Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2018

    added a subsection “Properties – As algebraic K-theory over the field with one element” (here)

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2018

    have send a question to MO for more: Stable Cohomotopy as KF_1

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 11th 2018

    Maybe that Berman thesis I mentioned could count as an answer.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2018

    I should have another look, maybe I missed it: what does the thesis achieve along these lines, beyond the vague comments?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 12th 2018

    Well, yes, only vague comments. He thanks Jack Morava for “suggesting that my results are related to the field with one element”.

    But what kind of “further developments” were you hoping for?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2018
    • (edited Sep 12th 2018)

    I was looking for more like Guillot’s result, strengthening the analogy between stable cohomotopy and K-theory. How about Chern-characters, Todd classes, A-roof genus, Riemann-Roch theorem over F_1? How about the comparison map between all structures, as we “extend scalars” from F_1 to C. This last question is related to what we are discussing in the thread on Burnside ring.

    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeSep 12th 2018
    • (edited Sep 12th 2018)

    Riemann-Roch theorem over F_1

    This is a major open problem, it is one of the main obstructions to a proof of the Riemann hypothesis via algebraic geometry over 𝔽 1\mathbb{F}_{1} (although for me what the analogue of the Frobenius should be is the main conceptual gap). I think Alain Connes may have some kind of results around Riemann-Roch over 𝔽 1\mathbb{F}_{1} in some setting, though, albeit not strong enough for the applications to the Riemann hypothesis.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2018

    This is a major open problem, it is one of the main obstructions to a proof of the Riemann hypothesis via algebraic geometry over 𝔽 1\mathbb{F}_1

    Thanks for saying this, I didn’t know. What would be a good source to read up on this?

  1. Though I have never focused on it in the same way as I have on other things, I’ve been ruminating off and on upon 𝔽 1\mathbb{F}_{1} for quite a long time, and mostly do not remember where I picked up things! However, perhaps the highlighted text towards the bottom of pg. 20 here would be the kind of thing you are looking for?

    I am not very familiar with Connes’ work, but it seems (regarding my remark about the Frobenius in #7) that he does have an idea of what the analogue of the Frobenius should be, so that the Riemann-Roch theorem actually is the principal obstruction in his setting.

  2. There have been a few developments since the article I linked to as well, see here.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2018

    Thanks! I am having a look…

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 23rd 2018
    • (edited Sep 23rd 2018)

    Riemann-Roch theorem over F_1

    In the process of some pondering on the place of 𝔽 1\mathbb{F}_1 in Sylow pp-group theory, John Baez pointed me to this paper by Kapranov and Smirnov which speaks of Riemann-Roch over 𝔽 1 n\mathbb{F}_{1^n} as counting residues mod nn of the number of integer points of some polyhedron.

    Hmm, so what’s K𝔽 1 nK \mathbb{F}_{1^n}?

    The Sylow thought, by the way, is that the pp-Sylow subgroup of any GL n(Z p k)GL_n(Z_{p^k}) is the maximal unipotent subgroup, and any group embeds in S nS_n which embeds in GL n(Z p k)GL_n(Z_{p^k}).

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 23rd 2018

    I haven’t seen a discussion of 𝔽 1 n\mathbb{F}_{1^n} in Connes-Consani; they have apparently proved the only finite semi-field that’s not a field is ({0,1},max,+)(\{0,1\},max,+), and they take this to be the “prime field” of characteristic 1.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2018

    added pointer to

    with respect to understanding the sphere spectrum as K(𝔽 1)K(\mathbb{F}_1)

    diff, v12, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2018

    finally added pointer to

    diff, v14, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020

    added pointer to:

    • C. T. Stretch, Stable cohomotopy and cobordism of abelian groups, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 90, Issue 2 September 1981 , pp. 273-278 (doi:10.1017/S0305004100058734)

    diff, v23, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020

    added pointer to

    • Sławomir Nowak, Stable cohomotopy groups of compact spaces, Fundamenta Mathematicae 180 (2003), 99-137 (doi:10.4064/fm180-2-1)

    diff, v23, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020

    added pointer to:

    • Ken-ichi Maruyama, ee-invariants on the stable cohomotopy groups of Lie groups, Osaka J. Math. Volume 25, Number 3 (1988), 581-589 (euclid:ojm/1200780982)

    diff, v23, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2021

    added pointer to:

    diff, v29, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2021

    added pointer to:

    diff, v29, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2023
    • (edited Dec 28th 2023)

    I see that the pdf-link math.mit.edu/~sglasman/bpq-beamer.pdf for this item is dead:

    • Saul Glasman, The multiplicative Barratt-Priddy-Quillen theorem and beyond, talk at AMS Sectional Meeting 1095 (2013) [webpage, pdf]

    (as is the whole website math.mit.edu/~sglasman).

    I haven’t found a backup copy yet…

    diff, v35, current