Trichotomies in mathematics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:50:51Z http://mathoverflow.net/feeds/question/120612 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120612/trichotomies-in-mathematics Trichotomies in mathematics Vesselin Dimitrov 2013-02-02T20:25:23Z 2013-02-13T09:42:01Z <p><strong>Added.</strong> Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the question, if legitimate at all, should have been restricted to interesting manifestations of a hyperbolic-parabolic-elliptic subdivision, then I can fully agree (although part of the idea was to interpret the question as you see fit); I left it open ended primarily because of the Weil trichotomy, which is of completely different kind and is so much more than a hierarchy, and in relation to which I was interested in hearing other people's opinions and elaborations. See, for instance, how Edward Frenkel, in a fascinating Bourbaki talk, builds upon the Weil trichotomy to introduce a parallel between Langlands and electro-magnetic dualities, which he uses as a springboard for the ideas from physics that have entered the arena of the geometric Langlands program. Or take the cherished three-sided parallel between the basic three-dimensional (from the point of view of etale cohomology) objects and their branched coverings: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathrm{Spec}(\mathbb{Z})$, and $S^3$, with primes in number fields corresponding to knots in threefolds, $\log{p}$ corresponding to hyperbolic length, etcetera.</p> <p>To those who were not convinced that there was a neat trichotomy of algebraic surfaces (arguing they should instead form a tetrachotomy by Kodaira dimension), let alone in higher dimensional algebraic geometry, I refer to Sándor Kovács's answer here, which demonstrates rather eloquently the fundamental trichotomy of birational geometry: </p> <p><a href="http://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle" rel="nofollow">http://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle</a></p> <p><strong>Original post.</strong> For many purposes, notably in classification hierarchies or in Weil's "big picture" of the fundamental unity in mathematics, it seems as if mathematical reality is more accurately captured by trichotomies than by two-sided dictionaries or questions of "either/or." The most basic is of course the trichotomy <em>negative</em> - <em>zero</em> - <em>positve</em> embodied by the complete ordered field $(\mathbb{R},&lt;)$ --- this is the Arrow of Time, if you will, or the conditioning of a dynamical system into states of past/present/future. As evidenced by some of the examples below, this trichotomy underlies varied, if crude, classification schemes in mathematics.</p> <p>Other trichotomies arise from closer examinations of a mathematical parallel. Mathematicians have always been fond of discovery by analogy; they take very seriously the intuitions supplied by different yet loosely connected fields. In doing so, they are guided by a tacit, platonic belief in the fundamental Unity of mathematics. An example is the similarity between finite geometries and Riemann surfaces. To explain this parallel, indeed to make sense of it, it is necessary to provide a "middle column" in the dictionary: the arithmetic geometry of number fields and arithmetic surfaces. This leads to the trichotomy that Weil explained so lucidly in a letter (which he wrote in 1940 in prison for his refusal to serve in the army) to his sister, the philosopher Simone Weil. This point of view led, as we know, to an entire new field of mathematical inquiry.</p> <p>Below I have listed some other cherished mathematical trichotomies. I am interested in seeing yet others, perhaps more specialized. <strong>This</strong> is my <strong>question</strong>: <em>add more trichotomies to the list</em>. Furthermore, I am interested in any reflections anyone might have, such as pertaining, for instance, to any of the following questions. Is 3 the most ubiquitous number in coarse classification schemes? Is it fair to say that a given trichotomy echoes the primeval trichotomy $(-,0,+)$? In a given trichotomy, is there a natural "middle column" of a corresponding three-sided dictionary? Is this "middle column" in any way the most fundamental, the most interesting, or the most elusive?