Robert Ghrist - US grants

PROPOSAL: DMS-9971629

PRINCIPAL INVESTIGATOR: Robert Ghrist

ABSTRACT:

The principal theme of this project is the development and refining of topological techniques for answering dynamical questions about inviscid fluid flows (the Euler Equations) in the full three-dimensional setting. Through a correspondence between Beltrami fields and the Reeb dynamics of contact structures, one can import techniques from contact topology and pseudoholomorphic curve theory to answer questions of existence of solutions and of forced recurrence. In addition, knot-theoretic techniques shed light on the global properties of the Euler equations on Riemannian manifolds. Goals of this project include the resolution of the Weinstein Conjecture on solid tori (which would imply that steady force-free fields in a tokamak have closed field lines), the spectrum of knot types which can/must appear as closed flowlines in steady Euler flows, and the implications of the tight/overtwisted dichotomy in the hydrodynamical context with respect to hydrodynamic instability and energy minimization.

One of the enduring challenges in science is the comprehension and taming of fluid behavior: from tornadoes to ocean currents, we live in a world of complicated fluid dynamics. All of these phenomena have mathematical manifestations --- projections onto a simplified abstract setting whose analysis yields clues to the workings of the physical world. This project will focus on the equations governing ``perfect'' fluids: models appropriate for certain non-viscous fluids and plasmas alike. Emerging techniques from global geometry and topology provide tools for attacking problems heretofore beyond the reach of analysis. Surprisingly, such seemingly unrelated questions as ``How do strings tie up into different knots'' and ``What does geometry in four dimensions look like'' are closely tied to questions of recurrence and stability in steady fluid flows.

0089631
Mischaikow

This award supports a three-year collaborative research project between Professor Konstantin Mischaikow of the Georgia Institute of Technology and Professor Hiroe Oka of Ryukoku University in Japan. A perennial challenge in science and engineering is the determination and description of coherent behaviors in systems with large (perhaps infinite) numbers of degrees of freedom. Given the ubiquity of dynamical systems as models for physical processes, it is perhaps not surprising that the primary obstacle to a coherent, effective program of dynamical systems applications is the deeply entangled manner in which low-dimensional phenomena are embedded within high-dimensional systems. The goal is to extend and develop new and effective topological techniques for extracting coherent low-dimensional dynamical phenomena from potentially high-dimensional systems. The investigators' primary tool is the index theory of Conley, which collates the topological features of the expanding portion of the dynamics into algebraic-topological data structures (e.g., homology and cohomology). The three areas of cooperative research for the theory are: 1) construction of an effective topological singular perturbation theory for reducing the dynamics of a fast-slow system to the slow manifold; 2) adaptation of the index theory to the context of complicated flows with an emphasis on particle paths and embeddings; and 3) reconstruction of dynamical information from time-series data, particularly as a tool for rigorous numerical integration of systems with noise.

The project brings together the efforts of two laboratories that have complementary expertise and research capabilities. Through the exchange of ideas and technology, this project will broaden our base of basic knowledge and promote international understanding and cooperation. The researchers plan to publish results of their research in scientific journals and report on the findings at scientific meetings.

DMS-0134408
Robert W. Ghrist

The efficacy of topological methods in contemporary applied
mathematics is primarily attributable to the fact that topological
features of a system are inherently robust and global. This
project focuses on a technology transfer from contemporary ideas
in topology, geometry, and dynamics to bear upon application
domains which include the following: First, Robotics: tools
from configuration space theory, CAT(0)complexes, and
computational topology will be directed toward specific
problems in reconfigurable robotics, sensor-based navigation
of mobile agents, and self-assembly systems. Second, Parabolic
coupled systems: a Morse-theoretic homotopy index for braids
will be used to solve parabolic variational problems arising in pattern-formation PDE's, discrete Lagrangian mechanics, and
coupled oscillators. A Floer-theoretic extension of the braid
index will also be developed for infinite dimensional systems.
Third, Hydrodynamics: tools from contact geometry and topology
will be directed toward solving global problems of the dynamics
and stability of Eulerian fluid flows in dimensions higher
than two.

