title: Selective Sampling for Example-based Word Sense Disambiguation

abstract:
  This paper proposes an efficient example sampling method for example-based
word sense disambiguation systems. To construct a database of practical size, a
considerable overhead for manual sense disambiguation (overhead for
supervision) is required. In addition, the time complexity of searching a
large-sized database poses a considerable problem (overhead for search). To
counter these problems, our method selectively samples a smaller-sized
effective subset from a given example set for use in word sense disambiguation.
Our method is characterized by the reliance on the notion of training utility:
the degree to which each example is informative for future example sampling
when used for the training of the system. The system progressively collects
examples by selecting those with greatest utility. The paper reports the
effectiveness of our method through experiments on about one thousand
sentences. Compared to experiments with other example sampling methods, our
method reduced both the overhead for supervision and the overhead for search,
without the degeneration of the performance of the system.


title: Efficient and Extensible Algorithms for Multi Query Optimization

abstract:
  Complex queries are becoming commonplace, with the growing use of decision
support systems. These complex queries often have a lot of common
sub-expressions, either within a single query, or across multiple such queries
run as a batch. Multi-query optimization aims at exploiting common
sub-expressions to reduce evaluation cost. Multi-query optimization has
hither-to been viewed as impractical, since earlier algorithms were exhaustive,
and explore a doubly exponential search space.
  In this paper we demonstrate that multi-query optimization using heuristics
is practical, and provides significant benefits. We propose three cost-based
heuristic algorithms: Volcano-SH and Volcano-RU, which are based on simple
modifications to the Volcano search strategy, and a greedy heuristic. Our
greedy heuristic incorporates novel optimizations that improve efficiency
greatly. Our algorithms are designed to be easily added to existing optimizers.
We present a performance study comparing the algorithms, using workloads
consisting of queries from the TPC-D benchmark. The study shows that our
algorithms provide significant benefits over traditional optimization, at a
very acceptable overhead in optimization time.


title:
Practical experiments with regular approximation of context-free
  languages

abstract:
  Several methods are discussed that construct a finite automaton given a
context-free grammar, including both methods that lead to subsets and those
that lead to supersets of the original context-free language. Some of these
methods of regular approximation are new, and some others are presented here in
a more refined form with respect to existing literature. Practical experiments
with the different methods of regular approximation are performed for
spoken-language input: hypotheses from a speech recognizer are filtered through
a finite automaton.


title: A System of Interaction and Structure

abstract:
  This paper introduces a logical system, called BV, which extends
multiplicative linear logic by a non-commutative self-dual logical operator.
This extension is particularly challenging for the sequent calculus, and so far
it is not achieved therein. It becomes very natural in a new formalism, called
the calculus of structures, which is the main contribution of this work.
Structures are formulae submitted to certain equational laws typical of
sequents. The calculus of structures is obtained by generalising the sequent
calculus in such a way that a new top-down symmetry of derivations is observed,
and it employs inference rules that rewrite inside structures at any depth.
These properties, in addition to allow the design of BV, yield a modular proof
of cut elimination.


title: On Reconfiguring Tree Linkages: Trees can Lock

abstract:
  It has recently been shown that any simple (i.e. nonintersecting) polygonal
chain in the plane can be reconfigured to lie on a straight line, and any
simple polygon can be reconfigured to be convex. This result cannot be extended
to tree linkages: we show that there are trees with two simple configurations
that are not connected by a motion that preserves simplicity throughout the
motion. Indeed, we prove that an $N$-link tree can have $2^{\Omega(N)}$
equivalence classes of configurations.


title: Semantics of Programming Languages: A Tool-Oriented Approach

abstract:
  By paying more attention to semantics-based tool generation, programming
language semantics can significantly increase its impact. Ultimately, this may
lead to ``Language Design Assistants'' incorporating substantial amounts of
semantic knowledge.


title:
Numeration systems on a regular language: Arithmetic operations,
  Recognizability and Formal power series

abstract:
  Generalizations of numeration systems in which N is recognizable by a finite
automaton are obtained by describing a lexicographically ordered infinite
regular language L over a finite alphabet A. For these systems, we obtain a
characterization of recognizable sets of integers in terms of rational formal
series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is
the complement of a polynomial language), then multiplication by an integer k
preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the
cardinality of A) for some integer t. Finally, we obtain sufficient conditions
for the notions of recognizability and U-recognizability to be equivalent,
where U is some positional numeration system related to a sequence of integers.


title: Subgraph Isomorphism in Planar Graphs and Related Problems

abstract:
  We solve the subgraph isomorphism problem in planar graphs in linear time,
for any pattern of constant size. Our results are based on a technique of
partitioning the planar graph into pieces of small tree-width, and applying
dynamic programming within each piece. The same methods can be used to solve
other planar graph problems including connectivity, diameter, girth, induced
subgraph isomorphism, and shortest paths.


title: Graph Ramsey games

abstract:
  We consider combinatorial avoidance and achievement games based on graph
Ramsey theory: The players take turns in coloring still uncolored edges of a
graph G, each player being assigned a distinct color, choosing one edge per
move. In avoidance games, completing a monochromatic subgraph isomorphic to
another graph A leads to immediate defeat or is forbidden and the first player
that cannot move loses. In the avoidance+ variants, both players are free to
choose more than one edge per move. In achievement games, the first player that
completes a monochromatic subgraph isomorphic to A wins. Erdos & Selfridge
(1973) were the first to identify some tractable subcases of these games,
followed by a large number of further studies. We complete these investigations
by settling the complexity of all unrestricted cases: We prove that general
graph Ramsey avoidance, avoidance+, and achievement games and several variants
thereof are PSPACE-complete. We ultra-strongly solve some nontrivial instances
of graph Ramsey avoidance games that are based on symmetric binary Ramsey
numbers and provide strong evidence that all other cases based on symmetric
binary Ramsey numbers are effectively intractable.
  Keywords: combinatorial games, graph Ramsey theory, Ramsey game,
PSPACE-completeness, complexity, edge coloring, winning strategy, achievement
game, avoidance game, the game of Sim, Polya's enumeration formula,
probabilistic counting, machine learning, heuristics, Java applet


title: What Next? A Dozen Information-Technology Research Goals

abstract:
  Charles Babbage's vision of computing has largely been realized. We are on
the verge of realizing Vannevar Bush's Memex. But, we are some distance from
passing the Turing Test. These three visions and their associated problems have
provided long-range research goals for many of us. For example, the scalability
problem has motivated me for several decades. This talk defines a set of
fundamental research problems that broaden the Babbage, Bush, and Turing
visions. They extend Babbage's computational goal to include highly-secure,
highly-available, self-programming, self-managing, and self-replicating
systems. They extend Bush's Memex vision to include a system that automatically
organizes, indexes, digests, evaluates, and summarizes information (as well as
a human might). Another group of problems extends Turing's vision of
intelligent machines to include prosthetic vision, speech, hearing, and other
senses. Each problem is simply stated and each is orthogonal from the others,
though they share some common core technologies


title: Question Answering System Using Syntactic Information

abstract:
  Question answering task is now being done in TREC8 using English documents.
We examined question answering task in Japanese sentences. Our method selects
the answer by matching the question sentence with knowledge-based data written
in natural language. We use syntactic information to obtain highly accurate
answers.


title:
One-Way Functions in Worst-Case Cryptography: Algebraic and Security
  Properties

abstract:
  We survey recent developments in the study of (worst-case) one-way functions
having strong algebraic and security properties. According to [RS93], this line
of research was initiated in 1984 by Rivest and Sherman who designed two-party
secret-key agreement protocols that use strongly noninvertible, total,
associative one-way functions as their key building blocks. If commutativity is
added as an ingredient, these protocols can be used by more than two parties,
as noted by Rabi and Sherman [RS93] who also developed digital signature
protocols that are based on such enhanced one-way functions.
  Until recently, it was an open question whether one-way functions having the
algebraic and security properties that these protocols require could be created
from any given one-way function. Recently, Hemaspaandra and Rothe [HR99]
resolved this open issue in the affirmative, by showing that one-way functions
exist if and only if strong, total, commutative, associative one-way functions
exist.
  We discuss this result, and the work of Rabi, Rivest, and Sherman, and recent
work of Homan [Hom99] that makes progress on related issues.


title:
On quantum and classical space-bounded processes with algebraic
  transition amplitudes

abstract:
  We define a class of stochastic processes based on evolutions and
measurements of quantum systems, and consider the complexity of predicting
their long-term behavior. It is shown that a very general class of decision
problems regarding these stochastic processes can be efficiently solved
classically in the space-bounded case. The following corollaries are implied by
our main result: (1) Any space O(s) uniform family of quantum circuits acting
on s qubits and consisting of unitary gates and measurement gates defined in a
typical way by matrices of algebraic numbers can be simulated by an unbounded
error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing
machine, and hence by space O(s) uniform classical (deterministic) circuits of
depth O(s^2) and size 2^(O(s)). The quantum circuits are not required to
operate with bounded error and may have depth exponential in s. (2) Any
(unbounded error) quantum Turing machine running in space s, having arbitrary
algebraic transition amplitudes, allowing unrestricted measurements during its
computation, and having no restrictions on running time can be simulated by an
unbounded error space O(s) ordinary probabilistic Turing machine, and hence
deterministically in space O(s^2).


title: Two-way finite automata with quantum and classical states

abstract:
  We introduce 2-way finite automata with quantum and classical states
(2qcfa's). This is a variant on the 2-way quantum finite automata (2qfa) model
which may be simpler to implement than unrestricted 2qfa's; the internal state
of a 2qcfa may include a quantum part that may be in a (mixed) quantum state,
but the tape head position is required to be classical.
  We show two languages for which 2qcfa's are better than classical 2-way
automata. First, 2qcfa's can recognize palindromes, a language that cannot be
recognized by 2-way deterministic or probabilistic finite automata. Second, in
polynomial time 2qcfa's can recognize {a^n b^n | n>=0}, a language that can be
recognized classically by a 2-way probabilistic automaton but only in
exponential time.


title: The Sources of Certainty in Computation and Formal Systems

abstract:
  In his Discourse on the Method of Rightly Conducting the Reason, and Seeking
Truth in the Sciences, Rene Descartes sought ``clear and certain knowledge of
all that is useful in life.'' Almost three centuries later, in ``The
foundations of mathematics,'' David Hilbert tried to ``recast mathematical
definitions and inferences in such a way that they are unshakable.'' Hilbert's
program relied explicitly on formal systems (equivalently, computational
systems) to provide certainty in mathematics. The concepts of computation and
formal system were not defined in his time, but Descartes' method may be
understood as seeking certainty in essentially the same way.
  In this article, I explain formal systems as concrete artifacts, and
investigate the way in which they provide a high level of certainty---arguably
the highest level achievable by rational discourse. The rich understanding of
formal systems achieved by mathematical logic and computer science in this
century illuminates the nature of programs, such as Descartes' and Hilbert's,
that seek certainty through rigorous analysis.


