Marc Culler - US grants

Professors Culler and Shalen will continue their program for estimating Margulis constants, injectivity radii, and volumes of hyperbolic manifolds by using Sullivan-Patterson measures associated to 2-generator Kleinian groups. They will investigate group actions on generalized trees, extending research of Culler- Vogtmann and Gillet-Shalen. They will also apply the theory of character varieties to the study of Heegaard splittings of non- Haken 3-manifolds. Culler will continue his work on cohomological properties of the outer automorphism group of the free group. Shalen will study questions about amalgamated free product decompositions and deficiencies of subgroups, both in 3-manifold groups and in more general finitely presented groups. He will also investigate possible generalizations of Casson's invariant and connections with SL(2,C) representations of 3-manifold groups. These are all very active topics in the topology of 3- dimensional manifolds, i.e. very natural geometric objects of the same dimension as the space in which we live.

A striking feature of the development of mathematics in the past decade has been the extent to which the diverse branches of mathematics have interacted, with methods from one branch being used to solve outstanding problems in another. Culler and Shalen's ongoing program for obtaining lower bounds for volumes of hyperbolic manifolds is in large measure an application of qualitative results from topology to quantitative estimates in geometry. Their program for the Property P conjecture would be likely to exploit K-theory and arithmetic geometry in attacking a problem from the topological theory of 3-manifolds. Culler's project of implementing Manin's program by means of Thurston's theory of pleated surfaces is a way of using synthetic hyperbolic geometry to obtain insight into number theory. Finally, Shalen's project involving fundamental groups of 2-complexes lies in the domain of geometric and combinatorial group theory but is motivated in part by the prospect of applications to the study of discrete subgroups of algebraic groups and Kac-Moody groups. Three-dimensional manifolds are the mathematical models of our physical universe. While the cosmological question of whether our universe is finite may be unresolved, it is certain that the study of finite-volume 3-manifolds is of fundamental importance in understanding the physical universe. The quantitative estimates obtained by Culler and Shalen may be regarded as being mathematical analogues of measurements of fundamental physical constants, such as the speed of light. The geometry, as well as the topology, of the universe is important in physics. While the non-euclidean geometries may have been considered curiosities in the nineteenth century, the development of general relativity shows that negatively curved spaces, such as hyperbolic 3-manifolds, are the appropriate models for the universe. Recent mathematical research has shown that they are ubiquitous and mathematically rich as well. The research proposed here is aimed at expanding our understanding of these important spaces.

The Department of Mathematics, Statistics and Computer Science at the University of Illinois at Chicago will purchase computational equipment which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects, including in particular research in Geometric Group Theory (Paul R. Brown); Topology of 3-Manifolds (Marc Culler); Algebraic Geometry (Anatoly Libgober); Theory of Finite Groups (Stephen Smith); and Number Theory (Jeremy Teitelbaum). Most of these projects are data-intensive and amenable to distributed computing techniques.

Proposal: DMS-9971660

Principal Investigator: Marc Culler and Peter Shalen

Abstract:
Culler and Shalen are proposing to study the set of boundary slopes of a knot, both in the case of an arbitrary closed orientable 3-manifold and in the case of a manifold with cyclic fundamental group. This is a topic of intrinsic interest, which Culler and Shalen believe is also related to some of the most difficult unsolved problems in 3-manifold theory. Their program for applying results about essential surfaces in knot exteriors to 3-manifold theory have led Culler and Shalen to a number of questions about the character variety of a hyperbolic knot group, which they propose to work on. These ideas also lead to an approach to the very broad problem of giving a quantitative version of Thurston's Dehn surgery theorem; this can be seen as the problem underlying most of the existent work on Dehn surgery. The character variety has a distinguished 1-dimensional irreducible component X which is the one that contains the character of the representation that determines the hyperbolic structure on the knot. One question which will be addressed by Culler and Shalen is that of determining properties that distinguish surfaces associated to ideal points of the main component of the character variety from other essential surfaces in the knot exterior. Another question involves the relationship between the hyperbolic volume of the knot complement and properties of a certain factor of the A-polynomial, this factor being the defining equation of a plane curve closely related to X. There is a general notion of hyperbolic volume which applies to a representation whose character lies on X. A third question is that of understanding what properties of an essential surface associated to an ideal point of X guarantee that this generalized volume tends to 0 at the ideal point. Yet another relevant question is that of understanding how often it happens that the curve X is invariant under complex conjugation, and what it means, in terms of the topology of the 3-manfiold, for this to happen. In a somewhat different direction, Culler and Shalen have developed methods for relating the boundary slopes and the genera of essential surfaces in certain situations. They are proposing to develop extensions of this theory. Their techniques in this area may be relevant to the general question of whether every knot in a homotopy sphere has at least one nonzero integer boundary slope.

