User robert haraway - MathOverflow

What is the difference between hard and soft analysis?

2010-11-20T02:01:16Z

I have heard references to "hard" vs. "soft" analysis. What is the difference? It seems to do with generality versus nitty-gritty estimates, but I haven't gotten any responses more clear than that.

A remark of Connes

2011-03-02T04:05:44Z

In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that

I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point is that as soon as you have a non-standard number, you get a non-measurable set. And in Choquet’s circle, having well studied the Polish school, we knew that every set you can name is measurable; so it seemed utterly doomed to failure to try to use non-standard analysis to do physics.

What does he mean; what is he referring to?

A conjecture of Thurston and possibly Weeks too

2013-03-15T02:03:24Z

What is the status of the following conjecture:

"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one obtains a manifold whose shortest closed geodesic has length $2.12255\ldots$ with angle of twist $\pm 1.80911 \ldots,$ and has self-replicating behaviour when removed."

This is Conjecture 6.1 from p. 2568 of Thurston's expository paper "How to see 3-manifolds" (Class. Quantum Grav. 15 (1998) 2545--2571).

Hyperbolic Brunnian links and rectangular cusp shapes

2013-01-18T21:34:46Z

My question is as follows.

Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?

Here is what I mean:

The Borromean rings form a famous link $B$ (a smooth closed 1-manifold in the three-sphere):

Link $B$

This link has the property that removing a single component yields an unlink. This property is called the Brunnian property. Links with the Brunnian property are Brunnian links.

Here are some other Brunnian links:

Link $S$

Link $R$

Link $N$

(Rolfsen '76, p. 67)

Link $OMG$

(Rolfsen '76, p. 67)

The complement $S^3 - L = K_L$ of any link $L$ (Brunnian or not) may admit a complete hyperbolic structure (cf. Thurston '97, pp. 131–132, p. 147) with finite volume (cf. SnapPy). By Mostow-Prasad rigidity, this structure (if it exists) is a topological invariant. We may then write $K_L$ as the image of a quotient $D: \mathbb{H}^3 \to K_L$ of hyperbolic space by a freely acting discrete group $G \simeq \pi_1(K_L)$ of isometries. Each component $C$ of the link admits a regular neighborhood $N(C)$ that is the image under $q$ of an open horoball $\tilde{N}$. In fact it admits many such. The maximal such neighborhood is the maximal cusp neighborhood $m_C$.

The lifts of $m_C$ to the universal cover are horoballs. Pick one such lift $M_C$. The closures of all the lifts abut the (horospherical) boundary $\partial M_C$ at a discrete set $\Lambda$ of points. Let $\Gamma$ be the subgroup of $G$ that preserves $\partial M_C$, a peripheral subgroup of $G$. This subgroup acts transitively and freely on this set of points, so we may identify $\Gamma$ with $\Lambda$ by picking one point $o \in \Lambda$ to represent the identity of $\Gamma$.

As it turns out, in the induced metric, the horosphere $\partial M_C$ is isometric to the Euclidean plane, and $\Gamma$ acts on this by isometries. So $\Gamma$ is a discrete, torsion-free, freely acting group of isometries of the Euclidean plane. Therefore it is either $0,$ $\mathbb{Z}$, or $\mathbb{Z} \oplus \mathbb{Z}$.

If $\Gamma \simeq 0$ or $\mathbb{Z}$, then the quotient of the maximal cusp neighborhood by $\Gamma$ (and therefore by $G$) would have infinite volume. Conversely, assuming that $\Gamma \simeq \mathbb{Z} \oplus \mathbb{Z}$, one can show easily that the quotient of the maximal cusp neighborhood has finite volume.

Assume therefore that $\Gamma \simeq \mathbb{Z} \oplus \mathbb{Z}$. Fix an isometry $\phi$ of $\partial M_B$ with $\mathbb{C}$ sending $o$ to $0$. The elements $\gamma_m,$ $\gamma_\ell$ of $\Gamma$ corresponding to the meridian and the longitude of the link component $C$ generate $\Gamma$, so we might as well also ensure that our choice of isometry sends one of them to a positive real number. SnapPy's convention is to send the longitude to a positive real number, apparently, so let's use that convention. Then we say the cusp shape of the link component $C$ is the ordered pair of complex numbers $(\phi(\gamma_m), \phi(\gamma_\ell))$.