</p> <p><strong>Trichotomies in mathematics: some examples.</strong></p> <ul> <li><p>The fabric of topology, geometry, and analysis is the real line $\mathbb{R}$. Tarski's eight axioms characterize it in terms of a complete binary total order &lt;, a binary operation +, and a constant 1. (Multiplication comes afterwards - it is implied by Tarski's axioms - and so does the Bourbaki definition of the reals as <em>the</em> complete ordered field). The sign trichotomies $(&lt;,=,>)$ and $(-,0,+)$ ensuing from those axioms have repercussions throughout all of mathematics.</p></li> <li><p>For example, there are three constant curvature spaces, leading to the three maximally symmetric geometries: hyperbolic, flat (or Euclidean), and elliptic (e.g. spherical forms).</p></li> <li><p>Locally symmetric spaces fall into three types: non-compact type, flat, and compact type.</p></li> <li><p>In complex analysis, there are three simply connected cloths: the Riemann surfaces $\Delta$, $\mathbb{C}$, and $\hat{\mathbb{C}}$.</p></li> <li><p>The connected component of the group of conformal automorphisms of a compact Riemann surface is one of the following three: <em>trivial</em>, $S^1 \times S^1$, $\mathrm{PGL}_2(\mathbb{C})$. </p></li> <li><p>The complexity of fundamental groups, as showcased first of all by topological surfaces: <em>genuinely non-abelian (perhaps we could say: anabelian)</em> - <em>abelian (or more generally, containing a finite index nilpotent subgroup)</em> - and <em>trivial (or more generally, finite)</em>. This is of course related to the subject of growth of finitely generated groups, brought forward by Lee Mosher's answer.</p></li> <li><p>In dynamics, a fixed point (or a periodic cycle) can be either repelling, indifferent, or attracting.</p></li> <li><p>In Thurston's work on surface homeomorphisms, elements of the mapping class group are classified according to dynamics into three types: pseudo-Anosov, reducible, and finite-order.</p></li> <li><p>In algebraic geometry, the positivity of the canonical bundle is central to the classification and minimal model problems. More generally, positivity is a salient feature of algebraic geometry. For a delightful discussion, see Kollar's review of Lazarsfeld's book "Positivity in algebraic geometry." (Bull. AMS, vol. 43, no. 2, pp. 279-284). The most basic example is the trichotomy of algebraic curves (rational, elliptic, general type).</p></li> <li><p>In birational algebraic geometry, at a <em>very</em> coarse level, there are three kinds of varieties out of which a general variety is made: rational curves, Calabi-Yau manifolds, and varieties of general type (or hyperbolic type, if you prefer). For example, an algebraic surface either: 1) admits a pencil of rational curves; or 2) admits a pencil of elliptic curves or is abelian or K3 (or a double quotient of a K3); or else 3) it is of general type. <em>Abelian</em> and <em>K3</em> are examples of Calabi-Yau manifolds.</p></li> <li><p>More concretely, consider smooth hypersurfaces $X \subset \mathbb{P}^n$. They divide into three types, according to how their degree $d$ compares with the dimension. If $d \leq n$, they contain plenty of rational curves (certainly uncountably many). If $d = n+1$, they are an example of a Calabi-Yau manifold, and <em>typically</em> contain a countably infinite number of rational curves. (The generating function of the number of rational curves of a given degree is then a very interesting function, of significance in the physics of quantum gravity.) And if $d \geq n+2$, then $X$ is of general type, and it is conjectured to <em>typically</em> contain only finitely many rational curves. (More precisely, Bombieri and Lang have conjectured that a variety of general type contains only finitely many maximal subvarieties not of general type).</p></li> <li><p>In diophantine geometry, rational points are supposed to come from rational curves and abelian varieties. The sporadic examples are believed to be finitely many. This leads to the following trichotomy for the growth rate of the number of rational points of bounded (big, i.e. exponential) height: <em>polynomial growth</em> - <em>logarithmic growth</em> - $O(1)$. Furthermore, even in dimension 1, it is for abelian varieties that the situation is the deepest and the most mysterious.