A fundamental problem in many areas of mathematics is to classify all examples of a certain type of mathematical object. The objects of study in this proposal are 3-manifolds, which are mathematical models of 3-dimensional spaces. Since our universe is a 3-dimensional space, the classification of 3-manifolds is directly related to our understanding of nature itself. The classification problem for 3-manifolds is far from solved, but the work of many mathematicians over the last 20 years has at least produced a conjectural answer. The work supported by this problem forms part of the effort to verify the conjectured geometric classification of 3-manifolds.

Culler and Shalen are continuing to investigate consequences of their
work on the character variety of a knot group. This involves studying
the relationship between the boundary slopes of a knot and the
topological properties of essential surfaces that realize the
slopes. It also includes their ongoing project with Dunfield and Jaco
about smallish knots in non-Haken manifolds, which is part of a
program to prove the Poincare Conjecture. A joint project by Agol,
Culler and Shalen, concerning the construction of covering spaces of a
triangulated 3-manifold by an inductive process, is also relevant to
this program. Agol is also continuing his work on geometric finiteness
of geometrically defined subgroups of knot groups, volume estimates
for non-fibered Haken manifolds and hyperbolic orbifolds, and the
complexity of algorithms in 3-manifold theory.

A fundamental problem in many areas of mathematics is to classify all
examples of a certain type of mathematical object. The objects of
study in this proposal are 3-manifolds, which are mathematical models
of 3-dimensional spaces. Since our universe is a 3-dimensional space,
the classification of 3-manifolds is directly related to our
understanding of nature itself. The classification problem for
3-manifolds is far from solved, but the work of many mathematicians
over the last 25 years has at least produced a conjectural
answer. Remarkably, the conjectures, if true, will provide a
unification of the most classical, rigid kind of geometry---both the
Euclidean version first studied by the ancient Greeks and the
non-Euclidean kind that constituted an exciting discovery in the 19th
century---and topology, a subject devoted to studying much more
flexible geometric structures, which until recently had developed
quite independently of the more classical theories. The work
supported by this grant forms part of the effort to verify the
conjectured geometric classification of 3-manifolds.

The work supported by this grant is bringing to bear on the study of
hyperbolic 3-manifolds methods from the analytic theory of Kleinian
groups, geometric and algebraic topology, algebra and combinatorics.
One group of projects aims at understanding, in a quantitative way,
how the volume of a hyperbolic manifold reflects its underlying
topology. Another project studies the relationship between the
algebraic rank of a hyperbolic 3- manifold and its Heegaard
genus. A third group of projects addresses the construction of
hyperbolic manifolds by the Dehn filling construction.

A hyperbolic manifold is a space which is locally modelled on the
non-euclidean geometry of Lobachevsky, Bolyai and Gauss, in which the
sum of the angles of a triangle is less than pi. Besides being of
fundamental importance for classical geometry and number theory,
hyperbolic manifolds have long been known to play a central role in
three-dimensional topology. This has been newly confirmed by
Perelman's announcement of a proof of Thurston's geometrization
conjecture.