In most systems of interest in science and engineering,
multiple cooperative tasks must be globally coordinated.
A common thread is that whether the tasks involve
macro-scale robots, micro-scale devices, coupled oscillators,
or fluid particles, there is an abstract space of configurations
lurking behind the physical phenomena. Unearthing and
examining those properties of physically-motivated
configuration spaces which capture the global features, the
topology, geometry, and dynamics, holds the promise of
providing global tools which transcend the physical
instantiation of the system at hand: ostensibly different
systems possess similar topological underpinnings.
The research component of this project is the development
of contemporary topological and global-geometric techniques
for analyzing the dynamics and coordination of systems of
interest in engineering and computer science. The overall
goal is an effective technology transfer from cutting-edge
perspectives in topology to bear upon systems in application
domains which include robotics, mechanics, and fluid dynamics.
This is combined with a blend of pedagogical service across
graduate, undergraduate, and high school levels, featuring
a focused research group on topological robotics and a
high-school outreach program of expository lectures on the
relevance and joy of mathematical research.

DMS-0138420
Jarek Rossignac

We propose to develop the theoretical foundations and a
set of practical computing tools for the automatic analysis
of time-evolving shapes. Given a family of surfaces that
represent the boundary of a 3D shape whose geometry and
topology change with time, we propose to construct a
higher-dimensional multiresolution representation, which
we have named Atlas Transition Diagram, abbreviated ATD,
that will identify and track the morphological and
topological features of the 3D shape as they evolve with
time and with resolution. Our ATD will also associate a
chart to each feature, thus providing a surface-parameterization
that naturally follows the branches and handles of each
shape in the family. The charts evolve smoothly with time
and resolution and are topologically glued together at
their common boundaries to provide a continuous mapping,
S(r,t,u,v;f); f), that, given a feature ff, a resolution rr,
a time tt, and two parameters u and v, will identify a
unique point on the surface and will allow us to trace
its evolution with rr and tt. The theoretical underpinnings
and algorithmic designs that will lead to a practical
implementation of an efficient system for building and
querying ATDs go far beyond simple extensions of Morse
theory, of surface segmentation approaches, and of
multi-resolution techniques, which have so far been
mainly explored for static surfaces in 3D.

Evolving surfaces are important to many scientific
and engineering disciplines, including medicine,
developmental biology, cell biology, computational
fluid dynamics and computer aided design. They may
for example represent the growth of a tumor, the shifting
in position of a vortex over an airplane wing, or the
budding of fingers on the hand of a human embryo. We
propose to develop and integrate a collection of
theoretical and algorithmic tools for the analysis
and automated visualization of such evolutions. These
tools will allow us to partition the evolving surface
into features upon which a high-level description of
the shape of the surface and of its evolution will be
based. Furthermore, they will allow us to track their
points, and thus surface properties, through time and
to better visualize their evolutions through texture
maps that continuously evolve with the features and
highlight their boundaries and natural orientation.
Finally, these tools will help us support queries
about the time and nature of topological changes
in the evolving surface, which may be important for
the automatic analysis and retrieval of scientific
datasets. To achieve these results, we propose to
build a surface representation that is controlled
independently or simultaneously by time and resolution
and to decompose the time/resolution domain into cells
where the surface topology (number of components and
through holes) and its partition into features remain
constant. To validate our theoretical contributions
and to increase their impact on the community, we plan
to develop a prototype implementation for animated
objects with triangulated boundaries. We expect to
make the source code of this implementation and its
programming interface publicly available. We envision
exploring collaborations with application developers
in Science, Engineering, Medicine and Biology to help
us refine and validate this approach.

