| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 25, 2012 |
| Latest Amendment Date: | April 25, 2012 |
| Award Number: | 1205781 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2012 |
| End Date: | June 30, 2015 (Estimated) |
| Total Intended Award Amount: | $150,151.00 |
| Total Awarded Amount to Date: | $150,151.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
400 HARVEY MITCHELL PKY S STE 300 COLLEGE STATION TX US 77845-4375 (979)862-6777 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
3368 TAMU College Station TX US 77843-3368 |
| 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): | PROBABILITY |
| Primary Program Source: |
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| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
We propose to study some known stability properties of the Gibbs measures in several mean-field spin glass models and to develop new stability properties for other models, as well as to explore what kind of information about the Gibbs measures can be deduced from their stability properties. One of the most important objectives in any given model is to understand the asymptotic structure of its Gibbs measure and in a number of models this structure is expected to be described by a version of the Parisi ultrametric Ansatz. For diluted models, such as diluted p-sat and p-spin models, the proposal concerns with a better mathematical understanding of a framework for the diluted Parisi ansatz described by a random measure on the space of measurable functions and, in particular, with finding new ways to utilize some recent stability results in these models. For perceptron type models, the proposal aims to develop new stability properties with applications to certain cavity computations.
Many models in the general area of Spin Glasses originate from the attempts to understand the behavior of various optimization problems from different branches of science (physics, computer science, biology) and, more specifically, their average or typical behavior rather than focusing on one fixed scenario. This is done by randomizing the parameters of the problem and then trying to answer several key questions using the methods from Statistical Physics and Probability Theory. In the seventies and eighties, the physicists developed a number of novel ideas to approach these very difficult questions, first, in the setting of the now famous Sherrington-Kirkpatrick model, and then later successfully applied these ideas to other models as well. The ideas of the physicists were for the most part heuristic and are often described by the German word "Ansatz" which means "an educated guess that is verified later by its results". In recent years, many of these ideas have been confirmed rigorously, especially, in the setting of the Sherrington-Kirkpatrick model. The goal of this project is to build upon recent progress and try to confirm other, even more bold, predictions of the physicists that are crucial for broader applicability of their ideas.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
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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 main motivations for the current project comes from theoretical statistical physics and, more specifically, from the theory of unusual magnetic alloys known as spin glasses. Physicists have studied various models describing these alloys and developed a general theory consisting of various striking assumptions and precise predictions on the behavior and structure of spin glasses, which later found applications well beyond the original models, especially, in the development of algorithms for solving various hard optimization problems. The purpose of this project was to advance rigorous mathematical understanding of the physicist's theories, and it was quite successful. In addition to explaining the origin of some of the central hypotheses, discoveries made in this project included new phenomena and new ideas that yielded results beyond what was previously known. A general long-standing philosophy in statistical physics is that one can extract a lot of information about a system by looking at how it reacts to small perturbations and, in this project, a rigorous mathematical version of this idea was a starting point. It has been successfully generalized and used to prove many interesting results about spin glass models. The ideas and methodologies of this project will be the key to the future development of the subject and, perhaps, will continue to bear fruit in the coming years.
Last Modified: 12/08/2015
Modified by: Dmitriy A Panchenko
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