Abstract

Award: DMS-0603270
Principal Investigator: Marc E. Culler, Steven Boyer

The subject of 3-manifold topology has a long history of deep and
interesting interactions with other parts of mathematics. The
complexity of these connections has increased with time, as the
subject developed, and it has exploded in recent years. The
coincidence of this explosion with the recent solution of several
of the major outstanding problems in the area make this a good
time for reflection on its relationship to the rest of
mathematics. This grant will support participants in a
conference, to be held at CRM in Montreal, which aims to bring
together a varied group of leading researchers whose work
demonstrates deep connections between 3-manifold topology and
other areas of mathematics. The speakers will include leading
researchers in areas such as: geometrization of 3-manifolds;
combinatorial group theory and coarse geometric properties of
groups; properties of random 3-manifolds and asymptotic
properties of hyperbolic surfaces; Floer homology,
Khovanov-Rozansky homology; 3-manifold invariants arising from
contact structures; character varieties; and Dehn surgery. The
conference will honor Peter Shalen, whose work has been a major
force in bringing many different aspect of mathematics to bear on
the study of 3-manifolds, and in expanding the influence of
3-manifold topology into other areas.

Throughout history the interaction between mathematics and
science has been characterized by the mysterious phenomenon that
mathematical ideas which are developed with no specific
application in mind turn out to be perfectly suited for future
applications to science. For example, differential geometry,
which provided the ideal mathematical foundation for Einstein's
general relativity theory, was developed much earlier by
mathematicians who could not have anticipated the way in which it
would be applied in physics. On a smaller scale, there is a
similar phenomenon involving different areas of mathematics. A
recent example of this, which is directly related to the topic of
this conference, is provided by Perelman's proof of Thurston's
Geometrization Conjecture. This conference aims to facilitate
these sorts of unexpected fruitful interactions between different
areas of mathematics. The central topic of the conference is the
study of 3-dimensional spaces. Speakers have been selected whose
work involves significant new applications to this subject of
ideas which originate from a wide range of other areas of
mathematics.

The conference web site is
http://www.crm.umontreal.ca/cal/fr/mois200606.html.

Culler and Shalen are continuing their study of the topological structure of hyperbolic 3-manifolds. The context for this work is the on-going unification of the geometric and topological theories of
3-manifolds. The results can be viewed in terms of the theorem, which is a consequence of work of Gromov, Thurston and Jorgensen, that the set of volumes of closed hyperbolic 3-manifolds is a well-ordered set of real numbers. The proposed research aims to understand the topological properties of the 3-manifolds with volume less than a given threshold value. Alternatively, the goal is to determine the volume which corresponds to the ordinal at which a given topological property first appears. The techniques used range from very classical topological methods, such as a refined version of the tower construction used in proving the Loop Theorem, to the most recent developments, including the proof of the Marden Tameness conjecture and Perelman's estimates on the change of volume under Ricci flow. Interesting connections with group theory and combinatorial topology are also involved.

The spaces which are being studied in this research project, namely 3-dimensional manifolds, serve as mathematical models of the spatial aspect of a possible physical universe. New connections between modern physics and the mathematical theory of 3-manifolds are being discovered at a rapidly accelerating pace. The mathematical theory has traditionally been divided into topology and geometry, where geometry focuses on quantities which can be measured, such as lengths, angles, areas or volumes, and topology focuses on global properties that are preserved even when the geometric features are distorted. However, these two aspects of the subject are closely related, and
there are many examples of results which relate geometric properties and topological properties. Recent mathematical achievements, beginning with the Mostow Rigidity theorem and continuing up to the recent proofs of Marden's Tameness Conjecture and Thurston's Geometrization Conjecture, are leading toward a unification of the geometrical and topological theories of 3-manifolds. The research supported by this grant concentrates on hyperbolic manifolds, a class which includes the vast majority of 3-manifolds with a homogeneous geometric structure, and aims to understand in a quantitative sense how topological complexity depends on the geometrical volume of the
manifold.

