# User talk:Gauge

feel free to place any constructive criticism or comments here. i am a friendly guy, so please no yelling. - Gauge 00:23, 4 Aug 2004 (UTC)

## Question about sylow theorem 1

Hi, I was wondering how you get pr divides |Ω| and pr+1 does not. It doesn't seem to follow directly from the definition of Ω —Preceding unsigned comment added by 68.239.152.13 (talk) 05:02, 13 July 2008 (UTC)

There's a little bit of combinatorial magic going on there. Just count the factors of p in ${\displaystyle {p^{k}m} \choose {p^{k}}}$. A professor of mine once said "finite group theory is just combinatorics". I think he was slightly biased, but this proof is a good example in his favor.  ;-) - Gauge (talk) 04:27, 28 October 2008 (UTC)

## Cauchy-Schwarz inequality

My faithful bot asked me to tell you that he is much happier now with downcased "which". :) Actually I do remember that particular article (I double check every article) but I was too lazy to actually downcase "which" myself. I think the style of the article is better now. And thanks for not reverting on me. :) Oleg Alexandrov 04:28, 22 Mar 2005 (UTC)

I'm glad the bot is happy; it was annoying me too  :-) I don't revert except in cases of vandalism. These bots are a great service, so thanks! - Gauge 05:30, 26 Mar 2005 (UTC)

## Wassup yo

Hi, Mike. I noticed your opponent on the UMBC entry rolled back my edit and branded me a vandal. As if! Mine was merely pointed social commentary, reflecting on the silliness of the place-name-quibbling. I'm a killer. - Deadbarnacle

I don't consider them to be "opponents". Rather, they are other contributors acting in good faith. - Gauge 20:12, 28 July 2005 (UTC)

## Hi

Just noticed that you're editing articles on my watchlist, so I thought I'd say hi! linas 04:35, 20 May 2005 (UTC)

Hi :-) I recently became interested in analytic number theory so I've been browsing and editing those pages quite a bit. In the modular forms article I was wondering if F could somehow be interpreted as a functor. Would you know anything about this? - Gauge 04:43, 20 May 2005 (UTC)
I've never had any formal education on category theory, and whenever I try to read a book on it, I fall asleep on page 3 because it always seems too simple. User:Charles Matthews would know; but he's also quite busy as he's the math overlord here on Wikipedia.

## Color charge

I can't quite give an accurate definition for color charge off the top of my head. I think you can say that the color charge of a Lie algebra (any Lie algebra) is the same as the root system of the algebra. To be more correct, we should really talk about a given irreducible and/or semi-simple rep. The reason that this is so is understood by working with the principal bundle of the associated Lie group. The connection (fiber bundle) is Lie-algebra valued. The Hamiltonian mechanics of a classical (not quantum!) particle coupled to the principal bundle is such that the momentum can be written as

${\displaystyle p=d+qA}$

where d is the exterior derivative on the base space (or the covariant derivative if the base space is curved) and A is the connection on the principal G-bundle. Here q is the charge, it is a vector that multiplies the Lie-algebra-valued A so that p becomes a vector in the space of the Lie algebra as well. It should be immediately recognized that p is the covarient derivative on the fiber bundle. When the underlying manifold (the base space) is physical space-time, then p can be understood to be the momentum of a particle with a given color charge, being acted on by the gauge field A.

The term charge comes from analogy to the electromagnetic field. Mathematicaly speaking, the elecromagentic field can be understood to be a line bundle or a principal bundle with fibre U(1). The strength of the electromagnetic field (literally, the strength of the electric field and the magnetic field) is given by the curvature ${\displaystyle F=dA}$ of the connection A. (In Minkowski spacetime, one of the four components of A is called the electric potential and the other three are called the magnetic vector potential). (More generally, the curvature of a principal bundle is given by ${\displaystyle F=dA+[A,A]}$ but the second term vanishes when the Lie algebra is Abelian. In the non-Abelian case, F is known as the Yang-Mills field.) The traditional coupling of an electrically charged particle is given thorough it's momentum, p=d+qA as above, the momentum being important as it gives the particle's motion through space, and, in the fourth component, through time. The momentum can essentially be understood to be the covariant derivative on the electromagnetic line bundle.

The root system for U(1) is trivial, it is just the scalar 1. Thus, the fiber-bundle interpretation immediately gives some insight to the problem of the quantization of the electric charge: the charge is not just any value, it is a particular value,and that value is 1.

In quantum chromodynamics, the relevant Lie group is SU(3) and the Lie algebra is the smallest irreducible representation su(3). Actually, su(3) has a pair of conjugate representations; they look identical except that they are complex conjugates. The root system can best be understood to be two pairs of three vectors each, oriented 120 degrees apart, forming a Star of David. The vectors are of length 1/2; one triangle belongs to one representation, the other to its conjugate. Quarks transform as one representation, anti-quarks transform as the conjugate representation. That is, the color charge of a quark is thus associated with one of the three vectors, and the charge of the anti-quark is associated with one of the vectors of the conjugate representation.