Proposal Title: PECASE: Topological Methods in Applied Mathematics Institution: University of Illinois at Urbana-Champaign

The efficacy of topological methods in contemporary applied mathematics is primarily attributable to the fact that topological features of a system are inherently robust and global. This project focuses on a technology transfer from contemporary ideas in topology, geometry, and dynamics to bear upon application domains which include the following: First, Robotics: tools from configuration space theory, CAT(0)complexes, and computational topology will be directed toward specific problems in reconfigurable robotics, sensor-based navigation of mobile agents, and self-assembly systems. Second, Parabolic coupled systems: a Morse-theoretic homotopy index for braids will be used to solve parabolic variational problems arising in pattern-formation PDE's, discrete Lagrangian mechanics, and coupled oscillators. A Floer-theoretic extension of the braid index will also be developed for infinite dimensional systems. Third, Hydrodynamics: tools from contact geometry and topology will be directed toward solving global problems of the dynamics and stability of Eulerian fluid flows in dimensions higher than two.
In most systems of interest in science and engineering, multiple cooperative tasks must be globally coordinated. A common thread is that whether the tasks involve macro-scale robots, micro-scale devices, coupled oscillators, or fluid particles, there is an abstract space of configurations lurking behind the physical phenomena. Unearthing and examining those properties of physically-motivated configuration spaces which capture the global features, the topology, geometry, and dynamics, holds the promise of providing global tools which transcend the physical instantiation of the system at hand: ostensibly different systems possess similar topological underpinnings. The research component of this project is the development of contemporary topological and global-geometric techniques for analyzing the dynamics and coordination of systems of interest in engineering and computer science. The overall goal is an effective technology transfer from cutting-edge perspectives in topology to bear upon systems in application domains, which include robotics, mechanics, and fluid dynamics. This is combined with a blend of pedagogical service across graduate, undergraduate, and high school levels, featuring a focused research group on topological robotics and a high-school outreach program of expository lectures on the relevance and joy of mathematical research.


This project was originally funded as a CAREER award, and was converted to a Presidential Early Career Award for Engineers and Scientists (PECASE) award in May 2004.

This proposal describes an aggressive program of research in computational topology with a focus on computing and characterizing shortest paths in a variety of domains relevant to applications in robotics, coordination, and locomotion. We will investigate efficient descriptions of shortest-path information and geodesic structures in spaces with different types of constraints. The three main goals of our project are (1) developing algorithms to compute optimal paths, cycles, and other one-dimensional substructures, primarily in two-dimensional surfaces; (2) applying tools from Alexandrov geometry and topology to more efficiently characterize and compute shortest paths in non-positively curved spaces; and (3) developing languages to characterize spaces of optimal paths for motion systems with mechanical and/or nonholonomic constraints. Our proposed work draws on techniques from low-dimensional geometric and algebraic topology, combinatorial group theory, computational geometry, and non-holonomic motion planning.

At a high level, our research focuses on techniques for computing the cheapest way to move from one point to another in a variety of interesting spaces. Consider, for example, a collection of robots moving around a factory floor. The positions of the robots can be encoded as a single point in a high-dimensional configuration space. The geometry of this space is governed by certain mechanical and/or kinematic constraints; for example, robots must never collide with each other, and they have a limited rate of acceleration. A shortest path in the configuration space describes a set of motions of the robots from one set of locations to another that is as efficient as possible. We plan to develop algorithms that compute such shortest paths quickly, by exploiting the overall ``shape" of the underlying space.

The grand challenge in efficiently harvesting and converting solar radiation into electricity lies in engineering materials on multiple length scales with architectures that direct the flow of energy and the transfer and transport of charge, as in naturally occurring light harvesting systems. Organic-inorganic hybrids, prepared from functional, electro-active organic and nanostructured inorganic materials, combine desirable and tunable chemical and physical properties of the constituent organic and inorganic building blocks in a single composite, making them promising systems for solar technologies. Hybrid materials incorporate the low-cost, large-area processing and high absorbance and quantum efficiencies of organic materials with the adjustable optical properties, high carrier conductivities, and good photostability of inorganic nanostructures. Solar photovoltaic and luminescent solar concentrator technologies will be dramatically advanced if the organic and inorganic building blocks of hybrid structures can be positioned and oriented on the nanometer scale to regulate the competitive processes of charge transfer and transport, emission, and energy transfer.