title: One-Level Prosodic Morphology

abstract:
  Recent developments in theoretical linguistics have lead to a widespread
acceptance of constraint-based analyses of prosodic morphology phenomena such
as truncation, infixation, floating morphemes and reduplication. Of these,
reduplication is particularly challenging for state-of-the-art computational
morphology, since it involves copying of some part of a phonological string. In
this paper I argue for certain extensions to the one-level model of phonology
and morphology (Bird & Ellison 1994) to cover the computational aspects of
prosodic morphology using finite-state methods. In a nutshell, enriched lexical
representations provide additional automaton arcs to repeat or skip sounds and
also to allow insertion of additional material. A kind of resource
consciousness is introduced to control this additional freedom, distinguishing
between producer and consumer arcs. The non-finite-state copying aspect of
reduplication is mapped to automata intersection, itself a non-finite-state
operation. Bounded local optimization prunes certain automaton arcs that fail
to contribute to linguistic optimisation criteria. The paper then presents
implemented case studies of Ulwa construct state infixation, German
hypocoristic truncation and Tagalog over-applying reduplication that illustrate
the expressive power of this approach, before its merits and limitations are
discussed and possible extensions are sketched. I conclude that the one-level
approach to prosodic morphology presents an attractive way of extending
finite-state techniques to difficult phenomena that hitherto resisted elegant
computational analyses.


title: Cox's Theorem Revisited

abstract:
  The assumptions needed to prove Cox's Theorem are discussed and examined.
Various sets of assumptions under which a Cox-style theorem can be proved are
provided, although all are rather strong and, arguably, not natural.


title: PushPush is NP-hard in 3D

abstract:
  We prove that a particular pushing-blocks puzzle is intractable in 3D. The
puzzle, inspired by the game PushPush, consists of unit square blocks on an
integer lattice. An agent may push blocks (but never pull them) in attempting
to move between given start and goal positions. In the PushPush version, the
agent can only push one block at a time, and moreover, each block, when pushed,
slides the maximal extent of its free range. We prove this version is NP-hard
in 3D by reduction from SAT. The corresponding problem in 2D remains open.


title: The Complexity of Poor Man's Logic

abstract:
  Motivated by description logics, we investigate what happens to the
complexity of modal satisfiability problems if we only allow formulas built
from literals, $\wedge$, $\Diamond$, and $\Box$. Previously, the only known
result was that the complexity of the satisfiability problem for K dropped from
PSPACE-complete to coNP-complete (Schmidt-Schauss and Smolka, 1991 and Donini
et al., 1992). In this paper we show that not all modal logics behave like K.
In particular, we show that the complexity of the satisfiability problem with
respect to frames in which each world has at least one successor drops from
PSPACE-complete to P, but that in contrast the satisfiability problem with
respect to the class of frames in which each world has at most two successors
remains PSPACE-complete. As a corollary of the latter result, we also solve the
open problem from Donini et al.'s complexity classification of description
logics (Donini et al., 1997). In the last section, we classify the complexity
of the satisfiability problem for K for all other restrictions on the set of
operators.


title:
The phase transition in random Horn satisfiability and its algorithmic
  implications

abstract:
  Let c>0 be a constant, and $\Phi$ be a random Horn formula with n variables
and $m=c\cdot 2^{n}$ clauses, chosen uniformly at random (with repetition) from
the set of all nonempty Horn clauses in the given variables. By analyzing \PUR,
a natural implementation of positive unit resolution, we show that
$\lim_{n\goesto \infty} \PR ({$\Phi$ is satisfiable})= 1-F(e^{-c})$, where
$F(x)=(1-x)(1-x^2)(1-x^4)(1-x^8)... $. Our method also yields as a byproduct an
average-case analysis of this algorithm.


title: A Geometric Model for Information Retrieval Systems

abstract:
  This decade has seen a great deal of progress in the development of
information retrieval systems. Unfortunately, we still lack a systematic
understanding of the behavior of the systems and their relationship with
documents. In this paper we present a completely new approach towards the
understanding of the information retrieval systems. Recently, it has been
observed that retrieval systems in TREC 6 show some remarkable patterns in
retrieving relevant documents. Based on the TREC 6 observations, we introduce a
geometric linear model of information retrieval systems. We then apply the
model to predict the number of relevant documents by the retrieval systems. The
model is also scalable to a much larger data set. Although the model is
developed based on the TREC 6 routing test data, I believe it can be readily
applicable to other information retrieval systems. In Appendix, we explained a
simple and efficient way of making a better system from the existing systems.


title:
Resolution of Indirect Anaphora in Japanese Sentences Using Examples 'X
  no Y (Y of X)'

abstract:
  A noun phrase can indirectly refer to an entity that has already been
mentioned. For example, ``I went into an old house last night. The roof was
leaking badly and ...'' indicates that ``the roof'' is associated with `` an
old house}'', which was mentioned in the previous sentence. This kind of
reference (indirect anaphora) has not been studied well in natural language
processing, but is important for coherence resolution, language understanding,
and machine translation. In order to analyze indirect anaphora, we need a case
frame dictionary for nouns that contains knowledge of the relationships between
two nouns but no such dictionary presently exists. Therefore, we are forced to
use examples of ``X no Y'' (Y of X) and a verb case frame dictionary instead.
We tried estimating indirect anaphora using this information and obtained a
recall rate of 63% and a precision rate of 68% on test sentences. This
indicates that the information of ``X no Y'' is useful to a certain extent when
we cannot make use of a noun case frame dictionary. We estimated the results
that would be given by a noun case frame dictionary, and obtained recall and
precision rates of 71% and 82% respectively. Finally, we proposed a way to
construct a noun case frame dictionary by using examples of ``X no Y.''


title:
Pronoun Resolution in Japanese Sentences Using Surface Expressions and
  Examples

abstract:
  In this paper, we present a method of estimating referents of demonstrative
pronouns, personal pronouns, and zero pronouns in Japanese sentences using
examples, surface expressions, topics and foci. Unlike conventional work which
was semantic markers for semantic constraints, we used examples for semantic
constraints and showed in our experiments that examples are as useful as
semantic markers. We also propose many new methods for estimating referents of
pronouns. For example, we use the form ``X of Y'' for estimating referents of
demonstrative adjectives. In addition to our new methods, we used many
conventional methods. As a result, experiments using these methods obtained a
precision rate of 87% in estimating referents of demonstrative pronouns,
personal pronouns, and zero pronouns for training sentences, and obtained a
precision rate of 78% for test sentences.


title: An Estimate of Referent of Noun Phrases in Japanese Sentences

abstract:
  In machine translation and man-machine dialogue, it is important to clarify
referents of noun phrases. We present a method for determining the referents of
noun phrases in Japanese sentences by using the referential properties,
modifiers, and possessors of noun phrases. Since the Japanese language has no
articles, it is difficult to decide whether a noun phrase has an antecedent or
not. We had previously estimated the referential properties of noun phrases
that correspond to articles by using clue words in the sentences. By using
these referential properties, our system determined the referents of noun
phrases in Japanese sentences. Furthermore we used the modifiers and possessors
of noun phrases in determining the referents of noun phrases. As a result, on
training sentences we obtained a precision rate of 82% and a recall rate of 85%
in the determination of the referents of noun phrases that have antecedents. On
test sentences, we obtained a precision rate of 79% and a recall rate of 77%.


title:
Resolution of Verb Ellipsis in Japanese Sentence using Surface
  Expressions and Examples

abstract:
  Verbs are sometimes omitted in Japanese sentences. It is necessary to recover
omitted verbs for purposes of language understanding, machine translation, and
conversational processing. This paper describes a practical way to recover
omitted verbs by using surface expressions and examples. We experimented the
resolution of verb ellipses by using this information, and obtained a recall
rate of 73% and a precision rate of 66% on test sentences.


title:
An Example-Based Approach to Japanese-to-English Translation of Tense,
  Aspect, and Modality

abstract:
  We have developed a new method for Japanese-to-English translation of tense,
aspect, and modality that uses an example-based method. In this method the
similarity between input and example sentences is defined as the degree of
semantic matching between the expressions at the ends of the sentences. Our
method also uses the k-nearest neighbor method in order to exclude the effects
of noise; for example, wrongly tagged data in the bilingual corpora.
Experiments show that our method can translate tenses, aspects, and modalities
more accurately than the top-level MT software currently available on the
market can. Moreover, it does not require hand-craft rules.


title: New Error Bounds for Solomonoff Prediction

abstract:
  Solomonoff sequence prediction is a scheme to predict digits of binary
strings without knowing the underlying probability distribution. We call a
prediction scheme informed when it knows the true probability distribution of
the sequence. Several new relations between universal Solomonoff sequence
prediction and informed prediction and general probabilistic prediction schemes
will be proved. Among others, they show that the number of errors in Solomonoff
prediction is finite for computable distributions, if finite in the informed
case. Deterministic variants will also be studied. The most interesting result
is that the deterministic variant of Solomonoff prediction is optimal compared
to any other probabilistic or deterministic prediction scheme apart from
additive square root corrections only. This makes it well suited even for
difficult prediction problems, where it does not suffice when the number of
errors is minimal to within some factor greater than one. Solomonoff's original
bound and the ones presented here complement each other in a useful way.


title:
Deduction over Mixed-Level Logic Representations for Text Passage
  Retrieval

abstract:
  A system is described that uses a mixed-level representation of (part of)
meaning of natural language documents (based on standard Horn Clause Logic) and
a variable-depth search strategy that distinguishes between the different
levels of abstraction in the knowledge representation to locate specific
passages in the documents. Mixed-level representations as well as
variable-depth search strategies are applicable in fields outside that of NLP.


title: Scalability Terminology: Farms, Clones, Partitions, Packs, RACS and RAPS

abstract:
  Defines a vocabulary for scaleable systems: Geoplexes, Farms, Clones, RACS,
RAPS, clones, partitions, and packs and dicusses the design tradeoffs of using
clones, partitons, and packs.


title: Adaptivity in Agent-Based Routing for Data Networks

abstract:
  Adaptivity, both of the individual agents and of the interaction structure
among the agents, seems indispensable for scaling up multi-agent systems
(MAS's) in noisy environments. One important consideration in designing
adaptive agents is choosing their action spaces to be as amenable as possible
to machine learning techniques, especially to reinforcement learning (RL)
techniques. One important way to have the interaction structure connecting
agents itself be adaptive is to have the intentions and/or actions of the
agents be in the input spaces of the other agents, much as in Stackelberg
games. We consider both kinds of adaptivity in the design of a MAS to control
network packet routing.
  We demonstrate on the OPNET event-driven network simulator the perhaps
surprising fact that simply changing the action space of the agents to be
better suited to RL can result in very large improvements in their potential
performance: at their best settings, our learning-amenable router agents
achieve throughputs up to three and one half times better than that of the
standard Bellman-Ford routing algorithm, even when the Bellman-Ford protocol
traffic is maintained. We then demonstrate that much of that potential
improvement can be realized by having the agents learn their settings when the
agent interaction structure is itself adaptive.


