| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 12, 2011 |
| Latest Amendment Date: | September 12, 2011 |
| Award Number: | 1105439 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Christopher Stark
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 15, 2011 |
| End Date: | August 31, 2016 (Estimated) |
| Total Intended Award Amount: | $149,259.00 |
| Total Awarded Amount to Date: | $149,259.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): |
TOPOLOGY, COFFES, EPSCoR Co-Funding |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The PI proposes to investigate the properties and applications of various models for random simplicial complexes. These objects are closely related to random graphs which have been of interest in mathematics for over fifty years, and have been widely used in computer science and engineering for the modeling of various networks and the internet. Viewing a graph as a one-dimensional simplicial complex, it is natural to investigate random simplicial complexes of higher dimension. In addition to basic algebraic, combinatorial, and topological properties of these objects, the PI will study a number of related objects of potential interest in applications such as robotics, including random configuration spaces and moment-angle complexes.
The theory of random graphs is a rapidly growing branch of discrete mathematics, bringing together ideas from graph theory, combinatorics, and probability theory. Random graphs have found use in computer science, engineering, physics, biology, chemistry, and the social sciences. These objects also serve within mathematics as accessible models for other, more complex random structures. Random simplicial complexes provide such a structure, one that will combine techniques and ideas from algebra and topology with those from the aforementioned fields.
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.
Outcomes included the resolution of two relatively long-standing open problems in disparate areas of mathematics. First, Cohen and Schenck positively resolved the "Chen ranks conjecture" made by Suciu in 2001. This conjecture, now theorem, provides a bridge for certain topological spaces between aspects of the algebraic structure of the fundamental group of the space and geometric and sometimes combinatorial aspects of jump loci of the cohomology ring of the space. For a number of groups of interest in various branches of mathematics and applications - groups of motions of particles, of loops, etc. - this provides a combinatorial means for determining the Chen ranks, numerical invariants of the group in question.
Second, Cohen and Vandembroucq determined the topological complexity of any nonorientable surface, the Klein bottle in particular, a measure of the number of different rules or instructions necessary to prescribe how to move from one point on the surface to another. The Klein bottle, and other (nonorientable) surfaces, familiar objects from classical geometry, may be realized as spaces of all possible configurations of certain robot arms. The topological complexity determines the complexity of motion planning - the number of instructions required to continuously pass between any two given configurations of the arm. While the study of the complexity of robot motion planning has been an active research area in topology since the introduction of topological complexity by Farber in 2003, this measure of complexity for the Klein bottle had previously defied calculation.
In addition to scientific outcomes, the project resulted in human resource development; six ? graduate students were involved in project-related summer reading and research, and one undergraduate student, who has since begun graduate study in mathematics at another institution, completed a project-related honors thesis. Additionally, Cohen attended a number of conferences and workshops, involved in participation, some organization, and dissemination of results.
Last Modified: 12/17/2016
Modified by: Daniel C Cohen
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