Jeffrey Brock - US grants

DMS-0204454
Jeffrey F. Brock

THE CLASSIFICATION PROBLEM FOR HYPERBOLIC 3-MANIFOLDS

The PI, Jeffrey Brock, will synthesize diverse techniques in the
deformation theory of hyperbolic 3-manifolds to address classification
problem for hyperbolic 3-manifolds. Brock will undertake joint work
with K. Bromberg that employs the theory of hyperbolic cone-manifolds
to show that each tame hyperbolic 3-manifold M is approximated by
geometrically finite 3-manifolds. This conjecture, known as the
Density Conjecture has recently been solved by Brock and Bromberg in
certain cases. Brock will also work toward completing joint work
with R. Canary and Y. Minsky to prove Thurston's ending lamination
conjecture, which predicts that a tame hyperbolic 3-manifold is
determined by its topology and its end invariants: combinatorial
invariants attached to the ``ends'' of a hyperbolic 3-manifold. In
new joint work with Bromberg, R. Evans, and J. Souto, Brock will study
the question of whether each algebraic limit of a sequence of
geometrically finite hyperbolic 3-manifolds is itself topologically
tame. This joint project has implications for a conjecture of Ahlfors
that the limit set of a finitely generated Kleinian group has either
measure zero or full measure in the Riemann sphere.

Classifying mathematical objects plays much the same scientific role
as classifying biological, chemical, or physical phenomena in the
development of these fields. For example, with the human genome
"cracked," scientists may now isolate specific genetic causes or
predispositions to diseases, greatly furthering the ability of science
to address these problems. In the proposed research, Brock will
endeavor to solve the classification problem for a "generic" class of
3-dimensional spaces, the "hyperbolic 3-manifolds." These
non-Euclidean spaces have geometry locally like our own Euclidean
space, but their large scale geometry is expanding exponentially:
for example, light rays (a metaphor for geodesics) emanating from a
point-source diverge exponentially rather than linearly. William
P. Thurston's revolutionary and pioneering work in the 1970's and
1980's showed that almost all 3-manifolds are hyperbolic, and went on
to raise as many questions about hyperbolic 3-manifolds as it
answered. From his contributions, a compelling conjectural picture of
the right classification of hyperbolic 3-manifolds has emerged as a
lasting problem for researchers in the field of geometry and topology.
Recent work of the PI and his collaborators has put the solution of
this problem within reach; the PI will make use his NSF support to
facilitate ongoing collaborations to solve this fundamental problem,
thereby making a "database" of hyperbolic 3-manifolds available for
wider use by other mathematicians and physicists alike.

The recent surge of results in the geometry and topology of
3-manifolds has provided many new tools for understanding
3-manifolds combinatorially and geometrically. In particular, the
question of how to gain an explicit understanding of the internal
geometry of a closed hyperbolic 3-manifold can be addressed with
new techniques developed by the P.I. together with Dick Canary and
Yair Minsky that apply to the infinite volume case. Such an
understanding would be a kind of effective version of Mostow rigidity,
wherein one not only knows the uniqueness of the hyperbolic structure
but additionally an explicit decription of its geometry. The P.I., in
joint work with Juan Souto, seeks to develop this kind of picture for
closed hyperbolic 3-manifolds admitting a Heegaard splitting,
given in terms of the Heegaard surface. Additionally, with Howard
Masur and Yair Minsky the P.I. will relate the internal geometry of
hyperbolic 3-manifolds homotopy equivalent to a surface to the
geometry of surfaces along a Weil-Petersson geodesic G in
Teichmueller space.


A recent trend in the study of geometry and topology is to develop
combinatorial models for geometric spaces. This kind of description
of a space or shape sacrifices a certain degree of precision in the
interest of capturing more of the large-scale structure, and often
general theorems guarantee that knowing this large-scale structure is
sufficient to completely determine the space. In a recent result of
the P.I. with R. Canary and Y. Minsky, such models were used to
classify all `constantly negatively curved,' or `hyperbolic'
3-dimensional spaces of infinite volume that are nevertheless tame
in a certain sense. This result solved a long-standing conjecture of
William thurston, where in a certain piece of data (akin to a kind of
DNA-sequence for the space) completely determines its structure. We
have developed a similar setup in the finite-volume case, and hope to
prove a similar classification theorem for such spaces. Information
of this large-scale type is more useful than existence results for
geometric structures, in that it gives one a more complete picture of
how such spaces behave. Such large-scale data arise and describe
phenomena in many contexts, whether it is the spaces themselves, or
their parameter spaces. The P.I., together with his collaborators,
will continue to develop a complete description of the large-scale
geometry of all hyperbolic 3-dimensional manifolds, as well as for
related spaces that parameterize them.