Hybrid organic-inorganic materials promise one of the best architectures for ultra-low-cost photovoltaic devices. Currently, the efficiency of hybrid photovoltaic devices is limited by the availability of red-absorbing, high-mobility organic and inorganic components (to match the solar spectrum and efficiently collect charge) and of composites with structures that achieve high surface area junctions, yet form well-connected organic and inorganic pathways. This project aims to produce significantly improved hybrid structures for photovoltaics. Improved hybrid materials may also enable creation of high-efficiency luminescent solar concentrators, which currently are limited in performance by materials challenges; organic and inorganic materials alone have not been found to satisfy the broad-spectrum collection, near-unity photoluminescence efficiency, low re-absorption, and good photostability required.

This project brings together advances in chemical synthesis, mathematical modeling, and self-organization to control the position and orientation of organic and inorganic building blocks, exploiting advances at the frontier of chemistry, materials science, and mathematics. We will combine precisely controlled 1) molecular and supramolecular dendrimeric systems tailored to assemble with different structural motifs and 2) nanocrystals of tunable size, shape, and composition that self-assemble into single and multi-component superlattices. Structural, optical, and electrical probes will be combined with mathematical modeling of the effects of interface geometry to optimize charge transfer and transport, emission, and energy transfer. The results will enable engineering of organic-inorganic materials that will be integrated in photovoltaic devices and luminescent solar concentrators.

More broadly, the research program will develop new synthetic methods and mathematical formalisms for the self-assembly of hybrid materials with tailored architectures that is important to provide materials with superior structural, electronic, and optical properties. These materials have applications in imaging, therapeutics, and information technology, in addition to energy harvesting. The project's emphasis on mathematical techniques for engineered self-assembling systems offers the potential for impact in robotics and biological systems. The project will also electronically and optically probe and establish mathematical models of the behavior of organic-inorganic heterojunctions key to their application in a range of electronic and optical devices.

Recent advances in artificial intelligence have led to significant progress in our ability to extract information from images and time sequences. Maintaining this rate of progress hinges upon attaining equally significant results in the processing of more complex signals such as those that are acquired by autonomous systems and networks of connected devices, or those that arise in the study of complex biological and social systems. This award establishes FINPenn, the Center for the Foundations of Information Processing at the University of Pennsylvania. The focus of the center is to establish fundamental theory to enable the study of data beyond time and images. The center's premise is that humans' rich intuitive understanding of space and time may not necessarily be applicable to the processing of complex signals. Therefore, matching the success in time and space necessitates the discovery and development of foundational principles to guide the design of generic artificial intelligence algorithms. FINPenn will support a class of scholar trainees along with a class of visiting postdocs and students to advance this agenda. The center will engage the community through the organization of workshops and lectures and will disseminate knowledge with onsite and online educational activities at the undergraduate and graduate level.

FINPenn builds on two observations: (i) To understand the foundations of data science it is necessary to succeed beyond Euclidean signals in time and space. This is true even to understand the foundations for Euclidean signal processing. (ii) Humans live in Euclidean time and space. To succeed in information processing beyond signals with Euclidean structure, operation from foundational principles is necessary because human intuition is of limited help. For instance, convolutional neural networks have found success in the processing of images and signals in time but they rely heavily on spatial and temporal intuition. To generalize their success to unconventional signal domains it is necessary to postulate fundamental principles and generalize from those principles. If the generalizations are successful they not only illuminate the new application domains but they also help establish the validity of the postulated principles for Euclidean spaces in the tradition of predictive science. The proposers further contend that the foundational principles of data sciences are to be found in the exploitation of structure and the associated invariances and symmetries that structure generates. The initial focus of the center is in advancing the theory of information processing in signals whose structure is defined by a group, a graph, or a topology. These three types of signals generate three foundational research directions which build on the particular strengths of the University of Pennsylvania on network sciences, robotics, and autonomous systems which are areas in which these types of signals appear often.