title: Avoiding Braess' Paradox through Collective Intelligence

abstract:
  In an Ideal Shortest Path Algorithm (ISPA), at each moment each router in a
network sends all of its traffic down the path that will incur the lowest cost
to that traffic. In the limit of an infinitesimally small amount of traffic for
a particular router, its routing that traffic via an ISPA is optimal, as far as
cost incurred by that traffic is concerned. We demonstrate though that in many
cases, due to the side-effects of one router's actions on another routers
performance, having routers use ISPA's is suboptimal as far as global aggregate
cost is concerned, even when only used to route infinitesimally small amounts
of traffic. As a particular example of this we present an instance of Braess'
paradox for ISPA's, in which adding new links to a network decreases overall
throughput. We also demonstrate that load-balancing, in which the routing
decisions are made to optimize the global cost incurred by all traffic
currently being routed, is suboptimal as far as global cost averaged across
time is concerned. This is also due to "side-effects", in this case of current
routing decision on future traffic.
  The theory of COllective INtelligence (COIN) is concerned precisely with the
issue of avoiding such deleterious side-effects. We present key concepts from
that theory and use them to derive an idealized algorithm whose performance is
better than that of the ISPA, even in the infinitesimal limit. We present
experiments verifying this, and also showing that a machine-learning-based
version of this COIN algorithm in which costs are only imprecisely estimated (a
version potentially applicable in the real world) also outperforms the ISPA,
despite having access to less information than does the ISPA. In particular,
this COIN algorithm avoids Braess' paradox.


title: Multivariate Regression Depth

abstract:
  The regression depth of a hyperplane with respect to a set of n points in R^d
is the minimum number of points the hyperplane must pass through in a rotation
to vertical. We generalize hyperplane regression depth to k-flats for any k
between 0 and d-1. The k=0 case gives the classical notion of center points. We
prove that for any k and d, deep k-flats exist, that is, for any set of n
points there always exists a k-flat with depth at least a constant fraction of
n. As a consequence, we derive a linear-time (1+epsilon)-approximation
algorithm for the deepest flat.


title:
Fast Hierarchical Clustering and Other Applications of Dynamic Closest
  Pairs

abstract:
  We develop data structures for dynamic closest pair problems with arbitrary
distance functions, that do not necessarily come from any geometric structure
on the objects. Based on a technique previously used by the author for
Euclidean closest pairs, we show how to insert and delete objects from an
n-object set, maintaining the closest pair, in O(n log^2 n) time per update and
O(n) space. With quadratic space, we can instead use a quadtree-like structure
to achieve an optimal time bound, O(n) per update. We apply these data
structures to hierarchical clustering, greedy matching, and TSP heuristics, and
discuss other potential applications in machine learning, Groebner bases, and
local improvement algorithms for partition and placement problems. Experiments
show our new methods to be faster in practice than previously used heuristics.


title: Comparative Analysis of Five XML Query Languages

abstract:
  XML is becoming the most relevant new standard for data representation and
exchange on the WWW. Novel languages for extracting and restructuring the XML
content have been proposed, some in the tradition of database query languages
(i.e. SQL, OQL), others more closely inspired by XML. No standard for XML query
language has yet been decided, but the discussion is ongoing within the World
Wide Web Consortium and within many academic institutions and Internet-related
major companies. We present a comparison of five, representative query
languages for XML, highlighting their common features and differences.


title: HMM Specialization with Selective Lexicalization

abstract:
  We present a technique which complements Hidden Markov Models by
incorporating some lexicalized states representing syntactically uncommon
words. Our approach examines the distribution of transitions, selects the
uncommon words, and makes lexicalized states for the words. We performed a
part-of-speech tagging experiment on the Brown corpus to evaluate the resultant
language model and discovered that this technique improved the tagging accuracy
by 0.21% at the 95% level of confidence.


title:
Mixed-Level Knowledge Representation and Variable-Depth Inference in
  Natural Language Processing

abstract:
  A system is described that uses a mixed-level knowledge representation based
on standard Horn Clause Logic to represent (part of) the meaning of natural
language documents. A variable-depth search strategy is outlined that
distinguishes between the different levels of abstraction in the knowledge
representation to locate specific passages in the documents. A detailed
description of the linguistic aspects of the system is given. Mixed-level
representations as well as variable-depth search strategies are applicable in
fields outside that of NLP.


title: Computation in an algebra of test selection criteria

abstract:
  One of the key concepts in testing is that of adequate test sets. A test
selection criterion decides which test sets are adequate. In this paper, a
language schema for specifying a large class of test selection criteria is
developed; the schema is based on two operations for building complex criteria
from simple ones. Basic algebraic properties of the two operations are derived.
  In the second part of the paper, a simple language-an instance of the general
schema-is studied in detail, with the goal of generating small adequate test
sets automatically. It is shown that one version of the problem is intractable,
while another is solvable by an efficient algorithm. An implementation of the
algorithm is described.


title: Quantum Bit Commitment Expansion

abstract: The paper was retracted.


title: Additive models in high dimensions

abstract:
  We discuss some aspects of approximating functions on high-dimensional data
sets with additive functions or ANOVA decompositions, that is, sums of
functions depending on fewer variables each. It is seen that under appropriate
smoothness conditions, the errors of the ANOVA decompositions are of order
$O(n^{m/2})$ for approximations using sums of functions of up to $m$ variables
under some mild restrictions on the (possibly dependent) predictor variables.
Several simulated examples illustrate this behaviour.


title:
Seeing the Forest in the Tree: Applying VRML to Mathematical Problems in
  Number Theory

abstract:
  We show how VRML (Virtual Reality Modeling Language) can provide potentially
powerful insight into the 3x + 1 problem via the introduction of a unique
geometrical object, called the 'G-cell', akin to a fractal generator. We
present an example of a VRML world developed programmatically with the G-cell.
The role of VRML as a tool for furthering the understanding the 3x+1 problem is
potentially significant for several reasons: a) VRML permits the observer to
zoom into the geometric structure at all scales (up to limitations of the
computing platform). b) VRML enables rotation to alter comparative visual
perspective (similar to Tukey's data-spinning concept). c) VRML facilitates the
demonstration of interesting tree features between collaborators on the
internet who might otherwise have difficulty conveying their ideas
unambiguously. d) VRML promises to reveal any dimensional dependencies among
3x+1 sequences.


title: On extremals with prescribed Lagrangian densities

abstract:
  Consider two manifolds~$M^m$ and $N^n$ and a first-order Lagrangian $L(u)$
for mappings $u:M\to N$, i.e., $L$ is an expression involving $u$ and its first
derivatives whose value is an $m$-form (or more generally, an $m$-density)
on~$M$. One is usually interested in describing the extrema of the functional
$\Cal L(u) = \int_M L(u)$, and these are characterized locally as the solutions
of the Euler-Lagrange equation~$E_L(u)=0$ associated to~$L$. In this note I
will discuss three problems which can be understood as trying to determine how
many solutions exist to the Euler-Lagrange equation which also satisfy $L(u) =
\Phi$, where $\Phi$ is a specified $m$-form or $m$-density on~$M$. The first
problem, which is solved completely, is to determine when two minimal graphs
over a domain in the plane can induce the same area form without merely
differing by a vertical translation or reflection. The second problem,
described more fully below, arose in Professor Calabi's study of extremal
isosystolic metrics on surfaces. The third problem, also solved completely, is
to determine the (local) harmonic maps between spheres which have constant
energy density.


title: Characteristic Numbers and Generalized Path Integrals

abstract:
  This is an extended write-up of a talk given in April, 1993 in honor of Raoul
Bott's 70th birthday. We first illustrate how some traditional topological and
geometric invariants obey ``gluing laws'' inspired by those in classical and
quantum field theory. Here we discuss characteristic numbers, particularly the
Euler number of a complex line bundle over an oriented surface. In the second
part of the paper we show how path integrals give rise to invariants which obey
gluing laws. The argument is formal in general, but for a particular example of
invariants related to finite groups it is rigorous. In that ``toy model'' we
show how to go further and generalize the path integral to obtain more exotic
gluing laws. This idea was used--at least in this toy model--to directly
construct quantum groups from 3 dimensional Chern-Simons invariants.


title:
On the Existence of Jenkins-Strebel Differentials Using Harmonic Maps
  from Surfaces to Graphs

abstract:
  We give a new proof of the existence (\cite{HM}, \cite{Ren}) of a
Jenkins-Strebel differential $\Phi$ on a Riemann surface $\SR$ with prescribed
heights of cylinders by considering the harmonic map from $\SR$ to the leaf
space of the vertical foliation of $\Phi$, thought of as a Riemannian graph.
The novelty of the argument is that it is essentially Riemannian as well as
elementary; moreover, the harmonic maps existence theory on which it relies is
classical, due mostly to Morrey (\cite{Mo}).


title: Moduli Spaces of Singular Yamabe Metrics

abstract:
  Complete, conformally flat metrics of constant positive scalar curvature on
the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were
constructed by R\. Schoen [S2]. We consider the problem of determining the
moduli space of all such metrics. All such metrics are asymptotically periodic,
and we develop the linear analysis necessary to understand the nonlinear
problem. This includes a Fredholm theory and asymptotic regularity theory for
the Laplacian on asymptotically periodic manifolds, which is of independent
interest. The main result is that the moduli space is a locally real analytic
variety of dimension $k$. For a generic set of nearby conformal classes the
moduli space is shown to be a $k-$dimensional real analytic manifold. The
structure as a real analytic variety is obtained by writing the space as an
intersection of a Fredholm pair of infinite dimensional real analytic
manifolds.


title: $n$-transitivity of certain diffeomorphism groups

abstract:
  Some groups of real analytic diffeomorphism act n-transitively for each
finite n.


title: Basic differential forms for actions of Lie groups

abstract:
  A section of a Riemannian $G$-manifold $M$ is a closed submanifold $\Sigma$
which meets each orbit orthogonally. It is shown that the algebra of
$G$-invariant differential forms on $M$ which are horizontal in the sense that
they kill every vector which is tangent to some orbit, is isomorphic to the
algebra of those differential forms on $\Sigma$ which are invariant with
respect to the generalized Weyl group of this orbit, under some condition.


