| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 27, 2021 |
| Latest Amendment Date: | May 27, 2021 |
| Award Number: | 2110398 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Yuliya Gorb
ygorb@nsf.gov (703)292-2113 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2021 |
| End Date: | October 31, 2024 (Estimated) |
| Total Intended Award Amount: | $119,972.00 |
| Total Awarded Amount to Date: | $119,972.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
1 UNIVERSITY OF NEW MEXICO ALBUQUERQUE NM US 87131-0001 (505)277-4186 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Dept. Mathematics & Statistics Albuquerque NM US 87131-0001 |
| 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): |
COMPUTATIONAL MATHEMATICS, ANALYSIS PROGRAM |
| 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 project will develop algorithms that are useful for modern physics and quantum information. In particular, the methods developed will be important for modeling topological lasers which are applicable to photonic chips and quantum circuitry. These applications pose challenges to the computational science and in particular to the linear algebra algorithms so they can work with joint measurement in multivariable setting and incommensurate observables recognizing the theoretical limits that exist. The project will consider a variety of multivariable linear algebra algorithms, mainly those filling an immediate need in computational quantum physics and quantum information, but also those that can increase the speed and accuracy of computer shape analysis. The project will involve students and provide training in interdisciplinary projects.
This project will develop numerical methods for collections of matrices and operators arising in quantum physics and image analysis. This includes algorithms that work with finite-dimensional approximations to infinite-dimensional systems, leading to better computer modeling of quasicrystals, amorphous systems and periodic systems with defects. Methods and algorithms to be developed will be useful in the study of topological insulators, including periodically-driven systems. The project will study various forms of spectra, including variations of the local density of states, a standard tool in many areas of physics and chemistry. The anticipated work on K-theory is expected to produce new and better tools that can be used by theoretical physicists in numerical studies. At the core of these methods is the study of joint approximate eigenvectors.
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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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.
This project lead to the creation of new mathematical methods and algorithms for use in the study of topological properties in materials. Varying topological properties can lead to stable states at boundaries or at anomalies. Methods were developed to better detect these states, and to predict the degree of stability of these states, mainly based on the spectral localizer method. The spectral localizer is thought of as a mathematical probe of mathematical models of materials. This is illustrated in the first attached image. Where precise calculations of bound states was required, we use the quadratic pseudospectrum. The quadratic pseudospectrum was developed in part to facilitate what computing a partial basis of joint approximate eigenvalues.
Crucially, Loring worked with physicists to modify the algorithms to detect topology to work with increasingly realistic models of materials. These improvement are critical for the study of photonic crystals, where one must directly interpret Maxwell’s equations using unbounded operators. We were able to modify the Clifford pseudospectrum, initially applicable only to systems with open boundary conditions, so it can now can handle periodic boundary conditions. This modification has become important in topological metals to deal with phase transitions in the bulk that have nothing to do with edge states. In one project an experiment was performed that demonstrates the possibility of creating experimental probes that are modeled on the spectral localizer. See the second attached image.
Much of the affiliated physics research was connected with Alex Cerjan and his research group at Sandia National Labs. This grant helped fund student researchers, from the University of New Mexico and Florida A&M University, who worked on computational projects involving the spectral localizer.
Last Modified: 11/30/2024
Modified by: Terry A Loring
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