Generalizations to supersymmetry and to curved base spaces follow the same pattern laid out here: one defines a principal bundle over some group, and couples fields by means of the covariant derivative on the bundle. For good measure, one recognizes that four-dimensional space-time can be represented by a pair of spinors, each spinor transforming under su(2) or its complex conjugate. Spinors naturally have an anti-symmetric algebra, by means of the Pauli exclusion principle. One fundamental problem of supersymmetry is that there are so many different Lie groups and couplings and representations one can choose from, leading to a bewildering number of fields and charges and the like.

Phew. Heh, reasonable for an imprecise definition? I'm gonna copy this into the article,I guess ... linas 05:24, 21 May 2005 (UTC)

Outstanding work! More than I expected, and certainly welcome. I will ponder this for a while. - Gauge 05:34, 21 May 2005 (UTC)

Crap. Actually, there is a glaring, embarassing error in there. I need to think it through. Wham-bam job, leads to errors. Dohh. Unfortunately I already copied into the color-charge page, so I fully expect to get whacked in a few hours when others notice. linas 06:53, 21 May 2005 (UTC)

Err, well, at the hand-waving level, I guess maybe it can pass muster. I'll fix the wording in the main article. For quantum fields, the field carries the charge, not q, so q becomes this dead scalar, and the field becomes a vector. linas 06:58, 21 May 2005 (UTC)

## Proof of the Sylow Theorems

The following proofs are based on combinatorial arguments of Wielandt and give much shorter proofs of the Sylow theorems than those found in most texts. In the following, we use a | b as notation for "a divides b" and a ${\displaystyle \nmid }$ b for the negation of this statement.

Theorem 1: A finite group G whose order |G| is divisible by a prime power pk has a subgroup of order pk.

Proof: Let |G| = pkm, and let pr be chosen such that no higher power of p divides m. Let Ω denote the set of subsets of G of size pk and note that |Ω| = ${\displaystyle {p^{k}m \choose p^{k}}\mathrm {,} }$ and furthermore that pr+1 ${\displaystyle \nmid }$ ${\displaystyle {p^{k}m \choose p^{k}}}$ by the choice of r. Let G act on Ω by left multiplication. It follows that there is an element A ∈ Ω with an orbit θ = AG such that pr+1 ${\displaystyle \nmid }$ |θ|. Now |θ| = |AG| = [G : GA] where GA denotes the stabilizer subgroup of the set A, hence pk | |GA| so pk ≤ |GA|. Note that the elements gaA for aA are distinct under the action of GA so that |A| ≥ |GA| and therefore |GA| = pk. Then GA is the desired subgroup.

Lemma: Let G be a finite p-group, let G act on a finite set Ω, and let Ω0 denote the set of points of Ω that are fixed under the action of G. Then |Ω| ≡ |Ω0| mod p.

Proof: Write Ω as a disjoint sum of its orbits under G. Any element x ∈ Ω not fixed by G will lie in an orbit of order |G|/|CG(x)| (where CG(x) denotes the centralizer), which is a multiple of p by assumption. The result follows immediately.

Theorem 2: If H is a p-subgroup of a finite group G and P is a Sylow p-subgroup of G then there exists a gG such that HgPg−1. In particular, the Sylow p-subgroups for a fixed prime p are conjugate in G.

Proof: Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H on Ω, we see that |Ω0| ≡ |Ω| = [G : P] mod p. Now p ${\displaystyle \nmid }$ [G : P] by definition so p ${\displaystyle \nmid }$0|, hence in particular |Ω0| ≠ 0 so there exists some gP ∈ Ω0. It follows that hgP = gP so g−1hgP = P, g−1hgP, and thus hgPg−1hH, so that HgPg−1 for some gG. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg−1| so that H = gPg−1 for some gG.

Theorem 3: The number of Sylow p-subgroups of a finite group G divides the order of G and is congruent to 1 mod p.

Proof: By Theorem 2, the number of Sylow p-subgroups in G is equal to [G : NG(P)], where P is any such subgroup, and NG(P) denotes the normalizer of P in G, so this number is a divisor of |G|. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω0 and observe that then Q = xQx−1 for all xP so that PNG(Q). By Theorem 2, P and Q are conjugate in NG(Q) in particular, and Q is normal in NG(Q), so then P = Q. It follows that Ω = {P} so that, by the Lemma, |Ω| ≡ |Ω0| = 1 mod p.

