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added pointer to:
added pointer to:
brief category:people-entry for hyperlinking references at B-meson and flavour anomaly
brief category:people-entry for hyperlinking references at B-meson and flavour anomaly
I have fixed the title of
and added the DOI
added pointer to
and tried to clean up the list of references a little, such as ordering it by date of appearance, and correcting the spelling to Katrin Wehrheim. There is more room for improvement here, but I’ll leave it at that.
brief category:people-entry for hyperlinking references at flavour anomaly and at leptoquark
brief category:people-entry for hyperlinking references at flavour anomaly
I needed a table exceptional spinors and division algebras – table, and so I have created one and included it into the relevant entries
a stub, just for completeness, to go alongside Spin(11,3) and D=12 supergravity
added a section “Examples” with the table of low dimensional rotation groups (which somehow had been omitted here, all along)
Renamed to match the spelling used by Faddeev in his English papers.
Linked to Dmitri Faddeev.
Created:
Dmitri Faddeev (Russian: Дми́трий Константи́нович Фадде́ев) was a Russian mathematician working in Galois theory, group cohomology, and numerical linear algebra. He is the father of Ludwig Faddeev.
He discovered group cohomology independently of Eilenberg and MacLane; the publication of his first paper on this subject was delayed till 1947 because of World War II.
On group cohomology:
Added this (in response to a MathOverflow query):
Jeffrey C. Morton, Cohomological Twisting of 2-Linearization and Extended TQFT, arXiv:1003.5603.
Section 2 and 3 of Gijs Heuts and Jacob Lurie’s Ambidexterity, in: Topology and Field Theories, doi.
Section 3 of Daniel Freed, Michael Hopkins, Jacob Lurie, Constantin Teleman, Topological quantum field theories from compact Lie groups, arXiv:0905.0731.
Created:
A generalization of homotopy groups.
Given a finitely generated abelian group and , we set
where is the th Peterson space of .
Franklin P. Peterson, Generalized Cohomotopy Groups. American Journal of Mathematics 78:2 (1956), 259–281. doi:10.2307/2372515
Joseph A. Neisendorfer, Homotopy groups with coefficients, Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338. doi:10.1007/s11784-010-0020-1.
brief category:people-entry for hyperlinking lecture notes at Hurewicz fibration
Created with the following content:
Given a finitely generated abelian group and , the th Peterson space of is the simply connected space whose reduced cohomology groups vanish in dimension and the th cohomology group is isomorphic to .
The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on and .
There are counterexamples both to existence and uniqueness without these conditions.
For example, the Peterson space does not exist if is the abelian group of rationals.
For all , we have a canonical isomorphism
where the left side denotes homotopy groups with coefficients and the right side denotes morphisms in the pointed homotopy category.
Removed the following discussion to the nForum:
Zoran Škoda: But there is much older and more general theorem of Hurewitz: if one has a map and a numerable covering of such that the restrictions for every in the covering is a Hurewicz fibration then is also a Hurewicz fibration. But the proof is pretty complicated. For example George Whitehead’s Elements of homotopy theory is omitting it (page 33) and Postnikov is proving it (using the equivalent “soft” homotopy lifting property).
Todd Trimble: Yes, I am aware of it. You can find a proof in Spanier if you’re interested. I’ll have to check whether the Milnor trick (once I remember all of it) generalizes to Hurewicz’s theorem.
Stephan: I wonder if this trick moreover generalizes (in a homotopy theoretic sense) to categories other that ; for example to the classical model structure on ?
Created with the following content:
Arthur Geoffrey Walker was a professor at the University of Liverpool, working in differential geometry, general relativity, and cosmology.
In particular, he is responsible for the Friedmann–Lemaître–Robertson–Walker metric in cosmology.
On vector fields as derivations:
some basics at FRW model (in cosmology)
Added:
Vector fields can be defined as derivations of the algebra of functions. See the article derivations of smooth functions are vector fields.
brief category:people-entry for hyperlinking references at coset space and Samelson product
brief category:people-entry for hyperlinking references at closed subgroup and at coset space
added pointer to the original reference (as kindly supplied by Dmitri over at Lie derivative):
Also cleaned up some text in this entry here.
