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Mathematics Research Statements' Journal
 
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Below are the 6 most recent journal entries recorded in Mathematics Research Statements' LiveJournal:

Wednesday, January 2nd, 2008
10:15 am
[patrickwonders]
Disitinct Structures on S7

I didn't initially post here because I am not currently doing anything approaching mathematical research. However, forvrkate suggested (way long ago), that I post about the thing I most want to understand. So, here it is (six months late)... [Note: this is mostly from memory with a little nudging from Mathworld, so don't take anything below as Gospel Truth.]

In 1959 (1958?), John Milnor showed that there are 28 distinct (up to diffeomorphism) structures on S7.

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On a random web page (which I can no longer find), Tony Smith claimed this was because there are exactly four real, normed division algebras.

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Elsewhere, though I cannot find that either at the moment, Tony Smith claimed there are only four real, normed division algebras because 0 is prime (run with it), 1 is prime, 2 is prime, and 3 is prime, but 4 is composite.

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So, I want to understand how Milnor established there are 28 distinct structures, what those are like, why this is connected to division algebras, what Spin(n) is like, and how the compositeness of four rules out division algebras of higher orders.



Current Mood: curious
Thursday, December 6th, 2007
8:56 pm
[lainlain29]
S.O.S., emergency! ALGEBRA AND THEORY OF NUMBERS

dear members of community, i need your help a lot!

the problem is - i can't find any info on topic
"theory of divisibility in the ring of polynomials with TWO variables and their usage"

i've looked through all catalogues of our uni library but in vain, there is only one Russian book which is not enough for my research.

if you can advise relevant links on free e-materials or send me the scanned books on topic (bereginya4891@ukr.net), i'll be very-very grateful!

THANKS IN ADVANCE!

Tuesday, June 5th, 2007
7:19 pm
[cnv]
Graph theory
Does anyone know of any literature about crossing numbers on the Mobius strip? If not, any idea why this hasn't been explored?

Thanks
Sunday, June 3rd, 2007
10:39 am
[hokie]
pseudoholomorphic curves and reeb dynamics
Let M be a 2n+1 manifold equipped with a nowhere integrable plane distribution ξ (i.e., a 2 dimensional distribution of the tangent bundle that is not even locally the tangent space to a surface. ξ is called a contact structure on M. Locally, ξ can always be given as the kernel of a 1-form λ with the property that λ ∧ dλn is a volume form. In other words, nonintegrability is expressed as the failure of the Frobenius condition. If M is coorientable, ξ is given globally by some λ failing the Frobenius condition, called a contact form. For the rest of this, we shall assume that M is coorientable and 3-dimensional.

There is a distinguished vector field associated to the contact manifold (M,λ). The Frobenius condition's failure tells us that dλ is nondegenerate on ξ, and thus ker dλ is one-dimensional in each tangent space. We can then form a vector field X by requiring iXdλ=0 and iXλ=1. X is called the Reeb vector field.

Currently, my research is oriented towards understanding more about the dynamics of the Reeb vector field on S3 for a choice of contact forms. The main tool that I use to this end is Gromov's theory of pseudoholomorphic curves in a symplectic manifold, introduced in 1985. A symplectic manifold is a 2n-manifold W equipped with a nondegenerate closed 2-form ω, and we assume that W also has been given an almost complex structure J, i.e., J:TW → TW and J2=-1. A pseudoholomorphic curve is a function from a Riemann surface (which can always be realized as a punctured sphere) into the symplectic manifold satisfying a Cauchy-Riemann-type operator.

We associate to any contact manifold (M,λ) the symplectic manifold (W=R x M, ω=d(etlambda)), where t is the R-coordinate. ω is a symplectic form by virtue of λ satisfying the contact property. Choosing an almost complex structure J on ξ gives a canonical choice for an almost complex structure on W, which I will also call J. Then a pseudoholomorphic curve is a map û: S2\Γ → W, where Γ is a finite set of punctures, satisfying the Cauchy-Riemann equation ûs+J(û)ût=0.