</p></li> <li><p>In topology, it seems as if the interesting dimensions fall into three qualitatively different ranges: $d = 3$, $d = 4$, and $d \geq 5$. (Although this might be stretching it a bit too much). Of these, four dimensions -- the "middle column" -- is the most mysterious, and also the most relevant for physics.</p></li> <li><p>The "Weil trichotomy," of course, goes at least as far back to Kronecker and Dedekind: <em>curves over</em> $\mathbb{F}_q$ - <em>number fields</em> - <em>Riemann surfaces</em>. Class field theory and Iwasawa theory are particularly eloquent examples of this trichotomy. Another example is of course the zeta function and the Riemann hypothesis.</p></li> <li><p>One would be tempted to extend the latter trichotomy to [<em>non-Archimedean world ($p$-adic, profinite) - global arithmetic - Archimedean world (geometry, topology, complex variables)</em>], if the middle column did not subsume (much of) the flanking columns. Likewise the triple [<em>$l$-adic cohomology-motive-Hodge structure</em>] would probably not be admissible. Here is a variation on the theme (you may find it to be rubbish, in which case throw it away). There are two ways of completing (or taking limits of) the regular polygons $C_n$. The first is to think of $C_n$ as $\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ and take the <em>direct limit</em> (in this case, union, or <em>synthesis</em>: $\rightarrow$), which is $\mathbb{Q}/\mathbb{Z}$. Completing, we get the circle $S^1 = \mathbb{R}/\mathbb{Z}$, which is the simplest manifold. The second is to think of $C_n$ as $\mathbb{Z}/n$ and take the projective limit (or <em>deconstruction</em>: $\leftarrow$), which is $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$, the profinite version of the circle. In this way, Archimedean (continuous) objects and $p$-adic objects may be seen as the two possible different limits (synthesis and deconstruction) of <em>the same</em> finite objects. Taking $C_n$ to be more general finite groups, we get <em>essentially</em> all the Lie groups, on the one hand; and all the profinite groups, on the other hand.</p></li> <li><p>That we live in three perceptible spatial dimensions does not, of course, fit our bill. But in 1984, Manin published an article ("New dimensions in geometry") in which, guided by ideas from number theory (Arakelov geometry) and physics (supersymmetry), he proposed that there are <em>three</em> kinds of geometric dimensions, modeled on the affine superscheme $\mathrm{Spec} \mathbb{Z}[x_i;\xi_j]$, an "object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}/2$-graded supercommutative rings." Here, $\xi_j$ are "odd," anticommuting variables, commuting with the "even" variables $x_i$. See the three coordinate axes $x, \xi$ and $\mathrm{Spec} \mathbb{Z}$ in his picture of "three-space-2000." The arithmetic axis $\mathrm{Spec} \mathbb{Z}$ is implicit in complex algebraic geometry, and is essential in problems such as the Ax-Grothendieck theorem and the construction of rational curves in Fano manifolds.</p></li> <li><p>In the theory of linear groups there is, loosely speaking, a trichotomy: $\mathbb{G}_m$ <em>(linear tori)</em> - <em>semisimple</em> - $\mathbb{G}_a$ <em>(unipotent)</em>.</p></li> <li><p>Algebraic groups: <em>reductive</em> - <em>abelian variety</em> - <em>unipotent</em>. Especially, the classification of one-dimensional groups: $\mathbb{G}_m$ - $E$ - $\mathbb{G}_a$. (<em>Thanks, Terry Tao!</em>)</p></li> <li><p>Variant: among <em>commutative</em> algebraic groups, there are: <em>multiplicative type</em> - <em>abelian varieties</em> - <em>additive type (unipotent)</em>.</p></li> <li><p>$\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ are the only finite-dimensional associative division algebras over the continuum. (<em>Thanks Paul Reynolds, Teo B, and Sam Lewallen!</em>)</p></li> <li><p>The most basic PDEs of physics: the wave equation (<em>hyperbolic</em>) - the heat and Schrodinger equations (<em>parabolic</em>) - the Laplace equation (<em>elliptic</em>). (<em>Thanks Alexandre Eremenko!</em>)</p></li> <li><p>An infinite finitely-generated group has $1,2$ or $\infty$ ends. (<em>Thanks shane.orourke and Artie Prendergast-Smith!