Since Thurston formulated his geometrization conjecture, the study of infinite volume hyperbolic 3-manifolds has risen to a prominent position in low-dimensional topology and geometry. For the past 30 years four major conjectures have guided this area: Marden's Tameness Conjecture, Thurston's Ending Lamination Conjecture, the Bers-Thurston-Sullivan Density Conjecture and Ahlfors' Measure Conjecture; all have been resolved in the last four years. The solutions of these conjectures have introduced new techniques into the field and opened the door to deeper investigation and the exploration of new directions. In this Focused Research Group, the principal
investigators propose to use these new techniques to deepen their understanding of the geometry of hyperbolic 3-manifolds, both of infinite and of finite volume, to explore further their still mysterious deformation theory, to pioneer new directions for research in the field, and to develop connections with related branches of low-dimensional geometry and topology.

Since the time of Poincare, topologists have pursued the idea that certain spaces called 3-manifolds might be simply described. In the 1970's, Thurston's geometrization conjecture showed topologists the power of bringing geometry to bear on this problem, and opened the possiblity for broad connections between topological, geometric and dynamical features that arise. Using technical tools arising from recent breakthroughs, the PIs hope to interconnect further these different perspectives on the field, and expose early career mathematicians and graduate students to the new range of problems emerging from this fertile area. The Focused Research Group will fund small conferences during its first and final year focused on emerging research areas, with introductory workshops to be run on the day prior to the beginning of the conference. This project will also support the research of the principal investigators' graduate students and provide travel funding for their interaction across institutions. Each of these efforts will allow young geometers and topologists both to learn about the exciting recent developments in the field and to explore the new directions opened up by these developments.

The classification of finitely generated Kleinian groups and their associated quotient hyperbolic 3-manifolds has generated significant new tools for studying many problems at the interfaces of Teichmüller theory, Kleinian groups and low-dimensional topology. In particular, the existence of a model manifold for a hyperbolic 3-manifold M that is uniformly bi-Lipschitz to M has foregrounded the extent to which many problems in the study of closed and finite volume hyperbolic 3- manifolds can be understood in terms of combinatorial structures associated to surfaces. Likewise, large-scale questions in the geometry of Teichmüller space have come into relief in terms of a new understanding of these combinatorics: the asymptotic geometry of geodesics in various metrics has been reconstituted and understood in a new language, yet the structure of the classical Weil-Petersson metric from this point of view remains largely unclear and tantalizingly open. Our proposed research will demonstrate how model manifolds serve as building blocks for hyperbolic structures on closed manifolds via Heegaard splittings, to develop control on the synthetic geometry and dynamics of the Weil-Petersson metric on Teichmüller space via the complex of curves, and to continue to reveal applications of the model manifolds to the topology of deformation spaces of hyperbolic 3-manifolds.

The idea of a "coarse model" in geometry proposes that one might sacrifice a certain degree of precision in the interest of capturing more large-scale structure. Frequently a coarse model plays a similar role to DNA in biology: it can determine fine features of a space despite its apparently coarse nature. In a recent result of the P.I.
with R. Canary and Y. Minsky, such models were used to classify all `constantly negatively curved,' or `hyperbolic' three-dimensional spaces of infinite volume that are `tame' in a certain sense. The classification result solved a long-standing conjecture of William Thurston, and opened the door to developing a more detailed and complete picture of geometries on manifolds previously considered understood. After Perelman's solution to Thurston's geometrization conjecture and the famous Poincaré conjecture, the groundwork is in place for a fundamental investigation of algebraic, geometric and topological properties of all spaces of 3-dimensions and how these properties interrelate.

The Institute for Computational and Experimental Research in Mathematics (ICERM) is a national institute whose mission is to support and catalyze fundamental research in the mathematical sciences with a thematic focus on the fruitful interplay between mathematics and computers, explored and developed through computation and experimentation. ICERM convenes leading scientists from academia and industry, together with students and early-career researchers, in programs that generate new mathematics and that accelerate the development of tools and technology arising from new mathematics. The institute has a special focus on training the next generation of computationally skilled mathematicians and preparing them to enter a variety of scientific careers. ICERM places a priority on increasing the participation of members of groups under-represented in mathematics. To support these goals, ICERM provides a sophisticated research infrastructure as well as access to high performance computing and current software resources.

