Added to *Klein geometry* a section *History* with quotations for where exactly Klein actually speaks about $G/H$.

(This key passages is a bit hidden in Klein’s text, appearing at a somewhat unexpected point somewhere in the middle of a 35 page document.)

]]>have added to *associated bundle* an exposition, here, of how in a context of $(2,1)$-topos theory, associated bundles are naturally thought of as homotopy pullbacks of action groupoid projections.

This is from one subsection of what I am currently typing into *geometry of physics – representations and associated bundles*. It may need more polishing, but I need to interrupt now for half an hour or so.

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 $n$Lab. Namely to a vector field, one associated the flow, not necessarily defined for all times, but for small times. Then for any $t$ 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 $t$ flow corresponds to the pushforward for time $-t$. 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 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”).

I do not understand the entry G-structure. G-structure is, as usual, defined there as the principal $G$-subbundle of the frame bundle which is a $GL(n)$-principal bundle. I guess this makes sense for equivariant injections along any Lie group homomorphism $G\to GL(n)$. The entry says something about spin structure, warning that the group $Spin(n)$ is not a subgroup of $GL(n)$. So what is meant ? The total space of a subbundle is a subspace at least. Does this mean that I consider the frame bundle first as a (non-principal) $Spin(n)$-bundle by pulling back along a fixed noninjective map $Spin(n)\to GL(n)$ and then I restrict to a chosen subspace on which the induced action of Spin group is principal ?

]]>added a chunk of some standard basics to *elliptic curve – Definition over a general ring*.

Also touched/briefly created various related entries, such as *Weierstrass equation*, *Weierstrass elliptic function*, *cubic curve*, *j-invariant* etc.

I have started a category:reference page

such as to be able to point to it for reference, e.g. from Kontsevich 15 etc.

]]>edited [[classifying topos]] and added three bits to it. They are each marked with a comment "check the following".

This is in reaction to a discussion Mike and I are having with Richard Williamson by email.

]]>added to *modular form* a brief paragraph with a minimum of information on modular forms *As automorphic forms*. Needs to be expanded.

started something at *twistor space*

stub for *constructible sheaf*

I have split off *complex projective space* from *projective space* and added some basic facts about its cohomology.

Our entries *formal scheme* and *formal spectrum* need attention. They fail to state their definitions clearly, and in parts to the extent that the main point is lost.

At *formal scheme* I have tried to clean up a bit right there in the Idea-section. But this needs to be expanded on.

What I really came for to this entry is that I wanted to make explicit the basic example which I now did add as

]]>at *morphism of finite presentation* I fixed a wrong statement by changing “finitely presented as a module” to “finitely presented as an algebra”. Created stub for finitely presented algebra in the course of this.

gave *formal Picard group* its own brief entry. Mainly I wanted to record the pointer to the lecture notes now given there.

I have polished a little at *geometry of physics – smooth sets*, in reaction to feedback that Arnold Neumaier provided over on PF here.

I ran across your site as i was finishing my book/Art project and found your philosophy of interest as resonating with my own.

I reach out here to introduce myself and project - below please find the abstract and the book which is freely observable on line

Many thanks

John

3.0 i: Abstract/Letter of Introduction

Greetings fellow sojourner seeking,

Please pardon my interruption; I wanted to take the

opportunity to briefly introduce myself through my art

and, as such, invite you to peruse my book project with

title: i, in the palm of my mind : the chapters are

available to observe and consider online at

www.eidolononesuch.com

In my book, I explore - through various writing and

art formats - a series of desperate tho related concepts

ranging from cosmology to consciousness and free will.

And I propose and explore some intriguing models

you may want to consider such as regarding time and

how it results in the apparent EPR paradox and, as well,

a structural and mechanistic model for how matter and

mass connect as well as provide some detail on the

relationships of time/gravity and other forces.

