User oliver - MathOverflow

Biography of Felix Hausdorff

2011-03-01T20:49:36Z

Felix Hausdorff was of course a great mathematician, who had major effects on several branches of mathematics. However he also wrote literature and philosophy and was affiliated with important German musicians. When Nazism came to power, Hausdorff failed to escape in time, lost his job, and finally committed suicide in order to avoid being sent to concentration camp.

The above summary of Hausdorff's life comes from reading his wikipedia page and other similar documents. I would like to learn more about him, but for some reason I don't seem to be able to find a book-length biography. (Can there really not be one?)

Question: What is a good source for a detailed biography of Felix Hausdorff?

Answer by Oliver for What is so "plactic" about the plactic monoid?

2012-01-22T04:09:51Z

The English translation of Symmetric Functions, Schubert Polynomials and Degeneracy Loci by Laurent Manivel contains the following footnote on the phrase "plactic ring":

From the Greek $\pi \lambda \alpha \xi$, flat place, stone plate, tablet. This terminology is due to Lascoux and Sch\"utzenberger.

There is no evidence provided for this assertion, but the fact that the author is French makes me suspicious that he has insider information. There is no discussion of why this name was chosen or what it is meant to suggest.

What is so "plactic" about the plactic monoid?

2011-09-24T01:49:01Z

The plactic monoid is the monoid consisting of all words from the alphabet $\mathbb{Z}^+$ modulo certain relations. It is important mainly because its elements enumerate semistandard Young tableaux.

I believe the plactic monoid was introduced by Knuth, but without that name. Lascoux and Schützenberger named it "le monoïde plaxique" in a French paper (1981) of the same name. (DISCLAIMER: I have never seen that paper; perhaps my second question is answered in it.)

Several questions:

1) How did plaxique $\rightarrow$ plactic? (This isn't the most obvious Anglicization; note that the MathSciNet entry for the original Lascoux/Schützenberger paper translates the title as "Plaxic'' monoids.) Who introduced the latter form of the word and why?

2) What is plactic/plaxique supposed to mean? As far as I know neither was a word in their respective languages before being applied to the word monoid/monoïde. I am entertaining an etymology from Greek $\pi \lambda \alpha \xi$ "flat surface," but I don't find it very compelling.

What does the σ in σ-algebra stand for?

2011-08-29T21:27:27Z

I was tutoring someone in analysis and realized I have no idea where this notation comes from (or analogous terms: σ-additive, σ-ring, etc). I would like to know why the letter σ was chosen. I can't think of anything relevant that starts with "S" in either English or French. My German is nearly nonexistent, but I didn't see an explanation while trying to read the German wikipedia page.

Bonus points if you can tell me who introduced this notation and when.

(By the way, I really don't like this notation very much. I think it would be much more reasonable if we just wrote "$\aleph_1$-algebra" instead. Or better yet, replaced "algebra" with a less overloaded word. But I might change my mind, if it turns out there is a good explanation for the σ!)

Why should I believe the Singular Cardinal Hypothesis?

2011-07-01T19:19:47Z

The Singular Cardinal Hypothesis (SCH) is the statement that $\kappa^{cf(\kappa)} = \kappa^+ \cdot 2^{cf(\kappa)}$ for every singular cardinal $\kappa$ (or various equivalent statements).

It is obviously implied by the Generalized Continuum Hypothesis. It is also implied by the Proper Forcing Axiom (under which $2^{\aleph_0} = \aleph_2$). Nonetheless it doesn't seem terribly compelling to me. But I am trying to learn to appreciate it!

Why should I believe SCH? (Now when I say "believe," it isn't clear exactly what I mean by that. There are obviously plenty of models of set theory in which SCH holds, and they are certainly worth studying, but somehow they aren't the models that feel most "realistic" in my little head.) What I want is an answer in the style of Maddy's Believing the axioms, explaining why I should "like" (or not) this hypothesis.