title:
The Spectral Flow of the Odd Signature operator and Higher Massey
  Products

abstract:
  We show how to compute the spectral flow of the odd signature operator $\pm
*d_{a_t}-d_{a_t}*$ along an analytic path of flat connections $a_t$ on a bundle
over a closed odd-dimensional manifold in terms of Massey products in the DGLA
of bundle-valued differential forms. To obtain this information, we set up a
sequence of cochain complexes $\{\calg^*_n,\delta_n\}$, for $n=0,1,2,\ldots$
and Hermitian forms $$Q_n:\calg_n\times\calg_n\ra \bbbC$$ whose signatures
determine the spectral flow through $t=0$. The complexes and Hermitian forms
are constructed using Massey products.


title: Teichmuller Space is Not Gromov Hyperbolic

abstract:
  We prove that the Teichmuller Space of Riemann Surfaces of genus g>1,
equipped with the Teichmuller metric, is not a Gromov Hyperbolic space.


title: A ``stable'' version of the Gromov-Lawson conjecture

abstract:
  We discuss a conjecture of Gromov and Lawson, later modified by Rosenberg,
concerning the existence of metrics of positive scalar curvature. It says that
a closed spin manifold $M$ of dimension $n\ge 5$ has such a metric if and only
if the index of a suitable ``Dirac" operator in $KO_n(C^* (\pi_1(M)))$, the
real $K$-theory of the group $C^*$-algebra of the fundamental group of $M$,
vanishes. It is known that the vanishing of the index is necessary for
existence of a positive scalar curvature metric on $M$, but this is known to be
a sufficient condition only if $\pi_1(M)$ is the trivial group, $\Bbb Z/2$, an
odd order cyclic group, or one of a fairly small class of torsion-free groups.
\par We note that the groups $KO_n(C^*(\pi))$ are periodic in $n$ with period
$8$, whereas there is no obvious periodicity in the original geometric problem.
This leads us to introduce a ``stable'' version of the Gromov-Lawson
conjecture, which makes the weaker statement that the product of $M$ with
enough copies of the ``Bott manifold" $B$ has a positive scalar curvature
metric if and only if the index of the Dirac operator on $M$ vanishes. (Here
$B$ is a simply connected $8$-manifold which represents the periodicity element
in $KO_8(pt)$.) We prove the stable Gromov-Lawson conjecture for all spin
manifolds with finite fundamental group and for many spin manifolds with
infinite fundamental group.


title: Proper group actions and symplectic stratified spaces

abstract:
  Let $(M,\omega)$ be a Hamiltonian $G$-space with a momentum map $F:M \to
{\frak g}^*$. It is well-known that if $\alpha$ is a regular value of $F$ and
$G$ acts freely and properly on the level set $F^{-1}(G\cdot \alpha)$, then the
reduced space $M_\alpha :=F^{-1}(G\cdot \alpha)/G$ is a symplectic manifold. We
show that if the regularity assumptions are dropped the space $M_\alpha $ is a
union of symplectic manifolds, and that the symplectic manifolds fit together
in a nice way. In other words the reduced space is a {\em symplectic stratified
space}. This extends results known for the Hamiltonian action of compact
groups.


title: Transformation de Fourier-Mukai sur les Surfaces Hyperk\"ahl\'eriennes

abstract:
  Given two compact hyperk\"ahler surfaces $X$ and $Y$ and a holomorphic vector
bundle $Q$ on $X\times Y$, which is a generalized instanton, one can define a
Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles
on $X$ to vector bundles on $Y$. If $X$ and $Y$ are dual complex tori, this
transform maps instantons on $X$ to instantons on $Y$. After a quick review of
these results, we define a Fourier-Mukai transform in the case when $X$ is a K3
surface, and study the behaviour of instantons on $X$ under this transform.
Hard copies may be mailed on request


title: A note on the index bundle over the moduli space of monopoles

abstract:
  Donaldson has shown that the moduli space of monopoles $M_k$ is diffeomorphic
to the space $\Rat_k$ of based rational maps from the two-sphere to itself. We
use this diffeomorphism to give an explicit description of the bundle on
$\Rat_k$ obtained by pushing out the index bundle from $M_k$. This gives an
alternative and more explicit proof of some earlier results of Cohen and Jones.


title: All regulators of flat bundles are torsion

abstract:
  I prove the Bloch conjecture:all secondary characteristic classes of flat
bundles over complex projective varietes are torsion, except the first.


title: Rationality of secondary classes

abstract:
  We prove the Bloch conjecture : $ c_2(E) \in H^4_\cald (X,\bbz(2))$ is
torsion for holomorphic rank two vector bundles $E$ with an integrable
connection over a complex projective variety $X$. We prove also the rationality
of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds.
We give a sharp higher-dimensional Milnor inequality for the volume regulator
of all representations to $PSO(1,n)$ of fundamental groups of compact
$n$-dimensional hyperbolic manifolds, announced in our earlier paper.


title: Jumps of the eta invariant

abstract:
  We study the eta-invariant, defined by Atiyah-Patodi-Singer a real valued
invariant of an oriented odd-dimensional Riemannian manifold equipped with a
unitary representation of its fundamental group. When the representation varies
analytically, the corresponding eta-invariant may have an integral jump, known
also as the spectral flow. The main result of the paper establishes a formula
for this spectral jump in terms of the signatures of some homological forms,
defined naturally by the path of representations. These signatures may also be
computed by means of a spectral sequence of Hermitian forms,defined by the
deformation data. Our theorem on the spectral jump has a generalization to
arbitrary analytic families of self-adjoint elliptic operators. As an
application we consider the problem of homotopy invariance of the
rho-invariant. We give an intrinsic homotopy theoretic definition of the
rho-invariant, up to indeterminacy in the form of a locally constant function
on the space of unitary representations. In an Appendix, written by
S.Weinberger, it is shown (using the results of this paper) that the difference
in the rho-invariants of homotopy-equivalent manifolds is always rational.


title: Int\'egration symplectique des vari\'et\'es de Poisson r\'eguli\`eres

abstract:
  A symplectic integration of a Poisson manifold $(M,\Lambda)$ is a symplectic
groupoid $(\Gamma,\eta)$ which realizes the given Poisson manifold, i.e. such
that the space of units $\Gamma_0$ with the induced Poisson structure
$\Lambda_0$ is isomorphic to $(M,\Lambda)$. This notion was introduced by A.
Weinstein in order to quantize Poisson manifolds by quantizing their symplectic
integration. Any Poisson manifold can be integrated by a local symplectic
groupoid but already for regular Poisson manifolds there are obstructions to
global integrability. The aim of this paper is to summarize all the known
obstructions and present a sufficient topological condition for integrability
of regular Poisson manifolds; we will indeed describe a concrete procedure for
this integration. Further our criterion will provide necessary and sufficient
if we require $\Gamma$ to be Hausdorff, which is a suitable condition to
proceed to Weinstein's program of quantization. These integrability results may
be interpreted as an generalization of the Cartan-Smith proof of Lie's third
theorem in the infinite dimensional case.


title: Quadratic Equations in Groups from the Global Geometry Viewpoint

abstract:
  I use harmonic maps and minimal surfaces to study quadratic equations in
groups.


title: Pr\'equantification de Certaines Vari\'et\'es de Poisson

abstract:
  A surjective submersion $\pi : M \to B$ carrying a field of simplectic
structures on the fibres is symplectic if this Poisson structure is minimal. A
symplectic submersion may be interpreted as a family of mechanical systems
depending on a parameter in $B$. We give some conditions to find a closed form
which represent the foliated form $\sigma$ gluing the symplectic forms on the
fibres. This is the first step to prequantize all these systems at once. We
will indeed exhibit an integrality condition which does not depend on the
closed form representing $\sigma$: if the fibres are 1-connected and
$H^3(B;Z)=0$, then there exists a $S^1$-principal fibre bundle with a
connection whose curvature represents $\sigma$ iff the group of spherical
periods of $\sigma$ is a discrete subgroup of R. The symplectic integration of
a Poisson manifold $(M,\Lambda)$ is a symplectic groupoid $(\Gamma,\eta)$ with
1-connected fibres such that the space of units with the induced Poisson
structure is isomorphic to $(M,\Lambda)$. This notion was introduced by A.
Weinstein in order to quantize Poisson manifolds by quantizing their symplectic
integration. We show that if the symplectic integration is prequantizable, then
there exists a unique prequantization which is trivial over $M$. We show that
the symplectic integration of a minimal Poisson manifold is prequantizable iff
the group of spherical periods is discrete. Moreover we prove that a {\em
totally aspherical} Poisson manifold (any vanishing cycle is trivial and the
$\pi_2$ of the leaves is zero) is prequantized in the sense of Weinstein by a
trivial fibre bundle.


title:
Theta and zeta functions for odd-dimensional locally symmetric spaces of
  rank one

abstract:
  This paper is a continuation of our work on theta and zeta functions In the
previous papers we considered the case of even dimensional rank one symmetric
spaces of non-compact type. The present is concerned with the odd-dimensional
case, i.e. with odd-dimensional real hyperbolic manifolds. It is the natural
appearence of eta invariants in connection with super theta and zeta functions
what makes this case particularly interesting. (authors name corrected)


title: Theta and zeta functions for locally symmetric spaces of rank one

abstract:
  We study the Selberg zeta and the theta function associated to bundles over
even-dimensional locally symmetric spaces of rank one.


title: A theta function for hyperbolic surfaces with cusps

abstract:
  For a Riemann surface with cusps we define a theta function using the
eigenvalues of the Laplacian and the singularities of the scattering
determinant. We provide its meromorphic continuation and discuss its
singularities.


title: Bundle gerbes

abstract:
  Just as $\Cstar$ principal bundles provide a geometric realisation of
two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a
geometric realisation of three dimensional integral cohomology through their
Dixmier-Douady class. I consider an alternative, related, geometric realisation
of three dimensional cohomology called a bundle gerbe. Every bundle gerbe gives
rise to a gerbe and most of the well-known examples examples of gerbes are
bundle gerbes. I discuss the properties of bundle gerbes, in particular bundle
gerbe connections and curvature and their associated Dixmier-Douady class.


title: Convexity Properties of the Moment Mapping Re-examined

abstract:
  Consider a Hamiltonian action of a compact Lie group on a compact symplectic
manifold. A theorem of Kirwan's says that the image of the momentum mapping
intersects the positive Weyl chamber in a convex polytope. I present a new
proof of Kirwan's theorem, which gives explicit information on how the vertices
of the polytope come about and on how the shape of the polytope near any point
can be read off from infinitesimal data on the manifold. It also applies to
some interesting classes of noncompact or singular Hamiltonian spaces, such as
cotangent bundles and complex affine varieties.


title: Witten deformation of Ray-Singer analytic torsion

abstract:
  Consider a flat vector bundle F over compact Riemannian manifold M and let f
be a self-indexing Morse function on M. Let g be a smooth Euclidean metric on
F. Set g_t=exp(-2tf)g and let \rho(t) be the Ray-Singer analytic torsion of F
associated to the metric g_t. Assuming that the vector field $grad f$ satisfies
the Morse-Smale transversality conditions, we provide an asymptotic expansion
for log(\rho(t)) for t\to\infty of the form a_0+a_1t+b log(t)+o(1). We present
explicit formulae for coefficients a_0,a_1 and b. In particular, we show that b
is a half integer.


