User robert haraway - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:38:54Z http://mathoverflow.net/feeds/user/10946 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46701/what-is-the-difference-between-hard-and-soft-analysis What is the difference between hard and soft analysis? Robert Haraway 2010-11-20T02:01:16Z 2013-05-05T11:06:57Z <p>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.</p> http://mathoverflow.net/questions/57072/a-remark-of-connes A remark of Connes Robert Haraway 2011-03-02T04:05:44Z 2013-04-09T11:40:45Z <p>In an interview (at <a href="http://www.alainconnes.org/docs/Inteng.pdf" rel="nofollow">http://www.alainconnes.org/docs/Inteng.pdf</a>) Connes remarks that</p> <blockquote> <p>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. </p> </blockquote> <p>What does he mean; what is he referring to?</p> http://mathoverflow.net/questions/124581/a-conjecture-of-thurston-and-possibly-weeks-too A conjecture of Thurston and possibly Weeks too Robert Haraway 2013-03-15T02:03:24Z 2013-03-15T02:26:16Z <p>What is the status of the following conjecture:</p> <blockquote> <p>"... [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."</p> </blockquote> <p>This is Conjecture 6.1 from p. 2568 of Thurston's expository paper "How to see 3-manifolds" (Class. Quantum Grav. <strong>15</strong> (1998) 2545--2571).</p> http://mathoverflow.net/questions/119291/hyperbolic-brunnian-links-and-rectangular-cusp-shapes Hyperbolic Brunnian links and rectangular cusp shapes Robert Haraway 2013-01-18T21:34:46Z 2013-01-23T17:40:32Z <p>My question is as follows.</p> <blockquote> <p>Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?</p> </blockquote> <p>Here is what I mean:</p> <p>The Borromean rings form a famous link $B$ (a smooth closed 1-manifold in the three-sphere):</p> <blockquote> <p><strong>Link $B$</strong></p> <p><img src="https://www2.bc.edu/robert-c-haraway/borro.jpg" alt="alt text"></p> </blockquote> <p>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.</p> <p>Here are some other Brunnian links:</p> <blockquote> <p><strong>Link $S$</strong></p> <p><img src="https://www2.bc.edu/robert-c-haraway/square.jpg" alt="alt text"></p> </blockquote> <hr> <blockquote> <p><strong>Link $R$</strong></p> <p><img src="https://www2.bc.edu/robert-c-haraway/rubberband.jpg" alt="alt text"></p> </blockquote> <hr> <blockquote> <p><strong>Link $N$</strong></p> <p><img src="https://www2.bc.edu/robert-c-haraway/brunnnothyp.jpg" alt="alt text"> (Rolfsen '76, p. 67)</p> </blockquote> <hr> <blockquote> <p><strong>Link $OMG$</strong></p> <p><img src="https://www2.bc.edu/robert-c-haraway/Brunnp67.jpg" alt="alt text"> (Rolfsen '76, p. 67)</p> </blockquote> <hr> <p>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. <a href="http://www.math.uic.edu/t3m/SnapPy/doc/" rel="nofollow">SnapPy</a>). 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 $m_C$.</p> <p>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 <em>peripheral subgroup</em> 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 $\Gamma$.</p> <p>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 $\mathbb{Z} \oplus \mathbb{Z}$.</p> <p>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.</p> <p>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 <em>cusp shape</em> of the link component $C$ is the ordered pair of complex numbers $(\phi(\gamma_m), \phi(\gamma_\ell))$.</p> <p>Here are SnapPy's computations for the cusp shapes of the components of the above Brunnian links (I have rounded the decimals):</p> <ul> <li>Link $B$: all of shape $(5\cdot 10^{-17}+0.569 i,\,1.140)$ (by symmetry)</li> <li>Link $S$: all of shape $(1\cdot 10^{-16}+0.550 i,\,1.181)$ (by symmetry)</li> <li>Link $N$: no hyperbolic structure</li> <li>Link $OMG$: <ul> <li>cusp 0: $(1\cdot 10^{-16}+0.258 i,\,2.514)$</li> <li>cusp 1: $(-9\cdot 10^{-16}+0.262 i,\,2.471)$</li> <li>cusp 2: $(1\cdot 10^{-16}+0.316 i,\,2.051)$</li> <li>cusp 3: $(1\cdot 10^{-15}+0.431 i,\,1.506)$</li> <li>cusp 4: $(2\cdot 10^{-17}+0.5848 i,\,1.111)$</li> <li>cusp 5: $(-5\cdot 10^{-16}+0.232 i,\,2.796)$</li> </ul></li> </ul> <p>That is to say, each of the above hyperbolic Brunnian links (almost certainly) has a rectangular cusp shape on every one of its components.