Culler and Shalen will continue their research on hyperbolic 3-manifolds. One of the main themes of their work is the connection between topologically defined invariants of such manifolds and their quantitative geometric invariants such as volume. This has involved interactions between very classical techniques in 3-manifold topology, some of which go back to Papakyriakopoulos's work in the 1950's, and more geometric methods such as the log(2k-1)-theorem of Anderson, Canary, Culler and Shalen, the work of Kojima and Miyamoto on hyperbolic manifolds with totally geodesic boundary, and the work of Agol, Dunfield, Storm and Thurston based on properties of the Ricci flow with surgeries. A second theme, which recently has grown out of the first, is the connection between the number-theoretic invariants of a manifold such as its trace field and quantitative geometric invariants such as its Margulis number. This aspect of the work depends on combining the earlier work with new group-theoretic observations, and has already brought into play such deep number-theoretic ingredients as the work of Siegel and Mahler on the unit equation in algebraic number fields.

Hyperbolic manifolds are geometric objects that arise in many branches of mathematics and in many applications of mathematics. The first hyperbolic manifold, called hyperbolic space, was discovered in the 19th century and settled---in the negative---the ancient problem of whether Euclid's fifth postulate could be deduced from his other postulates. Hyperbolic manifolds may be thought of as geometric objects which at small scales are indistinguishable from hyperbolic space, but whose large-scale behavior is more complicated. A major theme in modern geometry is the interaction between the quantitative properties of a geometric object, for example those defined in terms of distances, lengths, areas and volumes, and their "topological"
properties which are more qualitative and are unchanged when the object is deformed. In the case of hyperbolic manifolds, so much progress has been made in recent years in relating the quantitative and topological theories that they may be said to be completely unified at an abstract level. The present project involves making our understanding the connection in a more concrete way. Doing this turns out to involve deep ideas from many branches of mathematics.

This proposal is motivated by the unification between the topology and the geometry of three-dimensional manifolds. It is primarily focused on the quantitative geometry of hyperbolic 3-manifolds, specifically on estimating such quantitative invariants as volume, Margulis number and diameter in terms of topological data. This is an area in which connections with topology and several other branches of mathematics are playing unexpected roles. Classical techniques in 3-manifold topology, some of which go back to Papakyriakopoulos's work in the 1950s, become particularly powerful when applied in the context of hyperbolic geometry. These topological ideas interact with more geometric and analytic methods, such as the log(2k-1) Theorem of Anderson, Canary, Culler and Shalen; the isoperimetric inequality for hyperbolic space; the theory of algebraic and geometric convergence of Kleinian groups; the work of Kojima and Miyamoto on hyperbolic manifolds with totally geodesic boundary; and the work of Agol, Dunfield, Storm and Thurston which applies properties of the Ricci flow with surgeries to the study of Haken manifolds and Dehn filling. Furthermore, surprising interactions are emerging, via topology, between quantitative geometry of hyperbolic 3-manifolds and their number-theoretic aspects, specifically their trace fields; this has allowed applications of deep results in number theory to the subject.

Non-Euclidean geometry is a classical topic in pure mathematics which has seen remarkable developments in recent decades. The subject had its origin in the attempt, begun in ancient times, to prove that Euclid's fifth axiom could be deduced from his other axioms. It was shown in the course of the 19th century that this cannot be done: there is a mathematical structure called the hyperbolic plane (in two dimensions) or hyperbolic space (in three or more dimensions) which satisfies all of Euclid's axioms except the fifth, and in which the sum of the angles of a triangle is always less than 180 degrees. Remarkably, hyperbolic geometry turns out to be much richer than Euclidean geometry. This accounts for the astonishingly varied interactions that have developed since the 1960's between hyperbolic geometry and other branches of mathematics and science. Most of these interactions involve the study of hyperbolic manifolds, which are geometric objects that have the small-scale geometry of hyperbolic space but have a more complicated structure in the large. For example, a straight line in hyperbolic space, as in Euclidean space, always extends to infinity; but in a hyperbolic manifold, a path that is locally a straight line (called a geodesic) may exhibit globally "periodic" behavior like a circle. Some of the principal investigators' earliest work on hyperbolic manifolds produced a result about knots that has been applied to study the structure of recombinant DNA. They are at present investigating a variety of aspects of the geometry of hyperbolic manifolds and connections with some of the other topics mentioned above.