Here are SnapPy's computations for the cusp shapes of the components of the above Brunnian links (I have rounded the decimals):

Link $B$: all of shape $(5\cdot 10^{-17}+0.569 i,\,1.140)$ (by symmetry)
Link $S$: all of shape $(1\cdot 10^{-16}+0.550 i,\,1.181)$ (by symmetry)
Link $N$: no hyperbolic structure
Link $OMG$:
- cusp 0: $(1\cdot 10^{-16}+0.258 i,\,2.514)$
- cusp 1: $(-9\cdot 10^{-16}+0.262 i,\,2.471)$
- cusp 2: $(1\cdot 10^{-16}+0.316 i,\,2.051)$
- cusp 3: $(1\cdot 10^{-15}+0.431 i,\,1.506)$
- cusp 4: $(2\cdot 10^{-17}+0.5848 i,\,1.111)$
- cusp 5: $(-5\cdot 10^{-16}+0.232 i,\,2.796)$

That is to say, each of the above hyperbolic Brunnian links (almost certainly) has a rectangular cusp shape on every one of its components.

Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?

Answer by Robert Haraway for What are some examples of colorful language in serious mathematics papers?

2011-10-14T02:25:56Z

Here is a colorful rejoinder by D. Zagier (in his reprinted article on the dilogarithm) to colorful language by Ph. Elbaz-Vincent and H. Gangl:

[Ph. Elbaz-Vincent and H. Gangl] called these functions "polyanalogs," an amalgam of the words "analogue," "polylog," and "pollyanna" (an American term suggesting exaggerated or unwarranted optimism). Presumably the correct term for the case $m=2$ would then be "dianalog," which has a pleasing British flavo(u)r.

Unpublished Higman manuscript

2011-07-20T23:19:21Z

In his paper 'Natural constructions of the Mathieu groups,' Curtis references an unpublished manuscript of G. Higman with a "significant constuction which makes use of the outer automorphism of $S_6$ to construct $M_{12},$ and the outer automorphism of $M_{12}$ to construct $M_{24}.$"

Does anyone know to what Curtis is referring? Is there another source for these constructions?

Answer by Robert Haraway for Naturally occurring orderings

2011-05-07T03:32:11Z

Hrbacek, following Ballard, has recently put a certain partial ordering called $\sqsubseteq$ on absolutely everything in order to do nonstandard analysis a la Nelson (i.e. internal set theory):

Let $\mathscr{L}$ be the language of ZFC (and Tarski-Grothendieck if you insist). We say a well-formed formula of $\mathscr{L}$ is an $\in$-formula. We assert that ZFC holds for all well-formed $\in$-formulae.

Now we throw in $\sqsubseteq$ to $\mathscr{L}$ to get a bigger language $\mathscr{HB}.$ First, we assume that ZFC holds for all $\in$-formulae. Let us write $x \sqsubseteq_\alpha y$ as an abbreviation of $x \sqsubseteq \alpha \vee x \sqsubseteq y.$ Suppose we have a well-formed formula $P$ of $\mathscr{HB}.$ We write $P^\alpha$ for the replacement of every instance of $\sqsubseteq$ with $\sqsubseteq_\alpha.$ Let us also write $x \sqsubset y$ for $x \sqsubseteq y \wedge y \not\sqsubseteq x.$ We also write $2^A_{\mathrm{fin}}$ for the set of all finite subsets of $A.$ We also write $(\forall u \sqsubseteq v) P(u,v)$ for $(\forall u)(u \sqsubseteq v \implies P(u,v)),$ and so on for $\exists,$ and for $\in$ as well.

