Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
Book.FieldWithOneElement istorijaPaslėpti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2019 birželio 30 d., 17:38
atliko 
Pakeista 5 eilutė iš:
The field with one element, {$F_1$}, į:
The field with one element, {$F_1$}, is a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. 2019 birželio 30 d., 17:37
atliko 
Pridėta 48 eilutė:
* [[https://arxiv.org/pdf/0809.1564.pdf  Cyclotomy and Analytic Geometry over {$F_1$}]] 2019 birželio 30 d., 17:36
atliko 
Pakeista 44 eilutė iš:
* [[http://cage.ugent.be/~kthas/Fun/  F_un Mathematics]] į:
* [[http://cage.ugent.be/~kthas/Fun/  F_un Mathematics]] Lieven Le Bruyn 2019 birželio 30 d., 17:34
atliko 
Pakeistos 3840 eilutės iš
* [[http://www.neverendingbooks.org/thef_unfolklore  F_un folklore]] į:
* [[http://www.neverendingbooks.org/thef_unfolklore  F_un folklore]] Lieven Le Bruyn * [[http://www.neverendingbooks.org/f_unwithmanin  F_un with Manin]] Lieven Le Bruyn * See Lieven Le Bruyn: [[http://www.neverendingbooks.org/category/absolute  Absolute]] 2019 birželio 30 d., 17:33
atliko 
Pridėtos 3840 eilutės:
* [[http://www.neverendingbooks.org/thef_unfolklore  F_un folklore]] * [[http://www.neverendingbooks.org/f_unwithmanin  F_un with Manin]] * See: [[http://www.neverendingbooks.org/category/absolute  Absolute]] 2019 vasario 09 d., 11:51
atliko 
Pakeistos 8283 eilutės iš
į:
* A field relates two groups: an additive group (the level) and a multiplicative group (the metalevel of actions). As regards the action, the zero of the additive group is the negation of action  no action taken, whereas the one of the multiplicative group is the action that has no effect. Therein lies the distinction of the level and the metalevel. * I think that an affine geometry is not so much distinguished by its not having a zero (a zero or origin can always be defined) but by its not having a one. Perhaps a projective geometry has both a zero and an infinity and so a one is naturally available. 2019 vasario 05 d., 13:47
atliko 
Pridėta 62 eilutė:
* The “general linear group” in n dimensions over the field of one element is the symmetric group {$S_n$}. 2019 vasario 03 d., 22:34
atliko 
Pridėta 45 eilutė:
* [[https://arxiv.org/abs/math/0407093  Projective geometry over {$F_1$} and the Gaussian binomial coefficients]], [[http://math.mit.edu/~cohn/  Henry Cohn]], [[http://math.mit.edu/~cohn/research.html  his research]] Ištrinta 49 eilutė:
2019 vasario 03 d., 21:08
atliko 
Ištrintos 3839 eilutės:
Pridėta 55 eilutė:
* [[https://www.youtube.com/watch?v=1XRna0vUYdo  Video: Taylor Dupuy: The field with one element and the Riemann Hypothesis]] 2019 vasario 03 d., 17:30
atliko 
Ištrintos 811 eilutės:
The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q>1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes. Pridėta 25 eilutė:
* How do finite fields deal with the issue that Lie algebras deal with: how to link countings? Pridėtos 3940 eilutės:
[[https://www.youtube.com/watch?v=1XRna0vUYdo  field of one element video]] Pridėtos 7374 eilutės:
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489. Pridėtos 7882 eilutės:
* Limit as q>1 ** The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q>1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes. * I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling [[Gods Dance  God's dance]]. Pakeistos 8990 eilutės iš
į:
* The anharmonic group (see [[https://en.wikipedia.org/wiki/Crossratio  Crossratio]]) permutes 0,1 and infinity. Ištrintos 99102 eilutės:
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489. Pakeistos 118129 eilutės iš
Andrius  [[https://www.youtube.com/watch?v=1XRna0vUYdo  field of one element video]] The anharmonic group (see [[https://en.wikipedia.org/wiki/Crossratio  Crossratio]]) permutes 0,1 and infinity.  į:
Andrius 2019 vasario 03 d., 17:25
atliko 
Pakeistos 4041 eilutės iš
* [[https://en.wikipedia.org/wiki/Field_with_one_element  Field with one element]] į:
* [[https://en.wikipedia.org/wiki/Field_with_one_element  Wikipedia: Field with one element]] Pakeista 43 eilutė iš:
* [[http://arxiv.org/pdf/0909.0069  į:
* [[http://arxiv.org/pdf/0909.0069  Mapping F1land: An overview of geometries over the field with one element]] Preprint. J.L. Pena, O. Lorscheid. 2019 vasario 03 d., 17:23
atliko 
Pridėtos 38 eilutės:
[+Field with one element, {$F_1$}+] The field with one element, {$F_1$}, i a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. My impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't? Pakeista 14 eilutė iš:
į:
 Pakeista 30 eilutė iš:
į:
 Pakeistos 3334 eilutės iš
Readings į:
[+Readings+] Ištrintos 6170 eilutės:
Finite fields * Lyndon words  irreducible polynomials for finite fields * Duality of q and n in {$GL_n(F_q)$}. * Multiset of Lyndon words  reducible and irreducible. Homogeneous symmetric functions of eigenvalues. * Interpolation between homogeneous and elementary  between commutativity and anticommutativity. * Lyndon words are like prime numbers. * Dimension of free Lie algebras = number of Lyndon words of length n * What would be the qtheory for finite fields for matrix combinatorics? Pridėtos 6875 eilutės:
F1believers base their funny intuition on the following two mantras : * F1 forgets about additive data and retains only multiplicative data. * F1objects only acquire flesh when extended to Z (or C). What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18], deals with variables x, y, and q such that q commutes with x and y, and yx = qxy. [+Ideas+] Pakeistos 8491 eilutės iš
* F1 forgets about additive data and retains only multiplicative data. * F1 deals with variables x, y, and q such that q commutes with x and y, and yx = qxy. į:
Finite fields * Lyndon words  irreducible polynomials for finite fields * Duality of q and n in {$GL_n(F_q)$}. * Multiset of Lyndon words  reducible and irreducible. Homogeneous symmetric functions of eigenvalues. * Interpolation between homogeneous and elementary  between commutativity and anticommutativity. * Lyndon words are like prime numbers. * Dimension of free Lie algebras = number of Lyndon words of length n * What would be the qtheory for finite fields for matrix combinatorics? Pakeistos 95112 eilutės iš
Of course, there is no field F1 with only one element, but there is a trivial ring, convention, because the trivial ring has no nontrivial modules (if x is an element of n 1 theory, become simpler when one moves to projective geometry. (The papers [11] geometry.) Cohn, page 489 į:
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489. Pakeistos 119121 eilutės iš
It's a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which apparently has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. I hope to learn more about it and report. But my impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't? į:
2019 vasario 03 d., 17:18
atliko 
Pridėtos 1517 eilutės:
* Learn what is known about the field with one element. ** Learn the underlying algebraic geometry. ** Learn how the field with one element relates to the [[Riemann hypothesis]]. Pakeista 19 eilutė iš:
* Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to SchurWeyl duality. į:
* Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has {$[n!]_q$} elements. Relate this to SchurWeyl duality. Ištrintos 2022 eilutės:
* Learn how the field with one element relates to the [[Riemann hypothesis]]. * Learn what is known about the field with one element. 2019 vasario 03 d., 17:07
atliko 
Ištrinta 10 eilutė:
2019 vasario 03 d., 16:52
atliko 
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate { į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {$F_{1^n}$}. 2019 vasario 03 d., 16:51
atliko 
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_ į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_/{1^n/}$}. 2019 vasario 03 d., 16:51
atliko 
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_{1^n}$}. į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_\{1^n\}$}. 2019 vasario 03 d., 16:50
atliko 
Pakeistos 3536 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Field_with_one_element  Field with one element]] Pakeista 46 eilutė iš:
į:
* [[http://arxiv.org/pdf/0909.2522  (Non) Commutative Fun Geometry]] Lieven Le Bruyn Ištrinta 47 eilutė:
Pakeistos 4951 eilutės iš
* [[http://wwwusers.math.umn.edu/~dgrinber/ Darij Grinberg]], [[http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf  Research interests: CarlitzWitt vectors and functionfield symmetric functions.]] * [[https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf  Carlitz extensions]] Keith Conrad. * [[http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf  ohomology Determinants and Reciprocity Laws: Number Field Case.]] M.Kapranov, A.Smirnov. į:
** [[http://wwwusers.math.umn.edu/~dgrinber/ Darij Grinberg]], [[http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf  Research interests: CarlitzWitt vectors and functionfield symmetric functions.]] Pakeistos 5257 eilutės iš
http į:
Special aspects * [[https://arxiv.org/pdf/1801.01491.pdf  The Action of Young Subgroups on the Partition Complex]], Gregory Z. Arone, D. Lukas B. Brantnerthe. Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. * [[https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf  Carlitz extensions]] Keith Conrad. * [[http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf  Cohomology Determinants and Reciprocity Laws: Number Field Case.]] M.Kapranov, A.Smirnov. Pakeista 60 eilutė iš:
* Duality of q and n in į:
* Duality of q and n in {$GL_n(F_q)$}. Ištrinta 64 eilutė:
Ištrintos 7273 eilutės:
Pridėta 77 eilutė:
* P and NP. Field with 1 element  deterministic. Char q  nondeterministic. Pakeistos 8187 eilutės iš
F1believers base their funny intuition on the following į:
F1believers base their funny intuition on the following two mantras : * F1 forgets about additive data and retains only multiplicative data. * F1objects only acquire flesh when extended to Z (or C). Pakeista 132 eilutė iš:
į:
2019 vasario 03 d., 16:16
atliko 
Pridėta 33 eilutė:
* [[https://sbseminar.wordpress.com/2007/08/14/thefieldwithoneelement/  The Field With One Element]] Noah Snyder about Mike Shulman's class Pridėta 42 eilutė:
* [[https://arxiv.org/abs/0704.2030  New Approach to Arakelov Geometry]] Nikolai Durov. A theory of generalized rings and schemes. 2019 vasario 03 d., 12:03
atliko 
Pakeistos 3843 eilutės iš
į:
* Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs. General theories * [[http://arxiv.org/pdf/1508.04636  New foundations for geometry: Two nonadditive languages for arithmetical geometry]] Shai Haran. This point of view of the generallineargroup suggests also that for the field with one element F we have {$"GL_n(\mathbb{F}_1)"=S_n$}, the symmetric group, which embeds as a common subgroup of all the finite group {$GL_n(\mathbb{F}_p)$}, p prime (or the "field" {$\mathbb{F}\{\pm1\}$}, with {$"GL_n(\mathbb{F}\{\pm1\})"=\{\pm1\}^n \bowtie S_n$}. (The symmetry group for {$B_n$} and {$C_n$}.) * [[http://arxiv.org/pdf/1312.4191  Quantum {$F_{un}$}: The q=1 Limit of Galois Field Quantum Mechanics, Projective Geometry & the Field With One Element]] Chang, Lewis, Minic, Takeuchi. Model of passing from quantum to classical mechanics Pakeista 46 eilutė iš:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schurore.pdf  Do the symmetric functions į:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schurore.pdf  Do the symmetric functions have a functionfield analogue? Draft.]] Darij Grinberg. Ištrintos 5261 eilutės:
[[http://arxiv.org/pdf/1508.04636  New foundations for geometry]] Shai Haran. This point of view of the generallineargroup suggests also that for the field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u n ¸ Sn). (the symmetry group for Bn and Cn) Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs. http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics 2019 vasario 03 d., 11:47
atliko 
Pridėta 31 eilutė:
* [[http://math.ucr.edu/home/baez/week259.html  John Baez on the field with one element]] Pridėta 36 eilutė:
* [[http://arxiv.org/pdf/0909.0069 [PL] J.L. Pena, O. Lorscheid: Mapping F1land: An overview of geometries over the field with one element]] Preprint. Pakeista 41 eilutė iš:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schurore.pdf  Do the symmetric functions havea functionfield analogue? Draft. Darij Grinberg. į:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schurore.pdf  Do the symmetric functions havea functionfield analogue? Draft.]] Darij Grinberg. Pridėtos 4962 eilutės:
[[http://arxiv.org/pdf/1508.04636  New foundations for geometry]] Shai Haran. This point of view of the generallineargroup suggests also that for the field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u n ¸ Sn). (the symmetry group for Bn and Cn) Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs. http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics http://arxiv.org/pdf/0909.2522 Pakeistos 73115 eilutės iš