title: Modular operads

abstract:
  Modular operads are a special type of operad: in fact, they bear the same
relationship to operads that graphs do to trees (i.e. simply connected graphs).
One of the basic examples of a modular operad is the collection of
Deligne-Mumford-Knudsen moduli spaces $\bar{M}_{g,n}$ of stable pointed
algebraic curves; hence the word ``modular.'' In this paper, we introduce
various constructions on differential graded modular operads, notably a duality
which we call the Feynman transform, which extends Kontsevich's graph
complexes. Our main result is the calculation of the Euler characteristic of
the Feynman transform of a modular operad, using the theory of symmetric
functions: the result is a generalization of Wick's theorem for Gaussian
integrals.


title: The Moduli Space of Complete Embedded Constant Mean Curvature Surfaces

abstract:
  We examine the space of surfaces in $\RR^{3}$ which are complete, properly
embedded and have nonzero constant mean curvature. These surfaces are
noncompact provided we exclude the case of the round sphere. We prove that the
space $\Mk$ of all such surfaces with $k$ ends (where surfaces are identified
if they differ by an isometry of $\RR^{3}$) is locally a real analytic variety.
When the linearization of the quasilinear elliptic equation specifying mean
curvature equal to one has no $L^2-$nullspace we prove that $\Mk$ is locally
the quotient of a real analytic manifold of dimension $3k-6$ by a finite group
(i\.e\. a real analytic orbifold), for $k\geq 3$. This finite group is the
isotropy subgroup of the surface in the group of Euclidean motions. It is of
interest to note that the dimension of $\Mk$ is independent of the topology of
the underlying punctured Riemann surface to which $\Sig$ is conformally
equivalent. These results also apply to hypersurfaces of $\HH^{n+1}$ with
nonzero constant mean curvature greater than that of a horosphere and whose
ends are cylindrically bounded.


title: On the Dirichlet problem for harmonic maps with prescribed singularities

abstract:
  Let $\M$ be a classical Riemannian globally symmetric space of rank one and
non-compact type. We prove the existence and uniqueness of solutions to the
Dirichlet problem for harmonic maps into $\M$ with prescribed singularities
along a closed submanifold of the domain. This generalizes our previous work
where such maps into the hyperbolic plane were constructed. This problem, in
the case where $\M$ is the complex-hyperbolic plane, has applications to
equilibrium configurations of co-axially rotating charged black holes in
General Relativity.


title: Fano Manifolds, Contact Structures, and Quaternionic Geometry

abstract:
  Let Z be a compact complex (2n+1)-manifold which carries a {\em complex
contact structure}, meaning a codimension-1 holomorphic sub-bundle D of TZ
which is maximally non-integrable. If Z admits a K\"ahler-Einstein metric of
positive scalar curvature, we show that it is the Salamon twistor space of a
quaternion-K\"ahler manifold (M^{4n}, g). If Z also admits a second complex
contact structure, then Z= CP_{2n+1}. As an application, we give several new
characterizations of the Riemannian manifold HP_n=Sp(n+1)/( Sp(n)\times Sp(1)).


title: Scalar-Flat Kaehler Surfaces of All Genera

abstract:
  Let (M,J) be a compact complex 2-manifold which which admits a Kaehler metric
for which the integral of the scalar curvature is non-negative. Also suppose
that M does not admit a Ricci-flat K\"ahler metric. Then if M is blown up at
sufficiently many points, the resulting complex surface admits Kaehler metrics
with scalar curvature identically equal to zero.


title: The spectrum of an asymptotically hyperbolic Einstein manifold

abstract:
  This paper relates the spectrum of the scalar Laplacian of an asymptotically
hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary''
at infinity. It follows from work of R. Mazzeo that the essential spectrum of
such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$,
with no embedded eigenvalues; however, in general there may be discrete
eigenvalues below the continuous spectrum. The main result of this paper is
that, if the Yamabe invariant of the conformal structure on the boundary is
non-negative, then there are no such eigenvalues. This generalizes results of
R. Schoen, S.-T. Yau, and D. Sullivan for the case of hyperbolic manifolds.


title: On the nonHamiltonian interaction of two rotators

abstract:
  Classical dynamical equations describing a certain version of the
nonHamiltonian interaction of two rotators (Euler tops with completely
degenerate inertia tensors) are considered. The simplest case is integrated. It
is shown that the dynamics is almost periodic with periods depending on the
initial data.

categories: dg-ga chao-dyn math.DG nlin.CD nlin.SI solv-int


title: On multilinear operators commuting with Lie derivatives

abstract:
  Let $E_1,\dots ,E_k$ and $E$ be natural vector bundles defined over the
category $\Cal Mf_m^+$ of smooth oriented $m$--dimensional manifolds and
orientation preserving local diffeomorphisms, with $m\geq 2$. Let $M$ be an
object of $\Cal Mf_m^+$ which is connected. We give a complete classification
of all separately continuous $k$--linear operators $D\:\Ga
_c(E_1M)\x\dots\x\Ga_c(E_kM)\to \Ga (EM)$ defined on sections with compact
supports, which commute with Lie derivatives, i\.e\. which satisfy $$ \Cal
L_X(D(s_1,\dots ,s_k))=\sum _{i=1}^kD(s_1,\dots ,\Cal L_Xs_i,\dots,s_k), $$ for
all vector fields $X$ on $M$ and sections $s_j\in\Ga_c(E_jM)$, in terms of
local natural operators and absolutely invariant sections. In special cases we
do not need the continuity assumption. We also present several applications in
concrete geometrical situations, in particular we give a completely algebraic
characterization of some well known Lie brackets.


title: Dressing orbits of harmonic maps

abstract:
  We study the harmonic map equations for maps of a Riemann surface into a
Riemannian symmetric space of compact type from the point of view of soliton
theory. There is a well-known dressing action of a loop group on the space of
harmonic maps and we discuss the orbits of this action through particularly
simple harmonic maps called {\em vacuum solutions}. We show that all harmonic
maps of semisimple finite type (and so most harmonic $2$-tori) lie in such an
orbit. Moreover, on each such orbit, we define an infinite-dimensional
hierarchy of commuting flows and characterise the harmonic maps of finite type
as precisely those for which the orbit under these flows is finite-dimensional.


title: 3-manifolds with(out) metrics of nonpositive curvature

abstract:
  In the context of Thurstons geometrisation program we address the question
which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive
curvature. We show that non-geometric Haken manifolds generically, but not
always, admit such metrics. More precisely, we prove that a Haken manifold
with, possibly empty, boundary of zero Euler characteristic admits metrics of
nonpositive curvature if the boundary is non-empty or if at least one atoroidal
component occurs in its canonical topological decomposition. Our arguments are
based on Thurstons Hyperbolisation Theorem. We give examples of closed
graph-manifolds with linear gluing graph and arbitrarily many Seifert
components which do not admit metrics of nonpositive curvature.


title: On Rumin's Complex and Adiabatic Limits

abstract:
  This paper shows that when the Riemannian metric on a contact manifold is
blown up along the direction orthogonal to the contact distribution, the
corresponding harmonic forms rescaled and normalized in the $L^2$-norms will
converge to Rumin's harmonic forms. This proves a conjecture in Gromov ``
Carnot-Caratheodory spaces seen from within '', IHES preprint, 1994. This
result can also be reformulated in terms of spectral sequences, after Forman,
Mazzeo-Melrose. A key ingredient in the proof is the fact that the curvatures
become unbounded in a controlled way.


title:
A construction of singular solutions for a semilinear elliptic equation
  using asymptotic analysis

abstract:
  The aim of this paper is to prove the existence of weak solutions to the
equation $\Delta u + u^p = 0$ which are positive in a domain $\Omega \subset
{\Bbb R}^N$, vanish at the boundary, and have prescribed isolated
singularities. The exponent $p$ is required to lie in the interval $(N/(N-2),
(N+2)/(N-2))$. We also prove the existence of solutions to the equation $\Delta
u + u^p = 0$ which are positive in a domain $\Omega \subset {\Bbb R}^n$ and
which are singular along arbitrary smooth $k$-dimensional submanifolds in the
interior of these domains provided $p$ lie in the interval $((n - k)/(n-k-2),
(n-k+2)/(n-k-2))$. A particular case is when $p = (n+2)/(n-2)$, in which case
solutions correspond to solutions of the singular Yamabe problem. The method
used is a mixture of different ingredients used by both authors in their
separate constructions of solutions to the singular Yamabe problem, along with
a new set of scaling techniques.


title:
Hermitian structures and harmonic morphisms in higher dimensional
  Euclidean spaces

abstract:
  We construct new complex-valued harmonic morphisms from Euclidean spaces from
functions which are holomorphic with respect to Hermitian structures. In
particular, we give the first global examples of complex-valued harmonic
morphisms from ${\bf R}^n$ for each $n>4$ which do not arise from a K\"ahler
structure; it is known that such examples do not exist for $n \leq 4$.


title:
Geometry of Cyclic Quotients, I: Knotted Totally Geodesic Submanifolds
  in Positively Curved Spheres

abstract:
  We prove that there exists a metric of positive curvature in a three-sphere
which admits a given torus knot as a closed geodesic.We also sketch a
construction of a metric in a four sphere, very likely of positive curvature,
which admits a totally geodesic projective plane with Euler number four.
Surpisingly, the technique borrows a lot from the Mostow-Siu-Gromov-Thurston
constuction of exotic negatively curved manifolds.


title:
The Fr\"olicher-Nijenhuis bracket for derivation based non commutative
  differential forms

abstract:
  In commutative differential geometry the Fr\"olicher-Nijenhuis bracket
computes all kinds of curvatures and obstructions to integrability. In \cit!{3}
the Fr\"olicher-Nijenhuis bracket was developped for universal differential
forms of non-commutative algebras, and several applications were given. In this
paper this bracket and the Fr\"olicher-Nijenhuis calculus will be developped
for several kinds of differential graded algebras based on derivations, which
were indroduced by \cit!{6}.


title: Self-Dual Manifolds with Positive Ricci Curvature

abstract:
  We prove that the connected sums CP_2 # CP_2 and CP_2 # CP_2 # CP_2 admit
self-dual metrics with positive Ricci curvature. Moreover, every self-dual
metric of positive scalar curvature on CP_2 # CP_2 is conformal to a metric
with positive Ricci curvature.


title: Yamabe Spectra

abstract:
  We study the set of volumes of constant scalar curvature one metrics on an
atoroidal three-manifold.The infinum of this set is believed to be attained at
a hyperbolic metric. We prove that the supremum of this set is always infinity.
The technique is: minimal surfaces, Thurston norm in homology and new conformal
invariants.