</p> <blockquote> <p>Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?</p> </blockquote> http://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/78091#78091 Answer by Robert Haraway for What are some examples of colorful language in serious mathematics papers? Robert Haraway 2011-10-14T02:25:56Z 2011-10-14T02:25:56Z <p>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:</p> <blockquote> <p>[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.</p> </blockquote> http://mathoverflow.net/questions/70858/unpublished-higman-manuscript Unpublished Higman manuscript Robert Haraway 2011-07-20T23:19:21Z 2011-07-21T00:02:58Z <p>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 $M_{24}.$" </p> <p>Does anyone know to what Curtis is referring? Is there another source for these constructions?</p> http://mathoverflow.net/questions/61929/naturally-occurring-orderings/64169#64169 Answer by Robert Haraway for Naturally occurring orderings Robert Haraway 2011-05-07T03:32:11Z 2011-05-07T03:32:11Z <p>Hrbacek, following Ballard, has recently put a certain partial ordering called $\sqsubseteq$ on <em>absolutely everything</em> in order to do nonstandard analysis a la Nelson (i.e. internal set theory): </p> <p>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.</p> <p>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.</p> <p>Then Hrbacek's <strong>GRIST</strong> is the following condition on $\sqsubseteq,$ with four axiom schemata:</p> <p><strong>R</strong> elativization condition on $\sqsubseteq$: $\sqsubseteq$ is a total dense preordering with minimal element $\emptyset$ and no maximal element; i.e. the conjunction of </p> <ol> <li><p><em>Partial ordering</em>: $(\forall u,v,w)((v\sqsubseteq u \wedge w \sqsubseteq v) \implies w \sqsubseteq u) \wedge u \sqsubseteq u;$</p></li> <li><p><em>Preordering</em>: $(\forall u,v)(u \sqsubseteq v \wedge v \sqsubseteq u);$</p></li> <li><p><em>Minimality of $\emptyset$</em>: $(\forall u)(\emptyset \sqsubseteq u);$</p></li> <li><p><em>Illimitability</em>: $(\forall u) (\exists v) (u \sqsubset v);$</p></li> <li><p><em>Density</em>: $(\forall u,v) (u \sqsubset v \implies (\exists w)(u \sqsubset w \sqsubset v)).$</p></li> </ol> <p>Axiom schemata, in which we use words so as not to have our eyes completely glaze over: </p> <p>For any well-formed formula $P$ of $\mathscr{HB}$ depending on finitely many variables,</p> <ol> <li><p><strong>T</strong> ransfer: for all $u \sqsubseteq v$ and $x_1,\ldots, x_n \sqsubseteq u,$</p> <blockquote> <p>$P^u(x_1,\ldots,x_n) \iff P^v(x_1,\ldots,x_n)$</p> </blockquote></li> <li><p><strong>S</strong> 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 $v \sqsupseteq w \sqsupset u,$</p> <blockquote> <p>$(\forall y \sqsubseteq w)(y \in B \iff y \in A \wedge P^w(y,x_1,\ldots,x_n)).$</p> </blockquote></li> <li><p><strong>I</strong> dealization: For all $A \sqsubset v$ and all $x_1,\ldots,x_n,$</p> <blockquote> <p>$(\forall a \in 2^A_{\mathrm{fin}})\big([a\sqsubset v \implies (\exists y)(\forall x \in a) P^v(x,y,x_1,\ldots,x_n)] \iff[(\exists y)(\forall x \in A)[x \sqsubset u \implies P^v(x,y,x_1,\ldots,x_n)]\big).$</p> </blockquote></li> <li><p><strong>G</strong> ranularity: For all $x_1,\ldots,x_k,$ if $(\exists u) P^u(x_1,\ldots,x_k),$ then</p> <blockquote> <p>$(\exists u)[P^u(x_1,\ldots,x_k) \wedge (\forall v)(v \sqsubset u \implies \neg P^v(x_1,\ldots,x_n))]$</p> </blockquote></li> </ol> http://mathoverflow.net/questions/61878/geometric-interpretation-of-cartans-structure-equations/61880#61880 Answer by Robert Haraway for Geometric interpretation of Cartan's structure equations Robert Haraway 2011-04-15T23:32:30Z 2011-04-15T23:32:30Z <p>I think you may find an answer in <em>Differential Geometry</em> by Sharpe.