ICERM programs support and broaden the relationship between mathematics and computation by expanding the use of computational and experimental methods in mathematics, by supporting theoretical advances related to computation, and by addressing, through mathematical tools, research, and innovation, problems posed by the use of the computer. Experimentation has historically been a driver of advances in fundamental research in mathematics. New technology opens up new opportunities for research and discovery through experimentation; ICERM programs aim to identify and develop such opportunities. Mathematics interacts with an ever widening range of scientific and industrial enterprises, and computation is at the heart of this interaction. ICERM programs aim to develop resources for data-driven explorations in both pure and applied areas of mathematics.

ICERM serves as a national resource for mathematical activities, events, and programs that have strong computational/experimental components and that aim to catalyze development of new mathematics as well as new ways of doing mathematics. The institute runs thematic research programs together with associated international conferences, provides support for postdoctoral fellows and graduate students in residence during semester-long programs, hosts team-based research programs to train graduate and undergraduate students of mathematics in the use of experimental methodologies and computer-aided tools, and conducts a variety of independent workshops, outreach activities, and special events.

The notion of "bounded combinatorics" in the complex of curves on a surface controls geometry in a variety of settings. Understanding and bounding how simple closed curves on a surface can project to subsurfaces provides tools to construct models for hyperbolic 3-manifolds, classify Weil-Petersson geodesics, and elucidate the fine topological structure in boundaries of deformation spaces. Indeed, the existence and uniqueness of hyperbolic structures on 3-manifolds gives little information about their geometric features and their connection to topological properties of the manifold. Through his expansion of the use of the bi-Lipschitz Model Theorem (of the PI with Canary and Minsky), the PI will explore geometric models for arbitrary closed hyperbolic 3-manifolds and connect their structure to combinatorial features of the manifold. This ongoing project with collaborators Minsky, Namazi and Souto will produce explicit models for Heegaard splittings, and a new weak geometrization for 3-manifolds arising as infinite gluings with bounded combinatorics. Further, the PI will apply coarse methods in the study of the mapping class group via the curve complex and its associated "hierarchy paths" to understand the large scale structure of the Weil-Petersson metric on Teichmueller space, a fundamental object whose large scale geometry remains mysterious despite many investigations. Finally, the PI will exhibit further features of the deformation space of a hyperbolic 3-manifold, generalizing our study of the local topology of deformation spaces with Bromberg, Canary, and Minsky, and undertaking generalized studies of central compactness theorems for deformation spaces in the context of the curve complex.

In mathematics, the study of dynamical systems seeks to describe chaotic phenomena in simple terms. Sometimes dynamical systems can exhibit a kind of rigidity, where a small tweak or perturbation does not affect the long term behavior of the system. In the context of understanding our own three-dimesnsional universe and what kind of structures three-dimensional spaces can have, structures give rise to these rigid dynamical systems. The recent work of Grisha Perelman solving the famous Poincare Conjecture has ensured this study applies to virtually all three-dimensional spaces. When a space is rigid, one can understand it completely via "coarse" information, via so-called "models." In a recent result of the PI with R. Canary and Y. Minsky, such models were used to classify all constantly negatively curved, or "hyperbolic" three-dimensional spaces of infinite volume that are tame in a certain sense. The classification result solved a long-standing conjecture of William Thurston, and opened the door to developing a more detailed and complete picture of geometries on manifolds previously considered understood. The groundwork is in place for a fundamental investigation of algebraic and topological properties of all spaces of 3-dimensions and how these properties interrelate.

The conference Mapping Class Groups and Teichmuller Theory - in Ramat Hanadiv, Israel, May 7-14, 2014 will make possible the participation of a large number of early career mathematicians at an important conference to assess the state of the field and build new connections between mathematicians in the areas of geometry and topology. An important area in geometry is the study of the range of all possible shapes a given type of geometric object can take. For example, one might study the range of all possible shapes of triangles, circles, or other kinds of geometric objects. Teichmüller theory takes as its objects 2-dimensional surfaces, and studies the shapes of these surfaces. When a shape is symmetric, these symmetries are recorded by an algebraic notion called the "mapping class group". From topology, to geometry, to physics, the interaction of these notions continues to produce new fruitful topics of study. Indeed Teichmüller space (the "space of shapes") has its own geometry, and symmetries of the shapes appear as symmetries of Teichmüller space. This conference will seek to introduce new researchers to emerging aspects and connections in this burgeoning field.