A group is There exists (there is one=1) a unique (and only [[https://en.wikipedia.org/wiki/Orthogonal_group  Orthogonal group]] The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter [[https://en.wikipedia.org/wiki/Clifford_algebra  Clifford algebra]] A finite set can be thought of as a finite dimensional vector space over Zero is not a choice Intersection intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not http://arxiv.org/pdf/math/0407093v1 [5] H. Cohn. Projective Geometry over F1 and the Gaussian Binomial Coefficients. arXiv: 0407 [math.CO], 2009. Helpful and discusses 1 dimensional subspace of Pn(F). John Baez: http://math.ucr.edu/home/baez/week259.html http://arxiv.org/pdf/0909.0069 [PL] J.L. Pena, O. Lorscheid: Mapping F1land: An overview of geometries over the field with one element, preprint, arXiv: mathAG/0909.0069. [[http://arxiv.org/pdf/1508.04636  New foundations for geometry]] Shai Haran. This point of view of the generallineargroup suggests also that for the field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u n ¸ Sn). (the symmetry group for Bn and Cn) Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs. http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics http://arxiv.org/pdf/0909.2522 į:
Interpretations of mathematical structures in terms of {$F_1$}. * A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists! * A group is a Hopf algebra over the field with one element. * The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other. [[https://en.wikipedia.org/wiki/Orthogonal_group  Orthogonal group]] The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).... * [[https://en.wikipedia.org/wiki/Clifford_algebra  Clifford algebra]] Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. Cognitive ideas regarding {$F_1$}. * Zero is not a choice. The field needs to offer another choice. * There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs. There is none (negation=0). * Fields are "complete" mathematical structures (having all of the operations) but thus inevitably having a "gap" by which 0 and 1 are distinct. This is the quintessential gap and the prime numbers are likewise gaps in the factorization of numbers. * Intersection and union do not have inverses. * Intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not distributive over multtiplication. Pakeista 104 eilutė iš:
a module, then x = 1x = 0x = 0). Calling į:
a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, 2019 vasario 03 d., 11:15
atliko 
Pridėta 39 eilutė:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schurore.pdf  Do the symmetric functions havea functionfield analogue? Draft. Darij Grinberg. 2019 vasario 03 d., 11:13
atliko 
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_{1^n}$}. 2019 vasario 03 d., 11:12
atliko 
Pridėtos 2936 eilutės:
Introductory * [[https://cameroncounts.wordpress.com/2011/07/20/thefieldwithoneelement/  The field with one element]]. Peter Cameron. * [[http://www.neverendingbooks.org/DATA3/ncg.pdf  Geometry and the Absolute Point]] Overview * [[http://cage.ugent.be/~kthas/Fun/  F_un Mathematics]] Pakeistos 3943 eilutės iš
* [[http://wwwusers.math.umn.edu/~dgrinber/ Darij Grinberg]], http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf * http * http: * http į:
* [[http://wwwusers.math.umn.edu/~dgrinber/ Darij Grinberg]], [[http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf  Research interests: CarlitzWitt vectors and functionfield symmetric functions.]] * [[https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf  Carlitz extensions]] Keith Conrad. * [[http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf  ohomology Determinants and Reciprocity Laws: Number Field Case.]] M.Kapranov, A.Smirnov. * [[http://arxiv.org/abs/0911.3537  Characteristic one, entropy and the absolute point]]. Alain Connes, Caterina Consani. 2019 vasario 03 d., 10:46
atliko 
Pakeista 30 eilutė iš:
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. į:
* [[https://arxiv.org/pdf/1801.01491.pdf  The Action of Young Subgroups on the Partition Complex]], Gregory Z. Arone, D. Lukas B. Brantnerthe. Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. 2019 vasario 03 d., 10:22
atliko 
Pakeistos 3237 eilutės iš
į:
* http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf * https://cameroncounts.wordpress.com/2011/07/20/thefieldwithoneelement/ * http://cage.ugent.be/~kthas/Fun/ * http://arxiv.org/abs/0911.3537 Ištrintos 5664 eilutės:
http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf https://cameroncounts.wordpress.com/2011/07/20/thefieldwithoneelement/ http://cage.ugent.be/~kthas/Fun/ http://arxiv.org/abs/0911.3537 2019 sausio 19 d., 14:02
atliko 
Pakeistos 3031 eilutės iš
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. į:
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. 2019 sausio 19 d., 14:02
atliko 
Pridėta 28 eilutė:
Readings Pakeistos 3033 eilutės iš
į:
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. * [[http://wwwusers.math.umn.edu/~dgrinber/ Darij Grinberg]], http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf, See: Combinatorics and the field with one element. Witt vectors  padic integers Ištrintos 156165 eilutės:
Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. https://arxiv.org/pdf/1801.01491.pdf Darij Grinberg http://wwwusers.math.umn.edu/~dgrinber/ http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf See: Combinatorics and the field with one element. Witt vectors  padic integers 2019 sausio 19 d., 14:01
atliko 
Pridėtos 2728 eilutės:
* [[https://arxiv.org/abs/math/0407093  Projective geometry over {$F_1$} and the Gaussian binomial coefficients]], [[http://math.mit.edu/~cohn/  Henry Cohn]], [[http://math.mit.edu/~cohn/research.html  his research]] 2018 gruodžio 09 d., 17:08
atliko 
Pridėtos 2224 eilutės:
Investigate * Does 11=0 in the field with one element? 2018 lapkričio 11 d., 16:56
atliko 
Pakeistos 151157 eilutės iš
collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. https://arxiv.org/pdf/1801.01491. į:
collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. https://arxiv.org/pdf/1801.01491.pdf Darij Grinberg http://wwwusers.math.umn.edu/~dgrinber/ http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf See: Combinatorics and the field with one element. Witt vectors  padic integers 2018 lapkričio 08 d., 14:01
atliko 
Pridėta 12 eilutė:
* How the [[https://en.wikipedia.org/wiki/Hall_algebra  Hall algebra]] and Hall polynomials give rise to the symmetric functions and, in particular, Schur functions, when q goes to 0. 2018 rugsėjo 14 d., 11:12
atliko 
Pakeista 1 eilutė iš:
See: [[Math]], [[Binomial theorem]] į:
See: [[Math]], [[Binomial theorem]], [[Finite fields]] 2018 rugsėjo 13 d., 16:40
atliko 
Ištrintos 47 eilutės:
* Learn about finite fields, especially their combinatorics. Be able to contemplate F1n. * Learn how the field with one element relates to the [[Riemann hypothesis]]. Pakeistos 918 eilutės iš
Find qanalogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q>1. Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q>1 ? Relate Cayley's theorem to the field with one element. Study: * How Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and į:
Study * Find qanalogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q>1. * Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q>1 ? Learn * Learn about finite fields, especially their combinatorics. Be able to contemplate F1n. * Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to SchurWeyl duality. * Learn about the relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography  homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q>1. * Learn the underlying algebraic geometry. * Learn how the field with one element relates to the [[Riemann hypothesis]]. * Learn what is known about the field with one element. 2018 rugsėjo 13 d., 16:14
atliko 
Pakeistos 12 eilutės iš
See: [[Math]] į:
See: [[Math]], [[Binomial theorem]] I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling [[Gods Dance  God's dance]]. * Learn what is known about the field with one element. Learn the underlying algebraic geometry. * Learn about finite fields, especially their combinatorics. Be able to contemplate F1n. * Learn how the field with one element relates to the [[Riemann hypothesis]]. The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q>1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes. Pridėtos 1213 eilutės:
Find qanalogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q>1. 2018 vasario 15 d., 22:19
atliko 
Pridėtos 1524 eilutės:
Finite fields * Lyndon words  irreducible polynomials for finite fields * Duality of q and n in GLn(Fq). * Multiset of Lyndon words  reducible and irreducible. Homogeneous symmetric functions of eigenvalues. * Interpolation between homogeneous and elementary  between commutativity and anticommutativity. * Lyndon words are like prime numbers. * Dimension of free Lie algebras = number of Lyndon words of length n * P and NP. Field with 1 element  deterministic. Char q  nondeterministic. * What would be the qtheory for finite fields for matrix combinatorics? 2018 sausio 15 d., 18:05
atliko 
Pakeista 16 eilutė iš:
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių. į:
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių. Ar tai susiję su F(2)? Su forgotten basis? 2018 sausio 15 d., 17:57
atliko 
Pakeista 16 eilutė iš:
į:
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių. 2018 sausio 15 d., 13:58
atliko 
Pakeistos 127132 eilutės iš
 į:
 Partition complexes can be thought of as BruhatTits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of BruhatTits buildings. https://arxiv.org/pdf/1801.01491.pdf 2017 lapkričio 11 d., 21:08
atliko 
Pakeistos 911 eilutės iš
Relate Cayley's theorem to the field with one element į:
Relate Cayley's theorem to the field with one element. Study: * How Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to SchurWeyl duality. >><< 2017 spalio 04 d., 10:20
atliko 
Pridėtos 89 eilutės:
Relate Cayley's theorem to the field with one element 2017 spalio 04 d., 09:55
atliko 
Pridėtos 117118 eilutės:
The anharmonic group (see [[https://en.wikipedia.org/wiki/Crossratio  Crossratio]]) permutes 0,1 and infinity. 2016 gruodžio 03 d., 10:41
atliko 
Pridėtos 1011 eilutės:
A group is a Hopf algebra over the field with one element. 2016 rugsėjo 27 d., 20:02
atliko 
Pridėtos 1011 eilutės:
There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs. There is none (negation=0). 2016 rugsėjo 13 d., 11:27
atliko 
Pridėtos 1213 eilutės:
[[https://en.wikipedia.org/wiki/Clifford_algebra  Clifford algebra]] Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. 2016 rugsėjo 13 d., 11:21
atliko 
Pridėtos 1011 eilutės:
[[https://en.wikipedia.org/wiki/Orthogonal_group  Orthogonal group]] The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).... The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other. 2016 rugpjūčio 30 d., 22:21
atliko 
Pridėtos 67 eilutės:
Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q>1 ? 2016 rugpjūčio 15 d., 11:17
atliko 
Pridėtos 2930 eilutės:
John Baez: http://math.ucr.edu/home/baez/week259.html 2016 rugpjūčio 15 d., 11:15
atliko 
Pakeista 28 eilutė iš:
[math.CO], 2009. į:
[math.CO], 2009. Helpful and discusses 1 dimensional subspace of Pn(F). 2016 liepos 15 d., 12:45
atliko 
Pridėtos 814 eilutės:
http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf https://cameroncounts.wordpress.com/2011/07/20/thefieldwithoneelement/ http://cage.ugent.be/~kthas/Fun/ 2016 liepos 10 d., 16:43
atliko 
Pakeista 25 eilutė iš:
http://arxiv.org/pdf/1508.04636 į:
[[http://arxiv.org/pdf/1508.04636  New foundations for geometry]] Shai Haran. 2016 birželio 23 d., 15:35
atliko 
Pridėtos 9495 eilutės:
[[https://www.youtube.com/watch?v=1XRna0vUYdo  field of one element video]] 2016 birželio 23 d., 14:06
atliko 
Pridėtos 89 eilutės:
A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists! 2016 birželio 23 d., 10:21
atliko 
Pridėtos 1011 eilutės:
Intersection and union do not have inverses. 2016 birželio 23 d., 10:16
atliko 
Pridėtos 1011 eilutės:
intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not distributive over multtiplication. 2016 birželio 23 d., 09:54
atliko 
Pridėtos 89 eilutės:
Zero is not a choice. The field needs to offer another choice. 2016 birželio 23 d., 01:28
atliko 
Pakeista 46 eilutė iš:
a module, then x = 1x = 0x = 0). Calling į:
a module, then x = 1x = 0x = 0). Calling itj a field would not help solve Puzzle 1, Pridėta 59 eilutė:
Cohn, page 489 2016 birželio 23 d., 01:27
atliko 
Pridėtos 4257 eilutės:
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) 2016 birželio 23 d., 01:24
atliko 
Pridėtos 3638 eilutės:
What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18], deals with variables x, y, and q such that q commutes with x and y, and yx = qxy. 2016 birželio 23 d., 01:21
atliko 
Pakeista 9 eilutė iš:
http://arxiv.org/pdf/ į:
http://arxiv.org/pdf/math/0407093v1 2016 birželio 23 d., 01:18
atliko 
Pakeista 9 eilutė iš:
http://arxiv.org/pdf/0407. į:
http://arxiv.org/pdf/0407.7093 2016 birželio 23 d., 01:18
atliko 