title: Ricci Curvature and Betti Numbers

abstract:
  We derive a uniform bound for the total betti number of a closed manifold in
terms of a Ricci curvature lower bound, a conjugate radius lower bound and a
diameter upper bound. The result is based on an angle version of Toponogov
comparison estimate for small triangles in a complete manifold with a Ricci
curvature lower bound. We also give a uniform estimate on the generators of the
fundamental group and prove a fibration theorem in this setting.


title: $\Gamma$-Cohomology and the Selberg Zeta Function

abstract:
  We propose a new method for studying $n$- and $\Gamma$-cohomology of
globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one
semisimple Lie group, $\Gamma$ is a discrete subgroup of $G$ and $n=Lie(N)$. We
prove a conjecture of Patterson relating the singularities of Selberg zeta
functions with the $\Gamma$-cohomology of principal series representations if
$\Gamma$ is cocompact and torsion free.


title: Einstein Metrics and Mostow Rigidity

abstract:
  Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness
theorem is proved for Einstein metrics on compact quotients of irreducible
4-dimensional symmetric spaces of non-compact type. The proof also yields a
Riemannian version of the Miyaoka-Yau inequality.


title:
The singularities of Yang-Mills connections for bundles on a surface. I.
  The local model

abstract:
  Let $\Sigma$ be a closed surface, $G$ a compact Lie group, not necessarily
connected, with Lie algebra $g$, endowed with an adjoint action invariant
scalar product, let $\xi \colon P \to \Sigma$ be a principal $G$-bundle, and
pick a Riemannian metric and orientation on $\Sigma$, so that the corresponding
Yang-Mills equations $$d_A*K_A = 0$$ are defined, where $K_A$ refers to the
curvature of a connection $A$. For every central Yang-Mills connection $A$, the
data induce a structure of unitary representation of the stabilizer $Z_A$ on
the first cohomology group $\roman H^1_A(\Sigma,\roman{ad}(\xi))$ with
coefficients in the adjoint bundle $\roman{ad}(\xi)$, with reference to $A$,
with momentum mapping $\Theta_A$ from $\roman H^1_A(\Sigma,\roman{ad}(\xi))$ to
the dual $z^*_A$ of the Lie algebra $z_A$ of $Z_A$. We show that, for every
central Yang-Mills connection $A$, a suitable Kuranishi map identifies a
neighborhood of zero in the Marsden-Weinstein reduced space $\roman H_A$ for
$\Theta_A$ with a neighborhood of the point $[A]$ in the moduli space of
central Yang-Mills connections on $\xi$.


title:
The singularities of Yang-Mills connections for bundles on a surface.
  II. The stratification

abstract:
  Let $\Sigma$ be a closed surface, $G$ a compact Lie group, not necessarily
connected, with Lie algebra $g$, endowed with an adjoint action invariant
scalar product, let $\xi \colon P \to \Sigma$ be a principal $G$-bundle, and
pick a Riemannian metric and orientation on $\Sigma$ so that the corresponding
Yang-Mills equations are defined. In an earlier paper we determined the local
structure of the moduli space $N(\xi)$ of central Yang-Mills connections on
$\xi$ near an arbitrary point. Here we show that the decomposition of $N(\xi)$
into connected components of orbit types of central Yang-Mills connections is a
stratification in the strong (i.~e. Whitney) sense; furthermore each stratum,
being a smooth manifold, inherits a finite volume symplectic structure from the
given data. This complements, in a way, results of {\smc Atiyah-Bott} in that
it will in general decompose further the critical sets of the corresponding
Yang-Mills functional into smooth pieces.


title: Smooth structures on certain moduli spaces for bundles on a surface

abstract:
  Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra
$g$, $\xi \colon P \to \Sigma$ a principal $G$-bundle, let $N(\xi)$ denote the
moduli space of central Yang-Mills connections on $\xi$, for suitably chosen
additional data, and let $\roman{Rep}_{\xi}(\Gamma,G)$ be the space of
representations of the universal central extension $\Gamma$ of the fundamental
group of $\Sigma$ in $G$ that corresponds to $\xi$. We construct smooth
structures on $N(\xi)$ and $\roman{Rep}_{\xi}(\Gamma,G)$ and show that the
assignment to a smooth connection $A$ of its holonomies with reference to
suitable closed paths yields a diffeomorphism from $N(\xi)$ onto
$\roman{Rep}_{\xi}(\Gamma,G)$; moreover we show that the derivative of the
latter at the non-singular points of $N(\xi)$ amounts to a certain twisted
integration mapping. Finally we examine the infinitesimal geometry of these
moduli spaces with reference to the smooth structures and, for illustration, we
show that, on the moduli space of flat $\roman{SU}(2)$-connections for a
surface of genus two which, as a space, is just complex projective 3-space, our
smooth structure looks rather different from the standard structure.


title: Poisson structures on certain moduli spaces for bundles on a surface

abstract:
  Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra
$g$, and $\xi \colon P \to \Sigma$ a principal $G$-bundle. In earlier work we
have shown that the moduli space $N(\xi)$ of central Yang- Mills connections,
for appropriate additional data, is stratified by smooth symplectic manifolds
and that the holonomy yields a diffeomorphism from $N(\xi)$ onto a certain
representation space $\roman{Rep}_{\xi}(\Gamma,G)$, with reference to suitable
smooth structures $C^{\infty}(N(\xi))$ and
$C^{\infty}(\roman{Rep}_{\xi}(\Gamma,G))$ where $\Gamma$ denotes the universal
central extension of the fundamental group of $\Sigma$. Given an invariant
symmetric bilinear form on $g^*$, we construct here Poisson structures on
$C^{\infty}(N(\xi))$ and $C^{\infty}(\roman{Rep}_{\xi}(\Gamma,G))$ in such a
way that the mentioned diffeomorphism identifies them. When the form on $g^*$
is non-degenerate the Poisson structures are compatible with the
stratifications where $\roman{Rep}_{\xi}(\Gamma,G)$ is endowed with the
corresponding stratification and, furthermore, yield structures of a {\it
stratified symplectic space\/}, preserved by the induced action of the mapping
class group of $\Sigma$.


title: Curved Flats and Isothermic Surfaces

abstract:
  We show how pairs of isothermic surfaces are given by curved flats in a
pseudo Riemannian symmetric space and vice versa. Calapso's fourth order
partial differential equation is derived and, using a solution of this
equation, a M\"obius invariant frame for an isothermic surface is built.


title:
The number of functionally independent invariants of a
  pseudo--Riemannian metric

abstract:
  The number of functionally independent scalar invariants of arbitrary order
of a generic pseudo--Riemannian metric on an $n$--dimensional manifold is
determined.


title: Pseudoconvex and Disprisoning Homogeneous Sprays

abstract:
  The pseudoconvex and disprisoning conditions for geodesics of linear
connections are extended to the solution curves of general homogeneous sprays.
The main result is that pseudoconvexity and disprisonment are jointly stable in
the fine topology on the space of all homogeneous sprays of any degree of
homogeneity.


title:
Witten deformation of the analytic torsion and the spectral sequence of
  a filtration

abstract:
  Let F be a flat vector bundle over a compact Riemannian manifold M and let f
be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-2tf}g
and let \rho(t) be the Ray-Singer analytic torsion of F associated to the
metric g_t. Assuming that the vector field grad(f) satisfies the Morse-Smale
transversality conditions, we provide an asymptotic expansion for \log(\rho(t))
for t\to +\infty of the form
  a_0+a_1t+b\log\left(\frac t\pi\right)+o(1), where the coefficient b is a
half-integer depending only on the Betti numbers of F.
  In the case where all the critical values of f are rational, we calculate the
coefficients a_0 and a_1 explicitly in terms of the spectral sequence of a
filtration associated to the Morse function.
  These results are obtained as an applications of a theorem by Bismut and
Zhang.


title: Smoothing Riemannian Metrics with Ricci Curvature Bounds

abstract:
  We prove that Riemannian metrics with an absolute Ricci curvature bound and a
conjugate radius bound can be smoothed to having a sectional curvature bound.
  Using this we derive a number of results about structures of manifolds with
Ricci curvature bounds.


title: Integral geometry of plane curves and knot invariants

abstract:
  We study the integral expression of a knot invariant obtained as the second
coefficient in the perturbative expansion of Witten's Chern-Simons path
integral associated with a knot. One of the integrals involved turns out to be
a generalization of the classical Crofton integral on convex plane curves and
it is related with invariants of generic plane curves defined by Arnold
recently with deep motivations in symplectic and contact geometry. Quadratic
bounds on these plane curve invariants are derived using their relationship
with the knot invariant.


title: Nowhere Holomorphic Mappings

abstract: Retracted temporarily to fix some erratic things.


title: Differential Geometry of Composite Fibred Manifolds

abstract:
  In classical field theory, the composite fibred manifolds Y -> Z -> X
provides the adequate mathematical formulation of gauge models with broken
symmetries, e.g., the gauge gravitation theory. This work is devoted to
connections on composite fibred manifolds. In particular, we get the horizontal
splitting of the vertical tangent bundle of a composite fibred manifold,
besides the familiar one of its tangent bundle. This splitting defines the
modified covariant differential and implies the special fashion of Lagrangian
densities of field models on composite manifolds. The spinor composite bundles
are examined.


title:
Symplectic and Poisson structures of certain moduli spaces. II.
  Projective representations of cocompact discrete planar groups

abstract:
  Let $G$ be a Lie group with a biinvariant metric, not necessarily positive
definite. It is shown that a certain construction carried out in an earlier
paper for the fundamental group of a closed surface may be extended to an
arbitrary infinite orientation preserving cocompact planar discrete group of
euclidean or non-euclidean motions $\pi$ and yields (i) a symplectic structure
on a certain smooth manifold $\Cal M$ containing the space $\roman{Hom}(\pi,G)$
of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\Cal M$
preserving the symplectic structure together with a momentum mapping in such a
way that the reduced space equals the space $\roman{Rep}(\pi,G)$ of
representations. More generally, the construction also applies to certain
spaces of projective representations. For $G$ compact, the resulting spaces of
representations inherit structures of {\it stratified symplectic space\/} in
such a way that the strata have finite symplectic volume . For example, {\smc
Mehta-Seshadri} moduli spaces of semistable holomorphic parabolic bundles with
rational weights or spaces closely related to them arise in this way by {\it
symplectic reduction in finite dimensions\/}.