The study of mapping class groups and Teichmüller theory lies at the intersection of a variety of important mathematical topics, including geometry, topology, geometric group theory, and representation theory. New research threads in these different subject areas have emerged recently, and the time is right to revisit the connections between such topics to assess parallels between their development. While there are few mathematicians in Israel that work directly in the subject, there is a larger group of mathematicians that work in closely related areas, and great opportunity for enhancing connections between these working groups and those that exist elsewhere. The proposed conference on Mapping Class Groups and Teichmüller Theory, in Ramat Hanadiv, Israel, will provide a unique and new opportunity for scholars from a broad range of career stages to meet in Israel and interact with Israeli mathematicians in related areas, and will serve to broaden the geographic and intellectual base of the field.

Conference URL: http://www.math.technion.ac.il/cms/decade_2011-2020/year_2013-2014/mapping/

For much of the twentieth century, the challenge to describe the shape of three dimensional spaces was viewed as an algebraic problem: indeed it is through algebra that we first distinguish the (topological) structure of a sphere from that of a doughnut. Work of William Thurston brought the geometry of such spaces more clearly into view as a central feature to explore. The triumph of this approach was Perelman's proof of Thurston's geometrization conjecture, that each three-dimensional manifold could be naturally broken up into pieces, each with a uniform geometry. The study of these geometries is frequently reduced, via a notion of rigidity, to considering simple combinatorial structures arising from loops on surfaces. The proposed research will explore how these structures predict volume, diameter, length and other aspects of the geometry, relating to notions from quantum physics.

A longstanding connection between volume of hyperbolic three-manifolds and Weil-Petersson distance relied on a combinatorial comparison via the pants graph which organizes maximal multicurves on a surface, yet other connections were known to arise from work of Witten via renormalized volume. Recent work of Schlenker made this connection explicit in the context of quasi-Fuchsian manifolds, and PI's proposed work will develop this idea further to explicitly relate fibered 3-manifolds and translation distiance. More generally, a primary project in the proposed research is to solidify and extend the connection between combinatorics and geometry in closed manifolds, and to continue to investigate the structure of deformation spaces of Kleinian groups with these tools. As an example of the power of these techniques, bi-Lipschitz models for 'random' Heegaard splittings provide a full solution (with Rivin and Souto) to the conjecture of Dunfield and Thurston that random Heegaard splittings are almost surely hyperbolic and their volume grows linearly. Finally, the PI will pursue projects with Minsky, and with Modami, Leininger and Rafi on the role of combinatorics in the geometry of geodesics in the Weil-Petersson metric, investigating disparities between Teichmuller and Weil-Petersson geodesics in terms of unique ergodicity.

This project creates an institute at Brown University that brings together the disciplines of mathematics, statistics, and theoretical computer science to define and refine the foundational landscape of the emerging area of data science. The institute sponsors focused activities organized by small groups of researchers that cut across disciplinary boundaries. It connects with and informs a broad range of students at the undergraduate and graduate levels, working in a wide area of domain areas, from neuroscience, to genomics, to climate modeling, to public policy. Theoretical developments can improve diagnostic imaging and tumor classification, can develop improved models for neural structure, and can even inform findings regarding food stamps and recidivism in Rhode Island.

The mission of the institute is to foster development and principled application of theory and methods of big data to discover, refine, and validate underlying theoretical models that govern a system or data-generating process, which in turn improve predictions of new outcomes. Scientific projects in "causal and model-based inference," "data analysis on massive networks," and "geometric and topological methods to analyze and visualize complex data" drive home the role of the model, and its continuous refinement, in data analysis. Rather than seeking better "black-boxes" for analysis, the institute will emphasize the role of the "investigator-in-the-loop" interrogating the entirety of the data pipeline, seeking theoretical improvements and implications. It connects to the Brown Data Science Initiative and the Institute for Computational and Experimental Research in Mathematics (ICERM). Funds for the project come from CISE Computing and Communications Foundations, MPS Division of Mathematical Sciences, Growing Convergent Research, and EPSCoR. (Convergence can be characterized as the deep integration of knowledge, techniques, and expertise from multiple fields to form new and expanded frameworks for addressing scientific and societal challenges and opportunities. This project promotes Convergence by bringing together communities representing many disciplines including mathematics, statistics, and theoretical computer science as well as engaging communities that apply data science to practical research problems.)