This project is part of the National Science Foundation's Harnessing the Data Revolution (HDR) Big Idea activity.

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.

Recent advances in deep learning have led to many disruptive technologies: from automatic speech recognition systems, to automated supermarkets, to self-driving cars. However, the complex and large-scale nature of deep networks makes them hard to analyze and, therefore, they are mostly used as black-boxes without formal guarantees on their performance. For example, deep networks provide a self-reported confidence score, but they are frequently inaccurate and uncalibrated, or likely to make large mistakes on rare cases. Moreover, the design of deep networks remains an art and is largely driven by empirical performance on a dataset. As deep learning systems are increasingly employed in our daily lives, it becomes critical to understand if their predictions satisfy certain desired properties. The goal of this NSF-Simons Research Collaboration on the Mathematical and Scientific Foundations of Deep Learning is to develop a mathematical, statistical and computational framework that helps explain the success of current network architectures, understand its pitfalls, and guide the design of novel architectures with guaranteed confidence, robustness, interpretability, optimality, and transferability. This project will train a diverse STEM workforce with data science skills that are essential for the global competitiveness of the US economy by creating new undergraduate and graduate programs in the foundations of data science and organizing a series of collaborative research events, including semester research programs and summer schools on the foundations of deep learning. This project will also impact women and underrepresented minorities by involving undergraduates in the foundations of data science.

Deep networks have led to dramatic improvements in the performance of pattern recognition systems. However, the mathematical reasons for this success remain elusive. For instance, it is not clear why deep networks generalize or transfer to new tasks, or why simple optimization strategies can reach a local or global minimum of the associated non-convex optimization problem. Moreover, there is no principled way of designing the architecture of the network so that it satisfies certain desired properties, such as expressivity, transferability, optimality and robustness. This project brings together a multidisciplinary team of mathematicians, statisticians, theoretical computer scientists, and electrical engineers to develop the mathematical and scientific foundations of deep learning. The project is divided in four main thrusts. The analysis thrust will use principles from approximation theory, information theory, statistical inference, and robust control to analyze properties of deep networks such as expressivity, interpretability, confidence, fairness and robustness. The learning thrust will use principles from dynamical systems, non-convex and stochastic optimization, statistical learning theory, adaptive control, and high-dimensional statistics to design and analyze learning algorithms with guaranteed convergence, optimality and generalization properties. The design thrust will use principles from algebra, geometry, topology, graph theory and optimization to design and learn network architectures that capture algebraic, geometric and graph structures in both the data and the task. The transferability thrust will use principles from multiscale analysis and modeling, reinforcement learning, and Markov decision processes to design and study data representations that are suitable for learning from and transferring to multiple tasks.

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.

Algebraic Topology: Methods, Computation, and Science (ATMCS) is a biennial conference on algebraic topology and its applications. The tenth ATMCS meeting will be hosted by Oxford University, the week of June 20–24, 2022. The program includes invited talks, contributed talks, and a poster session. As many as 350 participants from around the world are expected at the conference. This project will support the travel of US-based participants to the meeting.

The areas represented at the meeting will include multivariate persistent homology, directed topology, stochastic topology, inverse problems, algorithms and software development, topological data analysis (TDA) in machine learning, network data analysis, and applications of TDA. By supporting the cross pollination of ideas, the conference will directly impact research in the United States. Participant support costs will be prioritized for early-career researchers. A conference website is available at https://atmcs.web.ox.ac.uk/.

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.