| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 15, 2015 |
| Latest Amendment Date: | June 9, 2017 |
| Award Number: | 1513441 |
| Award Instrument: | Continuing 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 15, 2015 |
| End Date: | June 30, 2018 (Estimated) |
| Total Intended Award Amount: | $165,000.00 |
| Total Awarded Amount to Date: | $165,000.00 |
| Funds Obligated to Date: |
FY 2017 = $55,000.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1 BERNARD BARUCH WAY NEW YORK NY US 10010-5585 (646)312-2211 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
NY US 10010-5585 |
| 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: |
01001718DB NSF RESEARCH & RELATED ACTIVIT |
| 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
A wide range of phenomena in nature, physics, and society can be viewed as the byproduct of many similar components or agents interacting in a seemingly random or disordered manner (e.g., stock markets, condensed matter in physics, prime numbers in mathematics). The focus of this project is the study of rare events, or extrema, that emerge in disordered systems. The vast number of agents and the disorder present in these complex systems make straightforward predictions impossible. The project goal is to develop tools of probability theory to improve our fundamental understanding of disordered systems and, ultimately, to lead to better statistical modeling of rare events, such as high volatility episodes in stock markets and certain kinds of phase transitions in physics.
Specifically, the aim of this project is to extend the theory of extreme values in probability to random systems with strong and complex correlation structure. The first objective is to derive fine asymptotics for the maxima of characteristic polynomials of random matrices and for the local maxima of the Riemann zeta function (which controls the distribution of prime numbers). These systems are examples of stochastic processes with logarithmically decaying correlations. The second part is a study of extrema of spin glasses, an important class of physical systems with more complex correlations that includes disordered magnets. The main objective there is to find a new approach to investigate the structure of the Gibbs states of the models. In addition, the nature and the existence of phase transitions for spin glasses in finite dimension (which are realistic models of disordered magnets) will be studied by extending methods based on fluctuations of relevant thermodynamic quantities.
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.
A wide range of phenomena in nature, physics, and society can be viewed as the byproduct of many similar components or agents interacting in a seemingly random or disordered manner (e.g. stock markets, condensed matter in physics, prime numbers in mathematics). The focus of this project was the study of rare events, or extrema, that emerge in a class of physical systems as well as in some important functions of mathematics that encode the location of the prime numbers.
The main outcome of the project was the development of a mathematical method to evaluate with high accuracy the extrema of a class of random processes called log-correlated processes that appear in physics and in mathematics. In particular, we were able to apply the methodology to the study of the Riemann zeta function, arguably the most important function in mathematics. The large values of the Riemann zeta functions have an impact on how many primes one can find in a large interval of natural numbers, but they are still widely misunderstood. The project spurred new interdisciplinary connections between number theorists and probabilists in the US to tackle this problem. In particular, it proposes new directions of research to the study of the large values of the Riemann zeta function.
Another parallel outcome of the project is the development of a new approach to the study of phase transition of spin glasses, an important class of physical systems with complex correlations that includes disordered magnets. The interests for these systems go beyond physics and include optimization problems with random constraints in computer sciences as well as in machine learning. The phase transitions of these systems are still elusive from a physics and mathematics point of view. We developed a new method to study these systems in finite dimensions based on the fluctuations of their energies. These methods can analyze the different critical behaviors of the systems as a function of the dimension, and ultimately can compare them with infinite-dimensional models where more properties are known.
The findings of the projects were published in top international journals of mathematics and of physics. They were also presented in international and national conferences of mathematics and physics.
Last Modified: 10/26/2018
Modified by: Louis-Pierre Arguin
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