PROJECT  SUMMARY  There  are  limited  formal  opportunities  for  biomedical  and  health  science  trainees  to  acquire  essential  data  science  skills.  Along  with  the  growth  in  biomedical  and  health  data,  there  will  be  a  need  for  researchers  to  develop  approaches  for  leveraging  them  to discover and validate hypotheses. The Training and Teaching for  Transforming  Big  Data  to  Knowledge  (T3BD2K)  Initiative  will  address  these  needs  by:  (1)  Developing  a  ten­week  short  course  to  teach  pragmatic  data  science  skills?  and  (2)  Coordinating  data  science  education  across  training  programs  in  Rhode  Island.  The  T3BD2K Initiative will directly impact all currently funded NIH  training  programs  and  be  developed  synergistically  with  the  Data  Science  Initiative  at  Brown  University.  A  significant  artifact  of  this  training  program  will  be  a  publicly  available  curriculum  and  associated  teaching  materials  aimed  at  biomedical  trainees  for  providing  pragmatic  training  in  biomedical  informatics  and  data  science  (e.g.,  biostatistics,  computer  science,  and applied mathematics), alongside fundamental principles of  team science, that could be utilized by students nationally. All course materials will be made available using a  Creative Commons CC­BY­4.0 license, and made available through publicly accessible systems (GitBooks and  GitHub).  Additionally,  all  lectures  will  be  video  captured,  with the goal of transitioning a traditional format mode  for  teaching  the  course  to  a  ?flipped?  classroom.  The  course  will  culminate  in  a  symposium that showcases  participants applications of data science in biomedical and health contexts, with project abstracts, posters, and  oral  presentations  being  made  available  through  a  publicly  accessible  data  repository  that  is  maintained  by  Brown  University  (the  Brown  Digital  Repository  [BDR]).  The  overall  success  of  the  T3BD2K Initiative will be  poised to transform pragmatic data science training and teaching in Rhode Island, with the potential to inform  biomedical  data  scientist  training  nationally.        

This project will build a transformative bridge between data science and neuroscience. These two young fields are driving cutting-edge progress in the technology, education, and healthcare sectors, but their shared foundations and deep synergies have yet to be exploited in an integrated way - a new discipline of "data neuroscience." This integration will benefit both fields: Neuroscience is producing massive amounts of data at all levels, from synapses and cells to networks and behavior. Data science is needed to make sense of these data, both in terms of developing sophisticated analysis techniques and devising formal, mathematically rigorous theories. At the same time, models in data science involving AI and machine learning can draw insights from neuroscience, as the brain is a prodigious learner and the ultimate benchmark for intelligent behavior. Beyond fundamental scientific gains in both fields, the project will produce additional outcomes, including: new collaborations between universities, accessible workshops, graduate training, integration of undergraduate curricula in data science and neuroscience, research opportunities for undergraduates that help prepare them for the STEM workforce, academic-industry partnerships, and enhanced high-performance computing infrastructure.

The overarching theme of this project is to develop a two-way channel between data science and neuroscience. In one direction, the project will investigate how computational principles from data science can be leveraged to advance theory and make sense of empirical findings at different levels of neuroscience, from cellular measurements in fruit flies to whole-brain functional imaging in humans. In the reverse direction, the project will view the processes and mechanisms of vision and cognition underlying these findings as a source for new statistical and mathematical frameworks for data analysis. Research will focus on four related objectives: (1) Distributed processing: reconciling work on communication constraints and parallelization in machine learning with the cellular neuroscience of motion perception to develop models of distributed estimation; (2) Data representation: examining how our understanding of the different ways that the brain stores information can inform statistically and computationally efficient learning algorithms in the framework of exponential family embeddings and variational inference; (3) Attentional filtering: incorporating the cognitive concept of selective attention into machine learning as a low-dimensional trace through a high-dimensional input space, with the resulting models used to reconstruct human subjective experience from brain imaging data; (4) Memory capacity: leveraging cognitive studies and natural memory architectures to inform approaches for reducing/sharing memory in artificial learning algorithms. The inherently cross-disciplinary nature of the project will provide novel theoretical and methodological perspectives on both data science and neuroscience, with the goal of enabling rapid, foundational discoveries that will accelerate future research in these fields.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.