title: Blow-up formulas for (-2)-spheres

abstract:
  Let $X$ be a simply connected 4-manifold containing a $(-1)$-sphere $e$.
Fintushel and Stern prove that $$ D_c(\exp(te)) = D_c(B(t)) on e^\perp if
c\cdot e is even, $$ $$ D_c(\exp(te)) = D_{c-e}(S(t)) on e^\perp if c \cdot e
is odd, $$ for some universal series $B(t),S(t) \in \Q[x][[t]]$ with $x$ the
class of a point. We show that their method can easily be extended to
$(-2)$-spheres $\tau$ to give blow up formulas like $$ D_c(\exp(t\tau)) =
D_c(B^2(t) + S^2(t)/2 \tau^2) on \tau^perp if c\cdot \tau is even. $$


title: Symplectic toric orbifolds

abstract:
  A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic
form $\omega$ on $M$, and an effective Hamiltonian action of a torus $T$ on
$M$, where the dimension of $T$ is half the dimension of $M$. We prove that
there is a one-to-one correspondence between symplectic toric orbifolds and
convex rational simple polytopes with positive integers attached to each facet.


title: On the Scalar Curvature of Complex Surfaces

abstract:
  Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown
that the smooth 4-manifold M admits a Riemannian metric of positive scalar
curvature iff (M,J) admits a KAEHLER metric of positive scalar curvature. This
extends previous results of Witten and Kronheimer.


title: Meromorphic Potentials and Smooth CMC Surfaces

abstract:
  This work is based on the approach developed by J.~Dorfmeister, F.~Pedit and
H.~Wu [GANG and KITCS preprint, Report KITCS94-4-1] to construct maps
$\Phi:D\rightarrow R^3$, $D$ being the unit disk in $C$, whose images are
surfaces of constant mean curvature. They start from certain meromorphic one
forms, so called meromorphic potentials, which take values in a twisted loop
algebra over $su(2)$. The authors give necessary and sufficient conditions on
the meromorphic potential for the constructed map $\Phi$ to be an immersion.
This revision corrects several misprints and minor TeX problems.


title: Exotic Holonomy on Moduli Spaces of Rational Curves

abstract:
  Bryant \cite{Br} proved the existence of torsion free connections with exotic
holonomy, i.e. with holonomy that does not occur on the classical list of
Berger \cite{Ber}. These connections occur on moduli spaces $\Y$ of rational
contact curves in a contact threefold $\W$. Therefore, they are naturally
contained in the moduli space $\Z$ of all rational curves in $\W$. We construct
a connection on $\Z$ whose restriction to $\Y$ is torsion free. However, the
connection on $\Z$ has torsion unless both $\Y$ and $\Z$ are flat. We also show
the existence of a new exotic holonomy which is a certain sixdimensional
representation of $\Sl \times \Sl$. We show that every regular $H_3$-connection
(cf. \cite{Br}) is the restriction of a unique connection with this holonomy.


title: Products and relations in symplectic Floer homology

abstract:
  We give a construction of a version of the Gromov-Witten classes, Q: H_*(J)
-> HF_*(M) otimes ... otimes HF^*(M), within the context of symplectic Floer
(co)homology. In particular, this gives a functorial approach to products and
relations in symplectic Floer (co)homology.


title: A Simple Geometric Representative for $\mu$ of a Point

abstract:
  For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a
simple geometric representative for $\mu$ of a point. Let $p$ be a generic
point in $X$. Then the set $\{ [A] | F_A^-(p) $ is reducible $\}$, with
coefficient -1/4 and appropriate orientation, is our desired geometric
representative.


title: The Twistor correspondence of the Dolbeault complex over $\C^n$

abstract:
  With respect to the Dolbeault complex over the flat manifold $\C^n$, an
explicit description of the inverse correspondence of the twistor
correspondence is given.


title: Spaces of Geodesics: Products, Coverings, Connectedness

abstract:
  We continue our study of the space of geodesics of a manifold with linear
connection. We obtain sufficient conditions for a product to have a space of
geodesics which is a manifold. We investigate the relationship of the space of
geodesics of a covering manifold to that of the base space. We obtain
sufficient conditions for a space to be geodesically connected in terms of the
topology of its space of geodesics. We find the space of geodesics of an
$n$-dimensional Hadamard manifold is the same as that of $\R^n$.


title: Simpson's Theory and Superrigidity of Complex Hyperbolic Lattices

abstract:
  We attack a conjecture of J. Rogawski: any cocompact lattice in $S U (2, 1)$
for which the ball quotient $X = B^2 / \Gamma$ satisfies $b_1 (X) = 0$ and
$H^{1, 1} (X) \cap H^2 (X, \bbq) \approx \bbq$ is arithmetic. We prove the
Archimedian suprerigidity for representation of $\Gamma$ is $S L (3, \bbc)$.


title:
Analytic and Reidemeister torsion for representations in finite type
  Hilbert modules

abstract:
  For a closed Riemannian manifold we extend the definition of analytic and
Reidemeister torsion associated to an orthogonal representation of fundamental
group on a Hilbert module of finite type over a finite von Neumann algebra. If
the representation is of determinant class we prove, generalizing the
Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal.
In particular, this proves the conjecture that for closed Riemannian manifolds
with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister
torsions are equal.


title: The inverse Penrose transform on Riemannian twistor spaces

abstract:
  With respect to the Dirac operator and the conformally invariant Laplacian,
an explicit description of the inverse Penrose transform on Riemannian twistor
spaces is given. A Dolbeault representative of cohomology on the twistor space
is constructed from a solution of the field equation on the base manifold.


title:
K\"ahlerian Killing Spinors, Complex Contact Structures and Twistor
  Spaces

abstract:
  We collect our recent results ([5] and [8]) and we get the equivalence of the
three notions of the title under some conditions. We then use this equivalence
in order to prove some consequences about Sasakian manifolds, complex almost
contact structures and complex k-contact structures.


title: Torus bundles and the group cohomology of GL(N,Z)

abstract:
  We prove the vanishing of a certain characteristic class of flat vector
bundles when the structure groups of the bundles are contained in GL(N,Z). We
do so by explicitly writing the characteristic class as an exact form on the
base of the bundle.


title: Cohomology of Kaehler manifolds with c_1=0

abstract:
  A linear constraint is given on the Betti numbers of a compact hyper-Kaehler
manifold, using an index formula for c_1c_{n-1} on an almost complex manifold.
The topology of some other manifolds with reduced holonomy is also discussed
briefly.


title: Twistor theory, complex homogeneous manifolds and $G$-structures

abstract:
  It is shown that any irreducible analytic 1-flat $G$-structure as well as any
analytic torsion-free affine connection with irreducibly acting holonomy group
can, in principle, be contstructed by twistor methods.


title: Coherent states and geodesics: cut locus and conjugate locus

abstract:
  The intimate relationship between coherent states and geodesics is pointed
out. For homogenous manifolds on which the exponential from the Lie algebra to
the Lie group equals the geodesic exponential, and in particular for symmetric
spaces, it is proved that the cut locus of the point $0$ is equal to the set of
coherent vectors orthogonal to $\vert 0>$. A simple method to calculate the
conjugate locus in Hermitian symmetric spaces with significance in the coherent
state approach is presented. The results are illustrated on the complex
Grassmann manifold.


title: A Groenewold-Van Hove Theorem for S^2

abstract:
  We prove that there does not exist a nontrivial quantization of the Poisson
algebra of the symplectic manifold S^2 which is irreducible on the subalgebra
generated by the components {S_1,S_2,S_3} of the spin vector. We also show that
there does not exist such a quantization of the Poisson subalgebra P consisting
of polynomials in {S_1,S_2,S_3}. Furthermore, we show that the maximal Poisson
subalgebra of P containing {1,S_1,S_2,S_3} that can be so quantized is just
that generated by {1,S_1,S_2,S_3}.


title:
On the number of geodesic segments connecting two points on manifolds of
  non-positive curvature

abstract:
  In this paper we show that on a complete Riemannian manifold of negative
curvature and dimension $n>1$ every two points which realize a local maximum
for the distance function are connected by at least $2n+1$ geometrically
distinct geodesic segments (i.e. length minimizing). Using a similar method, we
obtain that in the case of non-positive curvature, for every two points with
the same property as above the number of connecting distinct geodesic segments
is at least $n+1$.


title: Non-standard connections in classical mechanics

abstract:
  In the jet-bundle description of first-order classical field theories there
are some elements, such as the lagrangian energy and the construction of the
hamiltonian formalism, which require the prior choice of a connection. Bearing
these facts in mind, we analyze the situation in the jet-bundle description of
time-dependent classical mechanics. So we prove that this connection-dependence
also occurs in this case, although it is usually hidden by the use of the
``natural'' connection given by the trivial bundle structure of the phase
spaces in consideration. However, we also prove that this dependence is
dynamically irrelevant, except where the dynamical variation of the energy is
concerned. In addition, the relationship between first integrals and
connections is shown for a large enough class of lagrangians.


title:
Mayer-Vietoris Formula for Determinants of Elliptic Operators of
  Laplace-Beltrami Type (after Burghelea, Friedlander and Kappeler)

abstract:
  The purpose of this note is to provide a short cut presentation of a
Mayer-Vietoris formula due to Burghelea-Friedlander-Kappeler for the
regularized determinant in the case of elliptic operators of Laplace Beltrami
type in the form typically needed in applications to torsion.


title:
Invariant operators on manifolds with almost Hermitian symmetric
  structures, I. Invariant differentiation

abstract:
  This is the first part of a series of papers. The whole series aims to
develop the tools for the study of all almost Hermitian symmetric structures in
a unified way. In particular, methods for the construction of invariant
operators, their classification and the study of their properties will be
worked out. In this paper we present the invariant differentiation with respect
to a Cartan connection and we expand the differentials in the terms of the
underlying linear connections belonging to the structures in question. Then we
discuss the holonomic and non-holonomic jet extensions and we suggest methods
for the construction of invariant operators.


title:
Invariant operators on manifolds with almost Hermitian symmetric
  structures, II. Normal Cartan connections

abstract:
  In the first part of this series of papers we developed the invariant
differentiation with respect to a Cartan connection, we described this
procedure in the terms of the underlying principal connections, and we
discussed applications of this theory to the construction of natural operators.
In this part we will extend the results of \cite{Ochiai, 70} on the existence
and the uniqueness of the so called normal Cartan connections on manifolds with
almost Hermitian symmetric structures to first order structures which do not
admit a torsion free linear connection. Moreover, for each of these structures
we obtain explicit (universal) formulae for these canonical connections in the
terms of the curvatures of the underlying linear connections.


title: Secondary analytic indices

abstract:
  We define analytic indices which involve the eta form and the analytic
torsion form. We show that these indices are independent of the geometric
choices made in their definitions, and hence are topological in nature.


title: Witten-Helffer-Sjostrand Theory for a Generalized Morse Functions

abstract:
  In this paper, we extend the Witten-Helffer-Sj\"{o}strand theory from Morse
functions to generalized Morse functions. In this case, the spectrum of the
Witten deformed Laplacian $\Delta(t)$, for large t, can be seperated into the
small eigenvalues (which tend to 0 as $t\rightarrow\infty$), large and very
large eigenvalues (both of which tend to $\infty$ as $t\rightarrow\infty$). The
subcomplex $\Omega_{0}^{\ast}(M,t)$ spanned by eigenforms corresponding to the
small and large eigenvalues of $\Delta(t)$ is finite dimensional. Under some
mild conditions, it is shown that $(\Omega_{0}^{\ast}(M,t),d(t))$ converges to
a geometric complex associated to the generalized Morse function as
$t\rightarrow\infty$.


