Award Abstract # 1651581
CAREER: Visualizing Mathematical Structures in High-Dimensional Space

NSF Org: IIS
Division of Information & Intelligent Systems
Recipient: UNIVERSITY OF LOUISVILLE
Initial Amendment Date: April 3, 2017
Latest Amendment Date: April 20, 2021
Award Number: 1651581
Award Instrument: Continuing Grant
Program Manager: Hector Munoz-Avila
IIS
 Division of Information & Intelligent Systems
CSE
 Directorate for Computer and Information Science and Engineering
Start Date: April 1, 2017
End Date: December 31, 2023 (Estimated)
Total Intended Award Amount: $499,971.00
Total Awarded Amount to Date: $499,971.00
Funds Obligated to Date: FY 2017 = $298,330.00
FY 2020 = $100,093.00

FY 2021 = $101,548.00
History of Investigator:
  • Hui Zhang (Principal Investigator)
    hui.zhang@louisville.edu
Recipient Sponsored Research Office: University of Louisville Research Foundation Inc
2301 S 3RD ST
LOUISVILLE
KY  US  40208-1838
(502)852-3788
Sponsor Congressional District: 03
Primary Place of Performance: University of Louisville
2301 South Third Street
Louisville
KY  US  40292-0001
Primary Place of Performance
Congressional District:
03
Unique Entity Identifier (UEI): E1KJM4T54MK6
Parent UEI:
NSF Program(s): Info Integration & Informatics,
EPSCoR Co-Funding
Primary Program Source: 01001718DB NSF RESEARCH & RELATED ACTIVIT
01002021DB NSF RESEARCH & RELATED ACTIVIT

01002122DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 1045, 7364, 9150
Program Element Code(s): 736400, 915000
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.070

ABSTRACT

While the most efficient method for communicating math concepts is the use of real world objects or virtual manipulatives, creating illustrations of mathematical phenomena beyond three dimensions has been a particularly challenging task and many mathematical phenomena have thus only existed in the mathematician's mind. This research seeks to investigate the question of whether computer graphics techniques can help expert mathematicians and general public to visualize and communicate the higher-dimensional mathematical objects and their deformations. The ultimate goal of this research is to establish a mechanism by which an expert human viewer can manipulate a higher-dimensional geometric object that they can only see in part, i.e., via a slice or projection into two or three dimensions. Theoretical contributions of this research will impact and improve methods in mathematical visualization, particularly graph visualization, computer aided design, and large-scale spatial visualization, while the project deliverables will have direct and transformative impact on the ability of mathematicians to study higher-dimensional objects, and to communicate what they have learned in person, in presentations, and in archival works. The success of this project will ultimately translate into more rapid advancement in areas of pure and applied mathematics where higher dimensional geometry plays an important role, and provide mathematical visualization tool sets to facilitate college and K-12 students in their geometry courses. The project will make research outcomes including open source software freely available, and will disseminate mathematical sciences to the general public by rendering and presenting pedagogical animations at the University of Louisville Planetarium. The project also includes integrated educational and outreach activities for K-12, undergraduate, and graduate students.

The research will explore an interactive visualization paradigm that makes use of energy-driven self-deformable object models embedded in higher dimensions, supplemented by reduced-dimensional analogies for expert human viewers to guide the deformations towards their final goals. The investigators will begin by assigning a deformation energy to the higher-dimensional object, so that the aspects of the configuration that are unseen and unfamiliar can be controlled in a principled and well-posed manner by constraints or manipulations on the aspects of the configuration that are seen and familiar in our dimensions. Often times mathematical simulations are concerned with heavily vectorized operations performed over and over in a large number of iterations. The project will exploit hardware-enabled parallelism to accelerate mathematical simulations, and to extract key moments where successive terms differ by one critical change to represent and analyze various mathematical evolutions. By combining guided relaxation method and accelerated computation, this research can potentially make a novel contribution to building intuition about classes of geometric and topological problems that otherwise would be nearly impossible to communicate, perceptualize, and disseminate; and can potentially further the entire concept of the assistance and empowerment of human understanding by computer methods, specifically via the power of visual and computational spatial visualization tools. All outcomes of this project, including technical reports, research articles, links to educational and outreach activities, open source software, and pedagogical animations will be accessible from the project's web site (http://www.cecsresearch.org/vcl/nsf1651581/).