Some thoughts on why SCH isn't so unreasonable:

Somehow I find very compelling the result that SCH holds above the first strongly compact cardinal. This is what makes it seem most reasonable to me that SCH should hold everywhere.
SCH is implied by various contradictory axioms. Its negation is equiconsistent with the existence of a fairly large cardinal.
It is obvious that $\kappa^{cf(\kappa)} \geq \kappa^+ \cdot 2^{cf(\kappa)}$, so SCH is just saying that the cardinality on the left shouldn't be any larger than strictly necessary.
It makes cardinal arithmetic much easier! (But maybe this is an argument against SCH too...)

Why else should SCH seem reasonable?

Mathematical ideas named after places

2011-05-11T14:40:13Z

This question is quite unimportant, so feel free to close if you think it is inappropriate.

I've been thinking about how mathematicians come up with names for the ideas/objects they study, and how that differs from the practices of people in other fields.

It seems that almost always we do one of two things: 1) we pick a name that describes some feature of the object (sometimes not very well, e.g. flat modules, sets of second category), or 2) we name it after a person (who may or may not have studied that object).

Very rarely we name something after a place. (This is much more common in other fields.) I can think of only 3 examples:

*Japanese rings

*Polish spaces

*Tropical geometry

Does anyone know of any other examples in mathematics?

Answer by Oliver for Motivation behind Tutte's 1-factor theorem

2011-04-10T05:01:54Z

To make the condition seem natural, let's just try to copy Hall's Condition. So what's the first thing we do? We grab some set of vertices. In Hall's Condition this set is often called $X$ and we take it inside one of the partite sets, but we will call it $U$ and choose it arbitrarily since there there isn't any obvious way to restrict its choice in a general graph.

Then what is the next thing we do in Hall's Condition? We check that $X$ is big enough to take care of all the edges in a perfect matching that must come out of $X$. In Hall's Condition this means we need to check that $X$ has at least as many neighbors as there are vertices in $X$, since no edge of the perfect matching can lie inside $X$. In the general case, the analogous thing to check is that there are at least as many vertices in $U$ as there are odd components of $[V\backslash U]$ because every odd component must send an edge to $U$ in any perfect matching. (You are just trying to guarantee that a large set of edges must come out of $U$ in any perfect matching, and the number of odd components of $[V\backslash U]$ is the only obvious condition that does so. It's really the only thing you can say.) And well now you've stumbled upon Tutte's Condition.

The (sort of) tricky thing then is to realize that the condition is actually sufficient.

Answer by Oliver for Consistency strength needed for applied mathematics

2011-02-09T01:17:48Z

It seems like your interest is mainly in the philosophical side of your question, so I'd like to address that directly, although I'm not even close to being a philosopher of mathematics.

It is not strictly true that an empiricist viewpoint can only justify the consistency strength needed for immediate application. The empiricist argument that you give in your question for believing mathematics sounds a lot like Quine. Quine like you, argued for accepting the validity of (some) mathematics because of the utility of mathematics in the sciences. Other mathematics he did not accept because he could not think of scientific applications. For example, Quine advocated consistent use of the axiom of constructibility ($V=L$) throughout mathematics because he thought that doing so would suffice for the purposes of applied mathematics.

The problem with this viewpoint is that it draws a ragged edge through the heart of mathematics, denying the validity of important work that often motivates and interconnects with mathematics on the other (justifiable to Quine) side of the barrier. (For example, there is an MO question that I can't find right now about theorems that were first proved using the axiom of choice, and later proved with weaker hypotheses; similar examples exist of theorems first proved using large cardinal axioms, and later shown to follow from ZFC alone.)

It is possible to be an empiricist and also accept the validity of the entire mathematical enterprise. A strong proponent of such a view is Penelope Maddy. I particularly recommend her book Second Philosophy in this context. Her arguments are delicate, so I will avoid trying to summarize them. However, like Quine, she accepts the validity of some mathematics because of its importance in applications, while, unlike Quine, she accepts the rest of mathematics because of the inherent unity of mathematics and the unreasonability of any cutting of mathematics into philosophically justified and unjustified pieces.

