NumberFields@Home: Difference between revisions
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[https://numberfields.asu.edu/NumberFields/ '''''NumberFields@Home'''''] is a '''''[[wikipedia:Volunteer computing|volunteer distributed computing]]''''' project that needs your help to | BOINC project [https://numberfields.asu.edu/NumberFields/ '''''NumberFields@Home'''''] is a '''''[[wikipedia:Volunteer computing|volunteer distributed computing]]''''' project that needs your help to search for fields with special properties. | ||
== Why NumberFields@Home? == | == Why NumberFields@Home? == | ||
Fields are important mathematical constructs that have far reaching applications to many branches of mathematics. Many people are familiar with the fields of rational numbers, real numbers, and complex numbers. The fields we are concerned with in this project are number fields: subsets of the complex numbers which contain the root of a given polynomial and are minimal for then being closed under addition, subtraction, multiplication, and division (except for division by 0). In particular, we are interested in imprimitive degree 10 fields (called decic fields), which correspond to certain degree 10 polynomials. | Fields are important mathematical constructs that have far reaching applications to many branches of mathematics. Many people are familiar with the fields of rational numbers, real numbers, and complex numbers. The fields we are concerned with in this project are number fields: subsets of the complex numbers which contain the root of a given polynomial and are minimal for then being closed under addition, subtraction, multiplication, and division (except for division by 0). In particular, we are interested in imprimitive degree 10 fields (called decic fields), which correspond to certain degree 10 polynomials. | ||
== | Number theorists can mine the data for interesting patterns to help them formulate conjectures about number fields. Ultimately, this research will lead to a deeper understanding of the properties of numbers, the basic building blocks of all mathematics. Another application of number fields is in cryptography, where they are used in sophisticated factoring algorithms and as the basis for new cryptosystems. There are also distant applications to mathematical physics, including quantum mechanics and string theory. | ||
== Goals == | |||
One way to categorize fields is by the primes that ramify in them. For a given set of primes, the number of fields ramified at those primes is finite. The primary goal of the project is to find this finite set of fields for various sets of primes. Since the number of combinations of primes is unlimited, the project will remain open-ended for the foreseeable future. | One way to categorize fields is by the primes that ramify in them. For a given set of primes, the number of fields ramified at those primes is finite. The primary goal of the project is to find this finite set of fields for various sets of primes. Since the number of combinations of primes is unlimited, the project will remain open-ended for the foreseeable future. | ||
Another way to categorize fields is by their discriminant, which is an important invariant for a field. Given a fixed bound , there are only a finite number of fields whose discriminant is less than this bound. A secondary goal of the project is to determine the finite set of "minimum discriminant" imprimitive decic fields for the bound <math>B=1.2 \times 10^{11}</math>. | Another way to categorize fields is by their discriminant, which is an important invariant for a field. Given a fixed bound , there are only a finite number of fields whose discriminant is less than this bound. A secondary goal of the project is to determine the finite set of "minimum discriminant" imprimitive decic fields for the bound <math>B=1.2 \times 10^{11}</math>. This bound was chosen for it's potential to find more fields while keeping the computational load manageable. | ||
== Methods == | == Methods == | ||
Computing lower degree fields requires less processing power and have been more extensively tabulated. The degree 10 case is the first case requiring a massively parallel solution, and hence the reason for implementing a BOINC project. | |||
Number fields are related to automorphic forms which are part of the [[wikipedia:Langlands_program|'''''Langland's program''''']]. [https://www.quantamagazine.org/what-is-the-langlands-program-20220601/ '''''Explanations for the Langland's Program'''''] ([https://youtu.be/_bJeKUosqoY '''''see the video''''']): | |||
== Project team / Sponsors == | == Project team / Sponsors == | ||
Eric Driver. school of mathematics at Arizona State University. | |||
== Scientific results == | == Scientific results == | ||
* | * https://numberfields.asu.edu/NumberFields/FieldTables/FieldTables.html | ||
== Scientific publications == | == Scientific publications == | ||
https://boinc.berkeley.edu/pubs.php#NumberFields@Home | https://boinc.berkeley.edu/pubs.php#NumberFields@Home | ||
Revision as of 17:55, 11 February 2024
[[File:{{#setmainimage:Nf.jpg}}|alt=logo image|center|frameless]]
BOINC project NumberFields@Home is a volunteer distributed computing project that needs your help to search for fields with special properties.
Why NumberFields@Home?
Fields are important mathematical constructs that have far reaching applications to many branches of mathematics. Many people are familiar with the fields of rational numbers, real numbers, and complex numbers. The fields we are concerned with in this project are number fields: subsets of the complex numbers which contain the root of a given polynomial and are minimal for then being closed under addition, subtraction, multiplication, and division (except for division by 0). In particular, we are interested in imprimitive degree 10 fields (called decic fields), which correspond to certain degree 10 polynomials.
Number theorists can mine the data for interesting patterns to help them formulate conjectures about number fields. Ultimately, this research will lead to a deeper understanding of the properties of numbers, the basic building blocks of all mathematics. Another application of number fields is in cryptography, where they are used in sophisticated factoring algorithms and as the basis for new cryptosystems. There are also distant applications to mathematical physics, including quantum mechanics and string theory.
Goals
One way to categorize fields is by the primes that ramify in them. For a given set of primes, the number of fields ramified at those primes is finite. The primary goal of the project is to find this finite set of fields for various sets of primes. Since the number of combinations of primes is unlimited, the project will remain open-ended for the foreseeable future.
Another way to categorize fields is by their discriminant, which is an important invariant for a field. Given a fixed bound , there are only a finite number of fields whose discriminant is less than this bound. A secondary goal of the project is to determine the finite set of "minimum discriminant" imprimitive decic fields for the bound . This bound was chosen for it's potential to find more fields while keeping the computational load manageable.
Methods
Computing lower degree fields requires less processing power and have been more extensively tabulated. The degree 10 case is the first case requiring a massively parallel solution, and hence the reason for implementing a BOINC project.
Number fields are related to automorphic forms which are part of the Langland's program. Explanations for the Langland's Program (see the video):
Project team / Sponsors
Eric Driver. school of mathematics at Arizona State University.