Then Hrbacek's GRIST is the following condition on $\sqsubseteq,$ with four axiom schemata:

R elativization condition on $\sqsubseteq$: $\sqsubseteq$ is a total dense preordering with minimal element $\emptyset$ and no maximal element; i.e. the conjunction of

Partial ordering: $(\forall u,v,w)((v\sqsubseteq u \wedge w \sqsubseteq v) \implies w \sqsubseteq u) \wedge u \sqsubseteq u;$
Preordering: $(\forall u,v)(u \sqsubseteq v \wedge v \sqsubseteq u);$
Minimality of $\emptyset$: $(\forall u)(\emptyset \sqsubseteq u);$
Illimitability: $(\forall u) (\exists v) (u \sqsubset v);$
Density: $(\forall u,v) (u \sqsubset v \implies (\exists w)(u \sqsubset w \sqsubset v)).$

Axiom schemata, in which we use words so as not to have our eyes completely glaze over:

For any well-formed formula $P$ of $\mathscr{HB}$ depending on finitely many variables,

T ransfer: for all $u \sqsubseteq v$ and $x_1,\ldots, x_n \sqsubseteq u,$

$P^u(x_1,\ldots,x_n) \iff P^v(x_1,\ldots,x_n)$
S tandardization: for all $u \sqsupset \emptyset$ and for all $A, x_1, \ldots, x_n,$ there are $v \sqsubset u$ and $B \sqsubset v$ such that, for every $w$ with $v \sqsupseteq w \sqsupset u,$

$(\forall y \sqsubseteq w)(y \in B \iff y \in A \wedge P^w(y,x_1,\ldots,x_n)).$
I dealization: For all $A \sqsubset v$ and all $x_1,\ldots,x_n,$

$(\forall a \in 2^A_{\mathrm{fin}})\big([a\sqsubset v \implies (\exists y)(\forall x \in a) P^v(x,y,x_1,\ldots,x_n)] \iff[(\exists y)(\forall x \in A)[x \sqsubset u \implies P^v(x,y,x_1,\ldots,x_n)]\big).$
G ranularity: For all $x_1,\ldots,x_k,$ if $(\exists u) P^u(x_1,\ldots,x_k),$ then

$(\exists u)[P^u(x_1,\ldots,x_k) \wedge (\forall v)(v \sqsubset u \implies \neg P^v(x_1,\ldots,x_n))]$

Answer by Robert Haraway for Geometric interpretation of Cartan's structure equations

2011-04-15T23:32:30Z

I think you may find an answer in Differential Geometry by Sharpe.

Answer by Robert Haraway for M12 simple sporadic group

2011-03-23T20:49:35Z

If you want an intuitive presentation of M12, also take a look at Curtis's construction.

Answer by Robert Haraway for Non-standard enlargements, $\zeta(s)$ and analytic continuation

2011-03-03T05:08:00Z

I ~~just~~ saw this on arXiv: Nonstandard Mathematics and New Zeta and L-Functions, the PhD thesis of one Benjamin Clare of U. Nottingham.

This Ph.D. thesis, prepared under the supervision of Professor Ivan Fesenko, defines new zeta functions in a nonstandard setting and their analytical properties are developed. Further, p-adic interpolation is presented within a nonstandard setting which enables the concept of interpolating with respect to two, or more, distinct primes to be defined. The final part of the dissertation examines the work of M. J. Shai Haran and makes initial attempts of viewing it from a nonstandard perspective.

(Corrections welcome if my answer betrays any basic misunderstanding!)

Answer by Robert Haraway for nonstandard analysis book recommendation

2011-03-02T04:13:43Z

I have learned some internal set theory (IST) from Lutz and Goze's Nonstandard analysis: a practical guide with applications. It is jam-packed with lots of interesting material, and has a nifty proof of the inverse function theorem. However, since it is a bunch of lecture notes, it is not as coherent as some other books, such as Robert's Nonstandard analysis or Nelson's own papers, his own unfinished book at http://www.math.princeton.edu/~nelson/books/1.pdf, or the probability book mentioned above.

So if you want to get excited about IST and get fun ideas for using it, read Lutz and Goze. To understand it, read Nelson or Robert.

Answer by Robert Haraway for Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness

2011-03-02T03:29:49Z

You may be able to convey the significance of de Rham cohomology to really wide audiences through electromagnetism.

I don't claim to understand all the physics (or topology for that matter), but see my friend Rob Kotiuga's book at http://library.msri.org/books/Book48/index.html. See, for instance, chapter 1D: Nineteenth-Century Problems Illustrating the First and Second Homology Groups, or pp. 30--32, "Chain complexes in electrical circuit theory."

Answer by Robert Haraway for Examples of common false beliefs in mathematics.