gave the statement that derivations of smooth functions are vector fields a dedicated entry of its own, in order to be able to convieniently point to it
I am unhappy with Lie derivative. In the previous version it defined the Lie derivative as a secondary notion, using the differential and the Cartan homotopy formula (for which I finally created an entry). I have added a bit mentioning vector fields etc. and a formula using derivatives for forms but this is still not the right thing. Namely, in my understanding the Lie derivative is a fundamental notion and should not be defined using other differential operators, but by the “fisherman’s derivative” formula. Second it makes sense not only for differential forms but for any geometric quantities associated to the (co)frame bundle, and in particular to any kind of tensors, not necessarily contravariant or antisymmetrized. For this one has a prerequisite which will require some work in Lab. Namely to a vector field, one associated the flow, not necessarily defined for all times, but for small times. Then for any one has a diffeomorphism, which is used in the fisherman’s formula. But fisherman’s formula requires the pullback and the pullback is usually defined for forms while for general tensor fields one may need combination of pullbacks and pushforwards. However, for diffeomorphisms, one can define pullback in both cases, and pullback for time flow corresponds to the pushforward for time . To define such general pullback it is convenient to work with associated bundles for frame or coframe bundle and define it in the formalism of associated bundles. In the coframe case, this is in Sternberg’s Lectures on differential geometry (what returns me back into great memories of the summer 1987/1988 when I studied that book). So there is much work to do, to add details on those. If somebody has comments or shortcuts to this let me know.
However, there is a scientific question here as well: what about when frame bundle is replaced by higher jet bundles, and one takes some higher differential operator for functions and wants to do a similar program – are there nontrivial extensions of Lie derivative business to higher derivatives which does not reduce to the composition of usual Lie derivatives ?
Added:
Originally due to
Added a reference:
A coordinate-free treatment first appeared in
Added to noncommutative algebraic geometry a section “Relation to ordinary algberaic geometry” with what is really just a pointer to an article by Reyes:
The direct “naive” generalization of Grothendieck-style algebraic geometry via sheaves on a site (Zariski site, etale site etc.) of commutative rings-op to non-commutative rings does not work, for reasons discussed in some detail in (Reyes 12). This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”).
brief category:people-entry for hyperlinking references at equivariant differential topology and equivariant bundle and maybe elsewhere
Removing an old discussion:
Is there a reason that you moved these references up here? We need them especially for the stuff about morphisms below. —Toby
Eric: What would a colimit over an MSet-valued functor look like?
Toby: That depends on what the morphisms are.
Eric: I wonder if there is enough freedom in the definition of morphisms of multisets so that the colimit turns out particularly nice. I’m hoping that it might turn out to be simply the sum of multisets. According to limits and colimits by example the colimit of a Set-valued functor is a quotient of the disjoint union.
Toby: I think that you might hope for the coproduct (but not a general colimit) of multisets to be a sum rather than a disjoint union. Actually, you could argue that the sum is the proper notion of disjoint union for abstract multisets.
I created inner product of multisets, which I hope will help make some sense of some speculations over at the discussion of magnitude of metric spaces.
Thank you Toby for your help on my personal wikiweb.
brief category:people-entry for hyperlinking references at slice theorem
brief category:people-entry for hyperlinking references at slice theorem
I wanted the links to weak nuclear force and strong nuclear force in various entries to cease appearing grayish and ugly. So I created a minimal entry nuclear force.
This is a short article with a definition of the category of filters, now called the category of filters.
I apologise for vandalising by mistake the article on filters, and thanks to Richard Williamson for fixing this. I tried to “edit a current page .. in context on a relevant page” misunderstanding the intructions from HowTo:
How to start a new page
You do this in two steps, the first of which may have already been done:
Create a preliminary link (represented by a question mark) by editing > a current page and putting the name of the new page in double square brackets
Anonymous
am starting something. So far this is just a glorified pointer to today’s informative:
brief category:people-entry, for the purpose of hyperlinking references at Parametrized Higher Category Theory and Higher Algebra
I have tried to improve the list of references at stable homotopy theory and related entries a bit. I think the key for having a satisfactory experience with the non--categorical literature reflecting the state of the art, is to first have a general but quick survey, and then turn for the details of highly structured ring spectra to a comprehensive reference on S-modules or orthogonal spectra. So I have tried to make that better visible in the list of reference.
I find that for the first point (general but quick survey) Malkiewich 14 is the best that I have seen.
Of the highly structured models, probably orthogonal spectra maximize efficiency. A slight issue as far as references go is that the maybe best comprehensive account of their theory is Schwede’s Global homotopy theory, which presents something more general than beginners may want to see (on the other hand, beginners often don’t know what they really want). In any case, I have kept adding this book reference as a reference for orthogonal spectra, joint with the comment that the inclined reader is to chooce the collection of groups as trivial, throughout.