</p> http://mathoverflow.net/questions/59329/m12-simple-sporadic-group/59348#59348 Answer by Robert Haraway for M12 simple sporadic group Robert Haraway 2011-03-23T20:49:35Z 2011-03-23T20:49:35Z <p>If you want an intuitive presentation of M12, also take a look at <A href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&amp;s1=198799&amp;vfpref=html&amp;r=30&amp;mx-pid=1010366" rel="nofollow">Curtis's construction</A>.</p> http://mathoverflow.net/questions/50990/non-standard-enlargements-zetas-and-analytic-continuation/57212#57212 Answer by Robert Haraway for Non-standard enlargements, $\zeta(s)$ and analytic continuation Robert Haraway 2011-03-03T05:08:00Z 2011-03-17T17:55:28Z <p>I <strike>just</strike> saw this on arXiv: <a href="http://arxiv.org/abs/0808.1965" rel="nofollow"><em>Nonstandard Mathematics and New Zeta and L-Functions</em></a>, the PhD thesis of one Benjamin Clare of U. Nottingham.</p> <blockquote> <p>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.</p> </blockquote> <p>(Corrections welcome if my answer betrays any basic misunderstanding!)</p> http://mathoverflow.net/questions/18840/nonstandard-analysis-book-recommendation/57075#57075 Answer by Robert Haraway for nonstandard analysis book recommendation Robert Haraway 2011-03-02T04:13:43Z 2011-03-02T04:13:43Z <p>I have learned some internal set theory (IST) from Lutz and Goze's <em>Nonstandard analysis: a practical guide with applications.</em> 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 <em>Nonstandard analysis</em> or Nelson's own papers, his own unfinished book at <a href="http://www.math.princeton.edu/~nelson/books/1.pdf" rel="nofollow">http://www.math.princeton.edu/~nelson/books/1.pdf</a>, or the probability book mentioned above.</p> <p>So if you want to get excited about IST and get fun ideas for using it, read Lutz and Goze. To <em>understand</em> it, read Nelson or Robert.</p> http://mathoverflow.net/questions/57025/down-to-earth-uses-of-de-rham-cohomology-to-convince-a-wide-audience-of-its-usefu/57070#57070 Answer by Robert Haraway for Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness Robert Haraway 2011-03-02T03:29:49Z 2011-03-02T03:29:49Z <p>You may be able to convey the significance of de Rham cohomology to <em>really</em> wide audiences through electromagnetism. </p> <p>I don't claim to understand all the physics (or topology for that matter), but see my friend Rob Kotiuga's book at <a href="http://library.msri.org/books/Book48/index.html" rel="nofollow">http://library.msri.org/books/Book48/index.html</a>. 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."</p> http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/56140#56140 Answer by Robert Haraway for Examples of common false beliefs in mathematics. Robert Haraway 2011-02-21T04:02:18Z 2011-02-21T04:02:18Z <p>Inversion is an automorphism of a group. ('Cause it, like, preserves the conjugacy classes and all that...)</p> http://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/54249#54249 Answer by Robert Haraway for What are some examples of colorful language in serious mathematics papers? Robert Haraway 2011-02-03T22:22:32Z 2011-02-03T22:22:32Z <p>John (Horton) Conway unrelentingly gets away with colorful, even whimsical language in definitions, in explanations, in paper titles, even in some book titles (<em>The Sensual (Quadratic) Form</em>.) Even in <em>SPLAG</em>, there is the following:</p> <p>"...we earnestly recommend that you use</p> <p><strong>The Best Method:</strong> guess the correct answer, and then justify it." <em>SPLAG</em>, p. 302</p> <p><em>On Numbers and Games</em> 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</p> <p>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.