2011-02-21T04:02:18Z

Inversion is an automorphism of a group. ('Cause it, like, preserves the conjugacy classes and all that...)

Answer by Robert Haraway for What are some examples of colorful language in serious mathematics papers?

2011-02-03T22:22:32Z

John (Horton) Conway unrelentingly gets away with colorful, even whimsical language in definitions, in explanations, in paper titles, even in some book titles (The Sensual (Quadratic) Form.) Even in SPLAG, there is the following:

"...we earnestly recommend that you use

The Best Method: guess the correct answer, and then justify it." SPLAG, p. 302

On Numbers and Games is just rife with colorful stuff. (I'm surprised no one has pointed out this elephant in the room yet.) The next to last theorem of the book is

THEOREM 99: Any short all-small game G which has atomic weight zero is infinitesimal with respect to (double-up) and dominated by some superstar.

And the last words of the book are famously

"...a certain feeling of incompleteness prompts us to add a final theorem.

THEOREM 100. This is the last theorem in this book.

(The proof is obvious.)" ONAG p. 224

Answer by Robert Haraway for Escher, Conway, Kali, etc.

2011-01-03T02:59:28Z

Yes! I had to rummage around, but D. Huson has such a program; he now works at Uni Bielefeld in bioinformatics, so it's somewhat hard to find.

http://www-ab.informatik.uni-tuebingen.de/software/2dtiler/welcome.html

Answer by Robert Haraway for Favorite popular math book

2010-12-09T04:17:00Z

Title: The Magical Maze

Author: Ian Stewart

Short description: Various bits of counterintuitive, fundamental, and/or easily understood mathematics, to wit: modular arithmetic, Fibonacci numbers, depth-first search, the axiomatic method, the Monty Hall problem, the birthday paradoxes, wallpaper groups, cellular automata, dynamical systems, formalism, Godel incompleteness, Turing's halting problem, optimization problems (both continuous and discrete), algorithmic complexity, fractals, chaos, and Hausdorff dimension.

This book is, to a large extent, the reason I am in graduate school right now. I read it in the eighth grade, and now here I am.

Answer by Robert Haraway for Is there a ruler and compass construction of the common perpendicular of two geodesics in H^3?

2010-11-20T04:58:47Z

If you really want to use just straight-edge and compass, don't go into $3$-space at all! Coxeter points out the equivalence between the inversive plane and hyperbolic space in the paper The inversive plane and hyperbolic space. In this particular situation one can think of lines as pairs of points in the inversive plane. Two pairs of points correspond to perpendicular lines of hyperbolic space when the two pairs are harmonic. So the problem is:

Given two pairs of points $(a,b)$ and $(c,d),$ to construct using straightedge and compass the (unique) pair $(x,y)$ harmonic to both.

Fenchel asserts the existence of such a thing, and relates them to square roots, but without mentioning the geometric argument, so here goes.

We first start off easy, and assume that $a = \infty.$ We define the (two) geometric means of $(c,d)$ with respect to b to be given as follows. (Basically they are the complex square roots of $d$ where $b = 0$ and $c = 1$.) Bisect the angle $\angle c b d$ by a line $\ell.$ Let $C_c, C_d$ be the circles centered at $b$ through $c,d$ respectively. Intersect these with $\ell$ to get four points $p_c, p_c', p_d, p_d',$ so that $(p_c,p_d)$ separate $(p_c',p_d').$ Let $q$ be the midpoint of $p_c$ and $p_d.$ Draw the circle $D$ centered at $q$ through $p_c$ and $p_d.$ Draw the perpendicular $\ell_\perp$ to $\ell$ at $b,$ and intersect it with $D$ at a point $r$. Then draw the circle $E$ centered at $b$ through $r$, and intersect it with $\ell$ at the points $s_{(c,d)},t_{(c,d)}$. These are the geometric means of $(c,d)$ with respect to $b$.

Now, it should be that $(s_{(c,d)}, t_{(c,d)})$ is harmonic to both $(\infty, b)$ and $(c,d).$ (I haven't worked this out yet.) So that's what we were looking for. In the case that $a \neq \infty,$ just let $C$ be the circle centered at $a$ through $b,$ and denote by $I_C$ the inversion in $C.$ Then the points we're looking for are $I_C(s_{I_C(c,d)}, t_{I_C(c,d)}) = (\sigma_{(c,d)}, \tau_{(c,d)}).$ These give the endpoints for the common perpendicular to the hyperbolic lines with endpoints $(a,b)$ and $(c,d).$

It's probably all in Fenchel, anyway....