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

Note:  When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

Juan Lin, Hui Zhang "Visualizing Mathematical Knot Equivalence" Electronic Imaging, Visualization and Data Analysis 2019 , v.2019 , 2019 https://doi.org/10.2352/ISSN.2470-1173.2019.1.VDA-683 Citation Details
Lin, Juan and Zhang, Hui "Accelerating visual communication of mathematical knot deformation" Journal of Visualization , v.23 , 2020 https://doi.org/10.1007/s12650-020-00663-w Citation Details
Lin, Juan and Zhang, Hui "View Selection in Knot Deformation" 2019 IEEE International Conference on Big Data (Big Data) , 2019 10.1109/BigData47090.2019.9005987 Citation Details
Lin, Juan and Zhang, Hui "Visually Communicating Mathematical Knot Deformation" VINCI'2019: Proceedings of the 12th International Symposium on Visual Information Communication and Interaction , 2019 10.1145/3356422.3356438 Citation Details
Liu, Huan and Zhang, Hui "A Flipbook of Knot Diagrams for Visualizing Surfaces in 4Space" Computer Graphics Forum , v.41 , 2022 https://doi.org/10.1111/cgf.14545 Citation Details
Liu, Huan and Zhang, Hui "A Suggestive Interface for Untangling Mathematical Knots" IEEE Transactions on Visualization and Computer Graphics , v.27 , 2021 https://doi.org/10.1109/TVCG.2020.3028893 Citation Details
Ruan, Guangchen and Zhang, Hui "Parallelized Topological Relaxation Algorithm" 2019 IEEE International Conference on Big Data (Big Data) , 2019 10.1109/BigData47090.2019.9006309 Citation Details
Zhang, Hui "Performance Engineering for Scientific Computing with R" International Journal on Data Science and Technology , v.4 , 2018 10.11648/j.ijdst.20180402.11 Citation Details
Zhang, Hui and Liu, Huan "Relaxing topological surfaces in four dimensions" The Visual Computer , v.36 , 2020 https://doi.org/10.1007/s00371-020-01895-5 Citation Details

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

The research explored 1) an interactive visualization paradigm that makes use of energy-driven self-deformable object models embedded in 3- and higher dimensions, supplemented by reduced-dimensional analogies for expert human viewers to guide the deformations towards their final goals; 2) an suggestive Reidemester move interface for mathematically deforming knots, by combining analytics, recommendation interface, and visualization; 3) a novel interface to visualize topological surfaces embedded in 4-space by slicing and visualizing their internal structures.

The research began by assigning a deformation energy to the higher-dimensional object, so that the aspects of the configuration that are unseen and unfamiliar can be controlled in a principled and well-posed manner by constraints or manipulations on the aspects of the configuration that are seen and familiar in our dimensions. Often times mathematical simulations are concerned with heavily vectorized operations performed over and over in a large number of iterations. The project exploited hardware-enabled parallelism to accelerate mathematical simulations, and to extract key moments where successive terms differ by one critical change to represent and analyze various mathematical evolutions. By combining guided relaxation method and accelerated computation, this research can potentially make a novel contribution to building intuition about classes of geometric and topological problems that otherwise would be nearly impossible to communicate, perceptualize, and disseminate; and can potentially further the entire concept of the assistance and empowerment of human understanding by computer methods, specifically via the power of visual and computational spatial visualization tools.

Starting the third year, the investigator was inspired a series of books written by Scott Carter (Surfaces in 4-space; how surfaces intersect in space) and started to investigate a "slicing and plotting" interface to expose and visualize the underlying structures of surfaces embedded in 4D. This research direction showed very promising results, as it reduces a smooth 4D surface that appears to be a complicated and self-intersecting surface in 3D, into a series of key frames of 2D curves to represent its structure.

The funded project trained and educated 3 phd students, engaged 3 REU students in summer experience. The research published 4 journal articles, in Journal of Visualization, the Visual Computer, the Journal of Imaging Science and Technology, and Computer Graphics Forum. The project in NCE period also expanded our research interests to using 2D sketching interface for creating 4D surfaces, it also led a ISVC conference publicatoin.

All outcomes of this project, including technical reports, research articles, links to educational and outreach activities, open source software, and pedagogical animations are accessible from the project's web site (http://www.cecsresearch.org/vcl/nsf1651581/)


Last Modified: 11/01/2023
Modified by: Hui Zhang

Please report errors in award information by writing to: awardsearch@nsf.gov.

Print this page

Back to Top of page