C*-algebra $B$, such that $A$ is not a spectral subalgebra
of $B$. This answers a question posed in T.W. Palmer's paper [1].


title: Representable K-theory of Smooth Crossed Products by R and Z

abstract:
  We show that the Thom isomorphism and the Pimsner-Voiculescu exact sequence
both hold for smooth crossed products of Frechet algebras by R and Z
respectively. We also obtain the same results for L^{1}-crossed products of
Banach algebras by R and Z.


title:
An analytic structure emerging in presence of infinitely many odd
  coordinates

abstract:
  This is a contribution to the program of featuring even geometry as a
``collective effect in infinite-dimensional odd geometry,'' as suggested by
Manin. We show that the (Gel'fand) spectrum of the locally convex nonstandard
hull (in the sense of Luxemburg) of a grassmannian algebra with infinitely many
odd generators contains a nontrivial analytic part.