title: Characteristic Classes in Symplectic Topology

abstract:
  From the cohomological point of view the symplectomorphism group $Sympl (M)$
of a symplectic manifold is `` tamer'' than the diffeomorphism group. The
existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the
symplectic Chern-Weil theory, and the existence of Chern-Simons-type secondary
classes are first manifestations of this principles. On a deeper level live
characteristic classes of symplectic actions in periodic cohomology and
symplectic Hodge decompositions. The present paper is called to introduce
theories and constructions listed above and to suggest numerous concrete
applications. These includes: nonvanishing results for cohomology of
symplectomorphism groups (as a topological space, as a topological group and as
a discrete group), symplectic rigidity of Chern classes, lower bounds for
volumes of Lagrangian isotopies, the subject started by Givental, Kleiner and
Oh, new characters for Torelli group and generalizations for automorphism
groups of one-relator groups, arithmetic properties of special values of Witten
zeta-function and solution of a conjecture of Brylinski. The Appendix, written
by L. Katzarkov, deals with fixed point sets of finite group actions in moduli
spaces.


title:
Homotopy and Homology Vanishing Theorems and the Stability of Stochastic
  Flows

abstract:
  We relate stability properties (i.e. moment exponents) of a stochastic
dynamical system on a compact manifold $M$ to the homotopy and integral
homology groups of $M$. In the special case of gradient Brownian systems
associated to isometric immersions of $M$ in Euclidean space, these moment
exponents can be estimated in terms of the second fundamental form of the
immersion. This yields topological obstructions to isometric immersions
generalizing results of Lawson-Simons and others. Our work also places these
authors' work into the general framework of Weitzenb\"ock formulas.


title: The Ring of Invariants for Smooth Completions of Kac-Moody Lie Algebras

abstract:
  It is proved that the ring of invariants of the standard smooth completion of
a Kac-Moody Lie algebra is functionally generated by two elements: the
coefficient of the center and the Killing form.


title: Isometric Immersions of Space Forms and Soliton Theory

abstract:
  This paper studies isometric immersions of space forms by means of a
hierarchy of finite dimensional integrable systems in Lax form on loop
algebras.


title: Analytic subsets of Hilbert spaces

abstract:
  We show that every complete metric space is homeomorphic to the precise locus
of zeros of an entire analytic map from a Hilbert space to a Banach space. As a
corollary, every complete separable metric space is homeomorphic to the precise
locus of zeros of an entire analytic map between two separable complex Hilbert
spaces. AmS TeX 2.1.


title: Geometry of Kodaira moduli spaces

abstract:
  A general theorem on the existence of natural torsion-free affine connections
on a complete family of compact complex submanifolds in a complex manifold is
proved. Applications to twistor theory are discussed.


title:
Fundamental Group of Self-Dual Four-Manifolds with Positive Scalar
  Curvature

abstract:
  Main Theorem (3.3): Let $M$ be a compact four-dimensional manifold either
with curvature, positive on complex isotropic two-planes, or self-dual of
positive scalar curvature. If $\pi_1 (M)$ admits a nontrivial unitary
representation, and $M$ is orientable, then there exists a surjective
homomorphism from $\pi_1 (M)$ on $\bbz$.
  Corollary: If $\pi_1 (M)$ is finite, then either $\pi_1 (M) = 1$, or $\pi_1
(M) = \bbz_2$.
  Observe that finitely presented groups which do not admit a nontrivial
unitary representation, are extremely rare (see 3.4).


abstract:
  We provide a simple way to obtain the meromorphic extension of Eisenstein
series and Scattering matrices under conditions which generalize the case of
discrete groups acting convex cocompactly on hyperbolic spaces.


title: Projective structures on moduli spaces of compact complex hypersurfaces

abstract:
  It is shown that moduli spaces of complete families of compact complex
hypersurfaces in complex manifolds often come equipped canonically with
projective structures satisfying some natural integrability conditions.


title: Symetric Monopoles

abstract:
  We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant
under various symmetry groups. The analysis is largely in terms of the spectral
curves, the rational maps, and the Nahm equations associated with monopoles. We
consider monopoles invariant under inversion in a plane, monopoles with cyclic
symmetry, and monopoles having the symmetry of a regular solid. We introduce
the notion of a strongly centred monopole and show that the space of such
monopoles is a geodesic submanifold of the monopole moduli space.
  By solving Nahm's equations we prove the existence of a tetrahedrally
symmetric monopole of charge $3$ and an octahedrally symmetric monopole of
charge $4$, and determine their spectral curves. Using the geodesic
approximation to analyse the scattering of monopoles with cyclic symmetry, we
discover a novel type of non-planar $k$-monopole scattering process.


title: Symplectic Approach of Wess-Zumino-Witten Model and Gauge Field Theories

abstract:
  A systematic description of the Wess-Zumino-Witten model is presented. The
symplectic method plays the major role in this paper and also gives the
relationship between the WZW model and the Chern-Simons model. The quantum
theory is obtained to give the projective representation of the Loop group. The
Gauss constraints for the connection whose curvature is only focused on several
fixed points are solved. The Kohno connection and the Knizhnik-Zamolodchikov
equation are derived. The holonomy representation and $\check R$-matrix
representation of braid group are discussed.


title: Symplectic Surgery and the Spin-C Dirac operator

abstract:
  Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian
$G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector
bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose
equivariant index is a symplectic invariant of $E$. In the present paper, we
study gluing properties of the equivariant index under "symplectic cutting"
operations. Our main application is a proof of the Guillemin-Sternberg
conjecture, which says that if $E=L$ is a quantizing line bundle and $0$ a
regular value of $J$, the multiplicity of the trivial representation in the
equivariant index is equal to the Riemann-Roch number of the symplectic
quotient. This generalizes previous results for the case that $G=T$ is abelian.


title: Variational Aspects of the Seiberg-Witten Functional

abstract:
  The Seiberg-Witten equations that have recently found important applications
for four-dimensional geometry are the Euler-Lagrange equations for a functional
involving a connection $A$ on a line bundle $L$ and a section $\phi$ of another
bundle $W^+$ constructed from $L$ and a spinor bundle on a given
four-dimensional Riemannian manifold. We show the regularity of weak solutions
and the Palais-Smale condition for this functional.


title: A Decomposition of Smooth Simply-connected h-Cobordant 4-Manifolds

abstract:
  We prove that any two smooth h-cobordant simply-connected 4-manifolds can be
obtained by taking two manifolds with boundary, one of which is contractible,
and gluing them along the boundary via two different attaching maps.


title: Determinant Line Bundles Revisited

abstract:
  This is a note for the conference proceedings Topological and Geometrical
Problems related to Quantum Field Theory. We summarize our joint work with Dai
about eta invariants on manifolds with boundary. Then we apply these results to
prove the curvature and holonomy formulas for the natural connection on the
determinant line bundle of a family of Dirac operators. These were originally
proved by Bismut and the author--the proofs here are much simpler.


title: On Ribbon $R^4$'s

abstract:
  We consider ribbon $R^4$'s, that is, smooth open 4-manifolds, homeomorphic to
$R^4$ and associated to $h$-cobordisms between closed 4-manifolds. We show that
any generalized ribbon $R^4$ associated to a sequence of $h$-cobordisms between
non-diffeomorphic 4-manifolds is exotic. Notion of a positive ribbon $R^4$ is
defined and we show that a ribbon $R^4$ is positive if and only if it is
associated to a sequence of stably non-product h-cobordisms. In particular we
show that any positive ribbon $R^4$ is associate to a subsequence of the
sequence of non-product h-cobordisms from [BG].


title: Geometry of Lagrangian First-order Classical Field Theories

abstract:
  We construct a lagrangian geometric formulation for first-order field
theories using the canonical structures of first-order jet bundles, which are
taken as the phase spaces of the systems in consideration. First of all, we
construct all the geometric structures associated with a first-order jet bundle
and, using them, we develop the lagrangian formalism, defining the canonical
forms associated with a lagrangian density and the density of lagrangian
energy, obtaining the {\sl Euler-Lagrange equations} in two equivalent ways: as
the result of a variational problem and developing the {\sl jet field
formalism} (which is a formulation more similar to the case of mechanical
systems). A statement and proof of Noether's theorem is also given, using the
latter formalism. Finally, some classical examples are briefly studied.


title: Extended moduli spaces and the Kan construction

abstract:
  Let $Y$ be a CW-complex with a single 0-cell, let $K$ be its Kan group, a
free simplicial group whose realization is a model for the space $\Omega Y$ of
based loops on $Y$, and let $G$ be a Lie group, not necessarily connected. By
means of simplicial techniques involving fundamental results of {\smc Kan's}
and the standard $W$- and bar constructions, we obtain a weak $G$-equivariant
homotopy equivalence from the geometric realization $|\roman{Hom}(K,G)|$ of the
cosimplicial manifold $\roman{Hom}(K,G)$ of homomorphisms from $K$ to $G$ to
the space $\roman{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space
$BG$ of $G$ where $G$ acts on $BG$ by conjugation. Thus when $Y$ is a smooth
manifold, the universal bundle on $BG$ being endowed with a universal
connection, the space $|\roman{Hom}(K,G)|$ may be viewed as a model for the
space of based gauge equivalence classes of connections on $Y$ for all
topological types of $G$-bundles on $Y$ thereby yielding a rigorous approach to
lattice gauge theory; this is illustrated in low dimensions.


title: Constant mean curvature surfaces via integrable dynamical system

abstract:
  It is shown that the equation which describes constant mean curvature surface
via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its
simplest finite-dimensional reduction has two degrees of freedom, integrable
and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer
surfaces (i.e., helicoidal constant mean curvature surfaces).


title: The Radius Rigidity Theorem for Manifolds of Positive Curvature

abstract:
  Recall that the radius of a compact metric space $(X, dist)$ is given by
$rad\ X = \min_{x\in X} \max_{y\in X} dist(x,y)$. In this paper we generalize
Berger's $\frac{1}{4}$-pinched rigidity theorem and show that a closed, simply
connected, Riemannian manifold with sectional curvature $\geq 1$ and radius
$\geq \frac{\pi}{2}$ is either homeomorphic to the sphere or isometric to a
compact rank one symmetric space.


title:
Fuchsian groups of the second kind and representations carried by the
  limit set

abstract:
  We compute the cohomology of a Fuchsian group of the second kind with
coefficients in the hyperfunction vectors of the principal series
representations of $SL(2,R)$ supported on the limit set.