Answer by Oliver for Complexity of random knot with vertices on sphere

2011-02-05T16:51:16Z

I think you should look at this paper "The average crossing number of equilateral random polygons" by Y Diao, A Dobay, R B Kusner, K Millett and A Stasiak. http://iopscience.iop.org/0305-4470/36/46/002

It can't seem to get the full text right now, so I'm not sure exactly what their random knot model is. It certainly isn't quite what you wanted, as they require their polygons to be equilateral. In their model the crossing number grows as $n \; \ln n$.

If this doesn't answer your question, consider looking through the rest of Ken Millett's work. He is one of the main people working on random knots.

Answer by Oliver for Length of shortest possible knot

2011-01-30T21:57:59Z

The invariant you are talking about is usually called the "ropelength" of the knot. You can find some basic stuff at the wikpedia page http://en.wikipedia.org/wiki/Ropelength which also gives some good references. (Note that some people use unit circles, while other people use circles of diameter 1, so the reported ropelength differs by a factor of 2.)

The exact value of the ropelength is not known for any nontrivial knot. However in the case of the trefoil, there are some pretty good bounds. It is between 15.66 and 16.372 if we define ropelength using circles of diameter 1. The upper bound is believed to be tighter.

Notation: Exponent of a group

2010-11-01T00:26:01Z

The exponent of a group $G$ is the least positive $n$ such that $g^n = e$ for all $g \in G$. This is obviously a sensible name for the concept.

A notational awkwardness arises however when the group $G$ is abelian and written additively. I find it grating to refer to the least positive $n$ such that $\forall g \in G$ $ng = e$ as the exponent because there is nothing going on that even looks like exponentiation.

Is there an alternate terminology that can be used in this situation?

Can an infinite commutative ring have a finite (but nonzero) number of non-nilpotent zero-divisors?

2010-07-11T20:48:12Z

By a theorem of Ganesan, if a commutative ring not a domain has only finitely many zero-divisors, then the ring must be finite. (There are analogous results for non-commutative rings.)

There are plenty of examples of infinite rings with a finite number of nonzero nilpotents. There are also plenty of examples of infinite rings with an infinite number of zero-divisors, all of which are nilpotent.

However, I am unaware of any ring with an infinite number of zero-divisors, of which $0 < n < \infty$ are non-nilpotent.

Can anyone give an example or explain why this can't happen. I am mostly interested in the commutative case, but non-commutative examples would be interesting too.

How much can we say about the number of nilpotents in a finite local commutative ring?

2010-07-11T20:57:31Z

A commutative ring is local if it has a single maximal ideal. If the ring is finite, this implies that all elements are either units or nilpotents. Further, all finite local rings have prime power order.

Given a prime power $p^k$ and a positive integer $n < p^k$, under what conditions on $p, k, n$ does there exist a local ring $R$ with $|R| = p^k$ and $n$ nilpotents?

What are your favorite finite non-commutative rings?

2010-06-07T14:48:31Z

When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.

There are many convenient examples of commutative rings, both finite and infinite, and there are many convenient examples of infinite non-commutative rings. But I don't have a good collection of finite non-commutative rings to think about. I usually just think of a matrix ring over a finite field.

Do YOU have other examples that you particularly like/find easy to use/find to be a good source of counterexamples?

When forcing with a poset, why do we order the poset in the order that we do?

2010-03-11T18:58:57Z

In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfectly fine, but it seems intuitively backwards to me. If I were designing the notation for forcing, I would want the stronger condition to be larger. (Something I read says, I think, that Shelah uses the opposite convention that I find more intuitive. Is this so?)

Further, if we are forcing with a collection of partial functions (as we often do), we want the stronger condition to be the partial function with the larger domain. This leads us to a definition of the poset order whereby $f < g$ iff $f \supset g$. This seems notationally awkward.

Nonetheless, Cohen must have had some good reasons choosing the order that he did. What is/was the rational for Cohen's notational convention? Does it have benefits today, or is it just an artifact of a older approach to forcing?

Answer by Oliver for What are some examples of narrowly missed discoveries in the history of mathematics?