</p> <p>And the last words of the book are famously</p> <p>"...a certain feeling of incompleteness prompts us to add a final theorem.</p> <p>THEOREM 100. This is the last theorem in this book.</p> <p>(The proof is obvious.)" <em>ONAG</em> p. 224</p> http://mathoverflow.net/questions/48717/escher-conway-kali-etc/50981#50981 Answer by Robert Haraway for Escher, Conway, Kali, etc. Robert Haraway 2011-01-03T02:59:28Z 2011-01-03T02:59:28Z <p>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.</p> <p><a href="http://www-ab.informatik.uni-tuebingen.de/software/2dtiler/welcome.html" rel="nofollow">http://www-ab.informatik.uni-tuebingen.de/software/2dtiler/welcome.html</a></p> http://mathoverflow.net/questions/8609/favorite-popular-math-book/48706#48706 Answer by Robert Haraway for Favorite popular math book Robert Haraway 2010-12-09T04:17:00Z 2010-12-09T04:17:00Z <p><strong>Title</strong>: <em>The Magical Maze</em></p> <p><strong>Author</strong>: Ian Stewart</p> <p><strong>Short description</strong>: 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.</p> <p>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.</p> http://mathoverflow.net/questions/44086/is-there-a-ruler-and-compass-construction-of-the-common-perpendicular-of-two-geod/46711#46711 Answer by Robert Haraway for Is there a ruler and compass construction of the common perpendicular of two geodesics in H^3? Robert Haraway 2010-11-20T04:58:47Z 2010-11-20T05:05:46Z <p>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 <a href="http://www.ams.org/mathscinet-getitem?mr=199777" rel="nofollow">The inversive plane and hyperbolic space</a>. In this particular situation one can think of <em>lines</em> 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:</p> <blockquote> <p>Given two pairs of points $(a,b)$ and $(c,d),$ to construct using straightedge and compass the (unique) pair $(x,y)$ harmonic to both. </p> </blockquote> <p>Fenchel asserts the existence of such a thing, and relates them to square roots, but without mentioning the geometric argument, so here goes.</p> <p>We first start off easy, and assume that $a = \infty.$ We define the (two) <em>geometric means of $(c,d)$ with respect to b</em> 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 $b$.</p> <p>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 $(c,d).$</p> <p>It's probably all in Fenchel, anyway....</p> http://mathoverflow.net/questions/40529/books-for-hyperbolic-geometry-surfaces-with-exercises/46702#46702 Answer by Robert Haraway for Books for hyperbolic geometry ( surfaces ) with exercises ? Robert Haraway 2010-11-20T02:14:08Z 2010-11-20T02:14:08Z <p>Al Marden's <em>Outer circles</em> 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.</p> <p>Beardon's <em>Geometry of Discrete Groups,</em> Iversen's <em>Hyperbolic Geometry,</em> and Bonahon's <em>Low-dimensional Geometry,</em> and Katok's <em>Fuchsian Groups</em> 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.</p> http://mathoverflow.net/questions/1722/free-high-quality-mathematical-writing-online/46587#46587 Answer by Robert Haraway for Free, high quality mathematical writing online? Robert Haraway 2010-11-19T04:33:05Z 2010-11-19T05:01:53Z <p>A draft of Albert Marden's <em>Outer circles: an introduction to hyperbolic 3-manifolds</em> is online, on his website:</p> <p><a href="http://www.math.umn.edu/~am/book/outercircles.pdf" rel="nofollow">http://www.math.umn.edu/~am/book/outercircles.pdf</a></p> <p>Edward Nelson's <em>Radically elementary probability theory</em> is also online, on his website:</p> <p><a href="http://www.math.princeton.edu/~nelson/books/rept.pdf" rel="nofollow">http://www.math.princeton.edu/~nelson/books/rept.pdf</a></p> http://mathoverflow.net/questions/128088/lecture-on-fractals-for-middle-school-students