Pridėtos 811 eilutės:
http://arxiv.org/pdf/0407.7093v1 [5] H. Cohn. Projective Geometry over F1 and the Gaussian Binomial Coefficients. arXiv: 0407.7093v1 [math.CO], 2009. 2016 birželio 23 d., 01:10
atliko 
Pridėtos 811 eilutės:
http://arxiv.org/pdf/0909.0069 [PL] J.L. Pena, O. Lorscheid: Mapping F1land: An overview of geometries over the field with one element, preprint, arXiv: mathAG/0909.0069. 2016 birželio 23 d., 01:07
atliko 
Pridėtos 1418 eilutės:
Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs. 2016 birželio 23 d., 01:05
atliko 
Pakeistos 913 eilutės iš
http://arxiv.org/pdf/1508. į:
http://arxiv.org/pdf/1508.04636 This point of view of the generallineargroup suggests also that for the field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u n ¸ Sn). (the symmetry group for Bn and Cn) 2016 birželio 23 d., 01:00
atliko 
Pridėtos 1419 eilutės:
F1believers base their funny intuition on the following two mantras : • F1 forgets about additive data and retains only multiplicative data. • F1objects only acquire flesh when extended to Z (or C). 2016 birželio 23 d., 00:56
atliko 
Pakeista 11 eilutė iš:
http://arxiv.org/pdf/1312. į:
http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics 2016 birželio 20 d., 01:28
atliko 
Pridėtos 36 eilutės:
>>bgcolor=#FFFFC0<< Consider relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography  homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q>1. Pakeistos 910 eilutės iš
į:
Symmetry involves a dual point of view: for example, vertices are distinct and yet not distinguishable. 2016 birželio 19 d., 22:39
atliko 
Pridėtos 49 eilutės:
Consider relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography  homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q>1.  2016 birželio 19 d., 14:08
atliko 
Pridėtos 129 eilutės:
See: [[Math]] >>bgcolor=#EEEEEE<< Dear Harvey, Thank you for your invitations in your letter below and also earlier, "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"." http://www.cs.nyu.edu/pipermail/fom/2016April/019724.html You mentioned, in my words, that you are looking for an issue that working mathematicians are grappling with where the classical ZFC foundations are not satisfactory or sufficient. Would the "field with one element" be such issue for you? https://ncatlab.org/nlab/show/field+with+one+element Jacques Tits raised this issue in 1957 and it has yet to be resolved despite substantial interest, conferences, and long papers. Would that count as a "problem" for the Foundations of Mathematics? It seems that in the history of math it is very easy to simply say "that is not real math" as was the case with the rational numbers, imaginary numbers, infinitesimals, infinite series, etc. The issue is that there are many instances where a combinatorial interpretation makes sense in terms of a finite field Fq of characteristic q, which is all the more insightful when q=1. For example, the Gaussian binomial coefficients can be interpreted as counting the number of kdimensional subspaces of an ndimensional vector space over a finite field Fq. When q=1, then we get the usual binomial coefficients which count the subsets of size k of a set of size n. So this suggests an important way of thinking about sets. However, F1 would be a field with one element, which means that 0=1. But if 0 and 1 are not distinct, then none of the usual properties of a field make sense. Nobody has figured out a convincing interpretation for F1. And yet the concept seems to be pervasive, meaningful and fruitful. If there was an alternate foundations of mathematics which yielded a helpful, meaningful, fruitful interpretation of F1, would that count in its favor? And if it could do everything that FOM can do, then might it be preferable, at least for some? But especially if that interpretation was shown not to make sense in other FOMs? I am curious what you and others think. Andrius  I just learned of the "field of one element": https://en.wikipedia.org/wiki/Field_with_one_element It's a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which apparently has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. I hope to learn more about it and report. But my impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?  
FieldWithOneElementNaujausi pakeitimai 
Puslapis paskutinį kartą pakeistas 2019 birželio 30 d., 17:38