2010-03-26T02:49:27Z

Fulkerson came extremely close to a proof of the Perfect Graph Conjecture eventually proved by Lovasz. He was stuck on one relatively easy lemma, which he had apparently become convinced was false.

I don't know a reliable source for this, but I have heard that after learning that the conjecture was proven, Fulkerson went to his office and finished his own proof later that day.

Does the beth function have fixed points of arbitrarily large cofinality?

2010-03-06T17:12:09Z

Background

The beth function is defined recursively by: $\beth_0 = \aleph_0$, $\beth_{\alpha + 1} = 2^{\beth_\alpha}$, and $\beth_\lambda = \bigcup_{\alpha < \lambda} \beth_\alpha$. Since the beth function is strictly increasing and continuous, it is guaranteed to have arbitrarily large fixed points by the fixed-point theorem on normal functions.

The cofinality of an ordinal $\alpha$ is the smallest ordinal $\beta$ such that there are unbounded increasing functions $f : \beta \to \alpha$.

Comment by Oliver

2012-01-25T02:31:53Z

Regarding the question in your preamble, I think the unknotting number of $K$ could be thought of as a measure of "how many "moves" are needed to unravel $K$ using the 4th dimension".

Comment by Oliver

2011-09-27T15:19:55Z

Thanks for your good ideas and for the link to the paper especially!

Comment by Oliver

2011-09-07T01:34:46Z

Guess as to what you are asking: What is an efficient algorithm to identify pairs of nonadjacent vertices $u,v$ such that adding the edge $uv$ increases the chromatic number? If your graphs are general, you will not find such an algorithm, since as you point out, calculating chromatic numbers is NP-hard. However if your graphs all belong to some special class, then perhaps there will be a polynomial algorithm. Or if your graphs are all small, you shouldn't care about the asymptotics anyway.

Comment by Oliver

2011-09-07T01:27:29Z

Your operation 'Merge' is usually called edge-contraction. Your operation 'Separate' is usually called joining (!) the vertices $u,v$. I don't understand your game otherwise. Does the player have a goal?

Comment by Oliver

2011-08-29T22:28:42Z

@Michael You are certainly right from a pedagogical standpoint. I actually only get bothered when people use the term "κ-additive" for some cardinal κ in one sentence, and then use σ-additive in the next.

Comment by Oliver

2011-08-29T22:22:35Z

@Henry I doubt that the source language is French, since French has the alternate terminology "tribu."

Comment by Oliver

2011-08-17T22:14:26Z

Are you intending to use $y$ in two different ways? You introduce it as a bound variable representing an element of $x$, and then you write that you are looking for all $y \subset x$ such that $P(y)$.

Comment by Oliver

2011-08-06T03:41:08Z

This question seems fine to me. It is certainly of importance to the research community. And where else would Anonymous ask this?

Comment by Oliver

2011-07-02T04:57:18Z

@Andres: Please do--I would love to hear more of your thoughts about this!

Comment by Oliver

2011-07-02T04:55:56Z

As I said in the question, I agree that this is a very compelling piece of evidence in favor of SCH. Is it known if "strongly compact" is best possible here?

Comment by Oliver

2011-07-01T22:14:13Z

@Andreas Not explicitly, but I always got the impression that other people liked SCH and I never understood why. Maybe I was mistaken. Do you think it "seems reasonable"? Do you know people who do?

Comment by Oliver

2011-05-12T06:54:35Z

@Peter I agree that the quality of answers has declined significantly and I am voting to close. Still I don't think that in itself justifies down-voting.

Comment by Oliver

2011-05-05T18:14:33Z

Could you please edit this to make it more readable? Judging from the number of typos in your question, it doesn't look like you put very much effort into writing it. I don't see why I should exert more effect to figure out what you are asking.

Comment by Oliver

2011-03-05T05:18:08Z

The other difference in the experimental sciences is that the order in which authors are listed matters. As long as the student is first author, they will be given almost as much credit as if they were the only author.

Comment by Oliver

2011-03-02T03:16:53Z

Thank you for this! It was an interesting read, though not as thorough as I would have liked and a bit too focused on making a case for Hausdorff's importance.