## Manifold/rewrite/differentiable manifold

Hi Gauge,

manifold is being rewritten and top. and diff. manifold are being split off. They haven't received much editing however. --MarSch 29 June 2005 13:39 (UTC)

We're making a final push on the manifold/rewrite page before either replacing or merging the original page. Your eyes are requested, and your edits, if you like. KSmrq 00:10, 2005 August 11 (UTC)

## Regarding U of Baltimore county

I found that it is within the boundaries of the Catonsville CDP. However, Catonsville is just that - a CDP. Baltimore County is the only municipal government that serves the University therefore it should be also mentioned. Now, may I edit the article? (And of course state that it has a Baltimore mailing address! WhisperToMe 20:49, 11 July 2005 (UTC)

Please go ahead and copy what you have here into the discussion page before making the change. That will ensure that there is some rationale for the edit. Thanks for your patience. - Gauge 21:09, 11 July 2005 (UTC)
I already did so. Technically, it's not "changing" the location as I am only tacking on "Baltimore County". That, and the guy who is reverting is supposed to be blocked, as he is User:Boothy443. WhisperToMe 22:34, 11 July 2005 (UTC)

## Model categories

Hi, I've noticed you seem to be hip to model categories. If you have any time, I'd appreciate any comments you might have about my article-in-progress. As you might imagine this is taking a long time. Take a look and let me know what you think. I think we ought to work especially hard on this one. There is not much standardization on the way fibrations, etc., are discussed around here, and I think a solid model categories page might be the place to start asking for it. Thanks in advance. Dave Rosoff 21:44, August 8, 2005 (UTC)

## math tags

I just noticed your edits to geodesic ... Oh, please, please don't change math tags to normal markup; I've been going in exactly the opposite direction. For two reasons: html markup looks nasty in my browser, and also, I'm anticipating further developments in mathml, as per discussions in WP:WPM. Should we take this conversation there? linas 01:38, 21 August 2005 (UTC)

Yeah, I agree that there needs to be a standard. My understanding was that the current standard recommended using math tags only for displaying equations outside the body of the text, as the inline images cause text size/alignment problems. We'll move this discussion over to WP:WPM per your suggestion. Thanks. - Gauge 01:46, 21 August 2005 (UTC)

## some work in progress

#### Torus

The group Z2 acts on on the plane R2 by translation. An element (ab) of Z2 moves a point (xy) of R2 to the point (x+ay+b). The resulting quotient space R2 / Z2 is the torus.

#### Projective space

The two-element group Z2 := Z/2Z = {0, 1} has an action on spheres by reflections, with 0 acting as the identity and 1 acting as the reflection at the origin, mapping a point to its antipode. The resulting quotient space Sn/Z2 is projective space. Projective space is a topological manifold.
A finite cylinder is a manifold with boundary.

#### Cylinder

The Cartesian product S1 × R of a circle with the real line is the cylinder.
If a closed interval, say [0, 1], is used instead of the real line, the resulting manifold S1 × [0, 1] is a finite cylinder and has a boundary consisting of two disjoint circles.

#### Torus

The Cartesian product S1 × S1 of two circles is the torus. Since neither circle has a boundary, neither does the torus.

#### Cylinder

When the two edges of a strip are glued together the result is a cylinder.
The strip can be defined as the product manifold R × [0, 1]. Its boundary has two components, each one a copy of R, and
${\displaystyle (x,0)\mapsto (x,1)}$
defines a diffeomorphism between them.

#### Klein bottle

A sphere with a hole has one boundary component, a circle. This is also true of the Möbius strip with boundary. If the sphere and Möbius strip are glued together along these diffeomorphic boundaries, the result is the Klein bottle.

## instanton

Hi Gauge, I've done some work to cut down on the physics jargon. Please take a look. --MarSch 10:12, 5 September 2005 (UTC)

## Another mathematician

Gauge, I noticed your edit at Chern's page. I have written ar article about his colleague, Jim Simons, who co-authored the paper which resulted in the Chern-Simons theory.

Simons' article could use some attention in two areas: organizing his acedemic credentials, both as a student and faculty, and double-checking to make sure the mathematical "things" described in the article are worded correctly, and meaningfully. I'm essentially the only author of the article up until now.

If you know W'pedians who might like to tackle it, or try it yourself, you might find the subject interesting.