Comment by Robert Haraway Robert Haraway 2013-04-20T00:44:02Z 2013-04-20T00:44:02Z Thank <i>you</i> in advance! It was just such a lecture that got me interested in mathematics, and now I am in graduate school. http://mathoverflow.net/questions/16312/how-helpful-is-non-standard-analysis/126883#126883 Comment by Robert Haraway Robert Haraway 2013-04-08T21:02:56Z 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. http://mathoverflow.net/questions/4918/can-you-fool-snappea/4948#4948 Comment by Robert Haraway Robert Haraway 2013-02-13T02:47:41Z 2013-02-13T02:47:41Z @Ryan Budney: I believe you're referring to Mel Slugbate; cf. <a href="http://euclid.colorado.edu/~jnc/MelSlugbate.html" rel="nofollow">euclid.colorado.edu/~jnc/MelSlugbate.html</a> http://mathoverflow.net/questions/119291/hyperbolic-brunnian-links-and-rectangular-cusp-shapes/119301#119301 Comment by Robert Haraway Robert Haraway 2013-01-23T15:23:37Z 2013-01-23T15:23:37Z Thank you very much! http://mathoverflow.net/questions/87094/elementary-but-difficult-system-of-equations Comment by Robert Haraway Robert Haraway 2012-01-31T01:16:13Z 2012-01-31T01:16:13Z This question is not research-level, and would be better posted on math.stackexchange.com. http://mathoverflow.net/questions/77175/taking-zooming-in-on-a-point-of-a-graph-seriously Comment by Robert Haraway Robert Haraway 2011-10-04T21:11:29Z 2011-10-04T21:11:29Z This is explored seriously in Keisler's nonstandard calculus text <i>Elementary Calculus;</i> his treatment of the microscope methodology is based on work of Keith Stroyan. http://mathoverflow.net/questions/70858/unpublished-higman-manuscript/70861#70861 Comment by Robert Haraway Robert Haraway 2011-07-21T03:19:20Z 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? http://mathoverflow.net/questions/65498/what-is-the-structure-of-theories-that-contain-ineffable-objects Comment by Robert Haraway Robert Haraway 2011-05-20T18:08:42Z 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. http://mathoverflow.net/questions/58495/why-hasnt-mereology-suceeded-as-an-alternative-to-set-theory Comment by Robert Haraway Robert Haraway 2011-03-15T13:55:46Z 2011-03-15T13:55:46Z @Qiaochu: A comment from Eric Raymond on Plan 9 may be in order here: &quot;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.&quot; The same could be said of bases for doing mathematics. http://mathoverflow.net/questions/18840/nonstandard-analysis-book-recommendation/57098#57098 Comment by Robert Haraway Robert Haraway 2011-03-03T04:46:48Z 2011-03-03T04:46:48Z I was really disappointed to see how thin Robert's book is.... http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/42801#42801 Comment by Robert Haraway Robert Haraway 2011-02-21T03:58:38Z 2011-02-21T03:58:38Z This was actually used as an April Fool's Joke by Martin Gardner.... http://mathoverflow.net/questions/48717/escher-conway-kali-etc/50982#50982 Comment by Robert Haraway Robert Haraway 2011-01-03T03:50:26Z 2011-01-03T03:50:26Z It was probably before that link was up, or before I had heard of orbifold notation. http://mathoverflow.net/questions/48717/escher-conway-kali-etc/50981#50981 Comment by Robert Haraway Robert Haraway 2011-01-03T03:49:00Z 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. http://mathoverflow.net/questions/31646/does-algebraic-numbers-coloured-by-degree-form-a-fractal Comment by Robert Haraway Robert Haraway 2011-01-03T03:14:09Z 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: <a href="http://math.ucr.edu/home/baez/roots/" rel="nofollow">math.ucr.edu/home/baez/roots</a> There are some references at the bottom. http://mathoverflow.net/questions/10031/collapsing-the-medial-axis-of-a-polytope Comment by Robert Haraway Robert Haraway 2011-01-03T02:31:59Z 2011-01-03T02:31:59Z Your Euclidean proof should work for the Klein/Beltrami disk model, right?