Answer by Robert Haraway for Books for hyperbolic geometry ( surfaces ) with exercises ?

2010-11-20T02:14:08Z

Al Marden's Outer circles is overflowing with excellent exercises. Although the majority of the book is about 3-manifolds, the first two chapters are an introduction to hyperbolic geometry brimming with vim.

Beardon's Geometry of Discrete Groups, Iversen's Hyperbolic Geometry, and Bonahon's Low-dimensional Geometry, and Katok's Fuchsian Groups all have exercises. Since you requested stuff specifically on surfaces, Katok may be the way to go. However, you may want to turn to other books for explanations at times, for this book is terse. Iversen has an entire chapter on covering spaces that you might want to look at.

Answer by Robert Haraway for Free, high quality mathematical writing online?

2010-11-19T04:33:05Z

A draft of Albert Marden's Outer circles: an introduction to hyperbolic 3-manifolds is online, on his website:

http://www.math.umn.edu/~am/book/outercircles.pdf

Edward Nelson's Radically elementary probability theory is also online, on his website:

http://www.math.princeton.edu/~nelson/books/rept.pdf

Comment by Robert Haraway

2013-04-20T00:44:02Z

Thank you in advance! It was just such a lecture that got me interested in mathematics, and now I am in graduate school.

Comment by Robert Haraway

2013-04-08T21:02:56Z

That's related (and interesting), but doesn't directly address the question, viz. what results have been proven first using NSA.

Comment by Robert Haraway

2013-02-13T02:47:41Z

@Ryan Budney: I believe you're referring to Mel Slugbate; cf. euclid.colorado.edu/~jnc/MelSlugbate.html

Comment by Robert Haraway

2013-01-23T15:23:37Z

Thank you very much!

Comment by Robert Haraway

2012-01-31T01:16:13Z

This question is not research-level, and would be better posted on math.stackexchange.com.

Comment by Robert Haraway

2011-10-04T21:11:29Z

This is explored seriously in Keisler's nonstandard calculus text Elementary Calculus; his treatment of the microscope methodology is based on work of Keith Stroyan.

Comment by Robert Haraway

2011-07-21T03:19:20Z

Unfortunately, Bogopolski only gives the definition of the Mathieu groups $M_{22},$ $M_{23},$ and $M_{24}$ so far as I can tell, and then only as automorphism groups of Steiner systems. Can you cite a page detailing the Higman construction?

Comment by Robert Haraway

2011-05-20T18:08:42Z

I think you may be looking for something along the lines of Nelson's internal set theory, or Hrbacek's extension thereof, GRIST.

Comment by Robert Haraway

2011-03-15T13:55:46Z

@Qiaochu: A comment from Eric Raymond on Plan 9 may be in order here: "Compared to Plan 9, Unix creaks and clanks and has obvious rust spots, but it gets the job done well enough to hold its position. There is a lesson here for ambitious system architects: the most dangerous enemy of a better solution is an existing codebase that is just good enough." The same could be said of bases for doing mathematics.

Comment by Robert Haraway

2011-03-03T04:46:48Z

I was really disappointed to see how thin Robert's book is....

Comment by Robert Haraway

2011-02-21T03:58:38Z

This was actually used as an April Fool's Joke by Martin Gardner....

Comment by Robert Haraway

2011-01-03T03:50:26Z

It was probably before that link was up, or before I had heard of orbifold notation.

Comment by Robert Haraway

2011-01-03T03:49:00Z

No; what I should say is that there was a bad link on Geometry Junkyard to this, and so I had to hunt for Huson on the web, and finally found him doing bioinformatics at Bielefeld.

Comment by Robert Haraway

2011-01-03T03:14:09Z

John Baez has a page on a similar picture of Dan Christensen, and some feathery patterns lend additional credibility to this: math.ucr.edu/home/baez/roots There are some references at the bottom.

Comment by Robert Haraway

2011-01-03T02:31:59Z

Your Euclidean proof should work for the Klein/Beltrami disk model, right?