Specifically,

I propose time as a 3-dimensional quasicrystalline net

of collapse events against the fabric of a fully collapsing

modular host pattern and describe where said pattern resides

I describe how to extract time from the pattern and evolve

a universe such as ours from a point containing

modular maths such as elliptics

I propose a unique model for the electron

(and nucleon components) as a resonant folded

wave/anti-wave (4:1 ratio in the electron) pattern in

the motif of a tetrix: a tetrahedral pyramidal fractal fold

of mass/void and the consequences of such regarding

the observed matter dominance in our universe and

its evolution

I describe the w/~w basis of the 3-state equilibrium

of photon/electron interaction as the tetrix tetrahedral

fold equilibrates with a cubic fold allowing point collapse

of part of the electron folded wave and how this yields time

and the EPR/Bell's inequality observations as waves and

anti-waves transition through the collapse state and

invert to yield the changed state

I explore how neutrinos may be the mechanism of

motion/momentum for the folded-matter waves and

how motion may be realized on a fully collapsing,

modular host with implications for how gravity

differs fundamentally from other forces

I elucidate a system - derived from neutrino oscillation

- detailing how rotational inertia is related and maintained

relative to linear momentum in a universe where the only force

is simplification and describe how momentum is prone to condense

with the greater frame of reference

I describe how the universe relies on de-construction

of its core information pattern, where the 'particle' machines

ur-data is extracted, sending the information by two mechanisms

to distant location and re-combining to yield the collapsible cubic

architecture: this results in alteration, temporarily, of the pattern

that consequently inflates our universe and this process can

occur reversibly without wave inversion beneath the

time collapse events

I describe entanglement as superposition/co-collapse

of mirrored, adjacent waves thus creating linkage to

subsequent collapse with inversion required as

observed time events

I propose straight-forward bases for the dark matter

and dark energy observations and describe the cycling

of universe iterations at our local site in the uberverse

including provenance of the high negative entropy

that is energy

Those are in part I of the book's triptych; if you find

these topics of interest, there are others that I find even

more intriguing in the remainder:

For example, I describe a functional definition of soul and

point to cryptic components, such as our ur-brain memory:

a cross-generational repository/information system which

provides survival utility as a fundamental evolutionary tool

This is interpreted in relation to the concepts of free will/choice

And I introduce potential reasons to question whether digital as

fundamentally differing, with relevance, from IRL

Please enjoy as you wont - it is intended purely and simply as art

and as such, simply and purely to inspire your art

Many thanks for all your good works,

J.

* June/July 20/20, JSM ]]>

at *Atiyah Lie groupoid* was this old query box discussion, which hereby I am moving from there to here:

+– {: .query} What is all of this $diag$ stuff? I don't understand either $(P \times P)/_{diag} G$ or $(P_x \times P_x)_{diag} G$. —Toby

David Roberts: It’s to do with the diagonal action of $G$ on $P\times P$ as opposed to the antidiagonal (if $G$ is abelian) or the action on only one factor. I agree that it’s a bad notation.

*Toby*: How well do you think it works now, with the notation suppressed and a note added in words? (For what it's worth, the diagonal action seems to me the only obvious thing to do here, although admittedly the others that you mention do exist.)

*Todd*: I personally believe it works well. A small note is that this construction can also be regarded as a tensor product, regarding the first factor $P$ as a right $G$-module and the second a left module, where the actions are related by $g p = p g^{-1}$.

*Toby*: H'm, maybe we should write diagonal action if there's something interesting to say about it.
=–

I have expanded the Idea-section at *moduli stack of elliptic curves*, have tried to add more pertinent references, and have touched the subsection on “Over general rings” and on the derived version.

In the course of this I started to split off some entries such as *nodal cubic curve* (which now has a little bit of content) and *cuspidal cubic curve* (which does not yet).

I have added to *Teichmüller theory* a mini-paragraph Complex structure on Teichmüller space with a minimum of pointers to the issue of constructing a complex structure on Teichmüller space itself.

Maybe somebody has an idea on the following: The Teichmüller orbifold itself should have a neat general abstract construction as the full subobject on the étale maps of the mapping stack formed in smooth $\infty$-groupoids/smooth $\infty$-stacks into the Haefliger stack for complex manifolds : via Carchedi 12, pages 37-38.

Might we have a refinement of this kind of construction that would produce the Teichmüller orbifold directly as on objects in $\infty$-stacks over the complex-analytic site?

]]>The scan of the writeup of Grothendieck’s 73 Buffalo lecture that we point to at *functorial geometry* is really badly done. Is there a better scan or any other re-typing available?

added a minimum to *Lorentz group*.

have tried to brush-up the entry locally infinity-connected (infinity,1)-topos.

Kicked out a bunch of material that we had meanwhile copied over to their dedicated entries and tried to organize the remaining material a bit better. Need to work on locally infinity-connected site

]]>Now I am working on the next chapter of “geometry of physics”: *geometry of physics – supersymmetry*.

A fair bit of material is in place now, but much is missing still. This here is mainly in case you are watching the logs and are wondering. At this point, if anyone has any edits to suggest (typo fixing or more substantial) maybe best to not touch the file yet but to tell me about it. Thanks!

]]>I added a discussion of space in Kant’s Transcendental Aesthetics in Critique of Pure Reason.

By the way, the translation of the quote from Kant in the section “On Aristotelian logic” seem a bit strange: I think the original German sentence was “Begriffe aber beziehen sich als Prädikate möglicher Urtheile auf irgend einen noch unbestimmten Gegenstand” (“But conceptions, as predicates of possible judgements, relate to some representation of a yet undetermined object.”).

PS The automatic function to create this thread in the nforum did not word.

]]>