In 1993, Hofer proved that if the Reeb vector field has isolated periodic orbits, at each puncture a pseudoholomorphic curve is asymptotic to a periodic orbit of the Reeb field and that this convergence is exponential. Thus the existence of pseudoholomorphic curves and their properties allow us to understand something about the dynamics of the Reeb field. Using this fact, Hofer was able to prove the Weinstein conjecture for several classes of contact manifolds (the conjecture states that every Reeb vector field has at least one periodic orbit and was recent proven in all cases by Cliff Taubes).

Using this technology, Hofer, Wysocki, and Zehnder have in recent years proven many results about Reeb dynamics. For example, they proved in 2002 that for a generic choice of contact structure on S3, the Reeb field has precisely two or infinitely many periodic orbits. My research at the moment is concerned with trying to extend this result.
Thursday, May 31st, 2007
9:59 pm
[forvrkate]
Relating to Tarski's Finite Base Problem
ON NONFINITELY AXIOMATIZABLE ALGEBRAS
We’ll call a nonempty set equipped with finitely many finitary operations an algebra. The equational theory of an algebra is the set consisting of all equations true in that algebra. In the case of groups, one such equation might be the associative identity “(xy)z=x(yz)”; in the case of rings, one such equation might be the distributive law “x(y+z)=xy+xz”. We make a distinction between tautologies (things like “3+1=4”) and equations (things like “x+y=y+x”). The associative identity really ought to read “For any x, y, z in the set, we have (xy)z=x(yz).” It will be fine to think of equations as sentences that begin with the words “For every.”

If there exists a finite list of equations true in an algebra from which all equations true in the algebra can be deduced, we say the algebra is finitely based. If no such finite list exists, we say the algebra is nonfinitely based. (We also have another idea of an algebra being inherently nonfinitely based, which is the state of being nonfinitely based in an "infectious manner." An algebra being INFB is a stronger condition than just being NFB.)

For my current research, I am restricting my attention to finite algebras, that is, those whose underlying sets have only finitely many elements. You might wonder what sorts (if any) of these finite algebras are finitely based. It turns out that every finite group is finitely based (Oates & Powell, 1965), as is every finite ring (L’vov, 1973, and Kruse, 1971). So are all finite lattices (McKenzie, 1970), all commutative semigroups (Perkins, 1969), and all nontrivial Boolean algebras.

If all the standard sorts of algebras are finitely based, you might wonder which ones (if any) fail to be finitely based. The smallest example was discovered by Murskii; he found a 3-element algebra with a single binary operation that fails to be finitely based. His algebra is based on the set {0, a, b} in which the nonzero products are ab=a, ba=b, bb=b (all other products are zero). His algebra is the smallest NFB algebra due to Lyndon's 1951 result proving that all two-element algebras are finitely based. All one element algebras are finitely based by the equation "x=y."

In the 1990s, Ralph McKenzie proved that the question of whether an algebra is finitely based is undecidable. An immediate consequence of his result is that we cannot hope to find a list of necessary and sufficient conditions for an algebra to be nonfinitely based. My research focuses on figuring out how nonfinitely based algebras arise. For more about equational theory, see "Minimum bases for equational theories of groups and rings: the work of Alfred Tarski and Thomas C. Green" by George F. McNulty, found here.
9:41 pm
[forvrkate]
Hi and welcome! I created this community out of curiosity for what my fellow LJ research mathematicians study. It is my hope to compile brief [≤ 500 word] research summaries of various areas of mathematics.

  • Please write your statement/summary in 500 or fewer words.

  • In order to keep the statements short, please write your summary to an audience of readers familiar with undergraduate level abstract algebra and real analysis. If you'd prefer to write your statement at a level requiring more background, feel free to add links to articles pertaining to any requisite material.
  • If you have a question you'd like answered by someone in the field -- like, "What's up with the Hodge conjecture?" -- feel free to ask! As a corollary, if someone asks about something you know about, feel free to make a post in response.
  • Please tag your entry with an appropriate heading, such as "topology" or "numerical methods." Use keywords that are general enough to be used more than once, and specific enough to be interesting for the casual browser.
I'm not a rules aficionado, nor do I have much interest in being an active moderator. Play nice, share ideas, and hopefully we'll all learn something.
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