Regards,

paul klenk 08:10, 6 September 2005 (UTC)

## Article requests

I am very curious what brought you to request articles on the matrix logarithm and the Magnus series, as these subject are very close to my research. Of course, you don't need to tell, but like I said, I'm curious. -- Jitse Niesen (talk) 00:49, 11 September 2005 (UTC)

I was going through matrix exponential and noted the red links to these topics, so I added them as requests. I suppose my motivation was just curiosity. It certainly is not my field. - Gauge 19:00, 11 September 2005 (UTC)

Aha. I've a sneaking suspicion that I actually added these red links myself. Funny to see how my activities come back to haunt me. -- Jitse Niesen (talk) 20:30, 11 September 2005 (UTC)

## Science pearls

Hello,

Since you contributed in the past to the publications’ lists, I thought that you might be interested in this new project. I’ll be glad if you will continue contributing. Thanks,APH 11:01, 11 September 2005 (UTC)

## Simons

I've left a question on Talk:James Harris Simons about the Bernstein conjecture. paul klenk talk 06:45, 28 September 2005 (UTC)

Thanks for your answer about the Bernstein conjection and Jim Simons, which I noticed after a check following my long absence.
Regards, Paul

## Please vote on list of lists, a featured list candidate

Please vote at Wikipedia:Featured list candidates/List of lists of mathematical topics. Michael Hardy 20:28, 13 October 2005 (UTC)

## Woodin

Thanks for catching the typo on W. Hugh Woodin. It's there on his Mathematics Genealogy Project page, from which I copied and pasted; I would probably never have noticed. --Trovatore 18:20, 29 October 2005 (UTC)

## Operator topology

I'm glad you like the intro, and the feedback is very appreciated. Brian Tvedt 13:30, 12 January 2006 (UTC)

## Borel equivalence relations

I noticed you added a section to Borel equivalence relation about "Kuratowski's theorem". I'm not sure the connection to the subject matter is clearly expressed, particularly in the form in which the theorem is stated. There's a corollary that the relation of equality on one uncountable Polish space is always Borel reducible to equality on any other uncountable Polish space; stated that way it might be relevant, though maybe just as the start of the exposition of the various Borel cardinalities in the plan for the page (see the talk page; I admit I've been lazy about following through with the plan). --Trovatore 22:35, 23 January 2006 (UTC)

Feel free to move that content to another article if you don't think it fits there. I just happened to have a reference in front of me and decided to write it up somewhere. Your plan looks great to me; I hope you can eventually get around to doing it all. - Gauge 01:35, 24 January 2006 (UTC)

Removing stub tages from articles at will is not encouraged.Thank you.Prasi90 07:39, 25 February 2006 (UTC)

The stub tags were removed because it is no longer a stub. Thank you. - Gauge 07:47, 25 February 2006 (UTC)

## Algebraic spaces

In your article on algebraic spaces, the section "Facts about algebraic spaces" does not seem quite right. Shouldn't there be "algebraic space" instead of "scheme" in several places? I do not know algebraic spaces, so I would rather not edit this myself. Best, 130.237.198.87 18:25, 27 March 2006 (UTC)

My source for these was Artin's book. It looks okay to me since algebraic spaces are more general than schemes, although certainly I could have misstated something. - Gauge 17:34, 1 April 2006 (UTC)
I found the statement in Artin's book, and you have stated it exactly as it reads there. The thing is that I guess that Artin means something like "algebraic space of dimension 1,2" when he writes algebraic curve, surface. Then the statements make sense. However, I think it is usually understood that one refers to varieties when saying algebraic curve etc, and then they are schemes by definition, which makes some statements trivial, and some false. Anyhow, since I now have learnt what an algebraic space is, I'll edit the section when I figure out some good terminology. Spakoj 09:38, 5 April 2006 (UTC)
Sounds good to me. As long as the meaning is clear everything should be okay. - Gauge 22:46, 11 April 2006 (UTC)

## sheaves vs function spaces

I don't like too much your "precision" O -> O(Ω) in wave front set: in fact the text refers to the (pre)sheaf, and not just "one" of these function spaces.— MFH:Talk 23:34, 30 March 2006 (UTC)

I've attempted to correct it, but I'm not an expert so feel free to change it. - Gauge 17:58, 1 April 2006 (UTC)

## Italicise definitions

I noticed a recent edit you made. Bolding shold only be made to synonyms of the article title. Definitions are usually distinguished by italics. Dysprosia 07:46, 24 April 2006 (UTC)

The Wikipedia:Manual of Style (mathematics) does not explicitly specify how to highlight definitions, but does refer to Wikipedia:Technical terms and definitions in the sidebox. There, I find that bold is to be used for the first time a new term is defined (either plain bold or italic, depending on the "rare technical term" qualification). The article also states that italics can be used to demonstrate usage thereafter. Therefore, from what I can gather, this usage of bold is correct according to style guidelines. We can discuss this further at Wikipedia talk:WikiProject Mathematics if you want to propose a new convention. Thanks for your attention. - Gauge 20:08, 17 June 2006 (UTC)

## Mathematics CotW

Hey Gauge, I am writing you to let you know that the Mathematics Collaboration of the week(soon to "of the month") is getting an overhaul of sorts and I would encourage you to participate in whatever way you can, i.e. nominate an article, contribute to an article, or sign up to be part of the project. Any help would be greatly appreciated, thanks--Cronholm144 21:28, 13 May 2007 (UTC)