</em>)</p></li> <li><p>A random walk is either transient, null recurrent, or positive recurrent. (<em>Thanks Vaughn Climenhaga!</em>)</p></li> <li><p>Zeta functions can by dynamical (Artin-Mazur); arithmetical on schemes of finite type over $\mathbb{Z}$ (Riemann and Hasse-Weil); and geometric (Selberg's zeta function of a hyperbolic surface).</p></li> <li><p>In Model theory, there is an important trichotomy between super-stable theories, strict-stable (stable but not superstable) theories, and non stable theories. </p></li> <li><p>It seems fair to say that there are three kinds of three-dimensional simply connected <em>spaces</em>: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathbb{Spec}(\mathbb{Z})$ compactified at archimedean infinity, and $S^3$. This brings about the Mazur knotty dictionary and the fruitful analogy between primes and knots (especially hyperbolic knots).</p></li> </ul> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120617#120617 Answer by Mariano Suárez-Alvarez for Trichotomies in mathematics Mariano Suárez-Alvarez 2013-02-02T20:51:25Z 2013-02-02T20:51:25Z <p>The representation type of a finite dimensional algebra is either finite, tame or wild. This is a theorem of Drozd.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120620#120620 Answer by Mariano Suárez-Alvarez for Trichotomies in mathematics Mariano Suárez-Alvarez 2013-02-02T20:59:24Z 2013-02-02T20:59:24Z <p>The Gelfand-Kirillov dimension of an algebra is either zero, one, or at least $2$. </p> <p>This sounds silly, but one has to remember that the GKdim is a real number (or $\infty$) and that there are algebras with GKdim equal to $r$ for all real $r\geq2$.</p> <p>This is a combination of results of several people. See the books by McConnell and Robson, or by Krause and Lenagan.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120621#120621 Answer by shane.orourke for Trichotomies in mathematics shane.orourke 2013-02-02T21:18:48Z 2013-02-02T21:18:48Z <p>An infinite finitely generated group $G$ has either 1, 2 or infinitely many ends; the last case is equivalent to saying that $G$ splits as an amalgamated free product or an HNN extension over a finite subgroup, by a theorem of Stallings. (Of course the assumption that $G$ is infinite brushes a fourth possibility under the carpet, namely that the number of ends is zero.)</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120622#120622 Answer by Paul Reynolds for Trichotomies in mathematics Paul Reynolds 2013-02-02T21:24:59Z 2013-02-03T12:09:17Z <p>A simple yet useful one: An irreducible complex representation of a compact Lie group is either 'real', 'quaternionic', or 'complex'. That is, it is the complexification of a real irreducible, or it can be considered quaternionic through the existence of an equivariant conjugate-linear (real-)automorphism $j$ that squares to $-I$, or it is neither.</p> <p>The statement combines Schur's Lemma and the fact that there are three associative real division algebras, here seen through complex eyes.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120623#120623 Answer by Simon Lyons for Trichotomies in mathematics Simon Lyons 2013-02-02T21:38:48Z 2013-02-02T21:38:48Z <p>Every logical statement is either true, undecidable or false.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120624#120624 Answer by Vaughn Climenhaga for Trichotomies in mathematics Vaughn Climenhaga 2013-02-02T21:41:38Z 2013-02-02T21:47:33Z <p>In thermodynamic formalism for dynamical systems, a H&ouml;lder continuous potential function $\phi$ on a countable state topological Markov chain $(X,\sigma)$ is either positive recurrent, null recurrent, or transient. These correspond to the three possibilities for equilibrium states (shift-invariant measures maximising the quantity $h(\mu) + \int_X \phi\,d\mu$): existence of a finite equilibrium state is equivalent to positive recurrence; null recurrence is the boundary case where the equilibrium state becomes $\sigma$-finite but not finite, and transience is the case where there is no equilibrium state (all the weight has gone to infinity).</p> <p>These can be characterised in terms of a particular sequence $a_n>0$: positive recurrence is equivalent to $\limsup a_n > 0$, null recurrence is equivalent to $a_n\to 0$ and $\sum a_n=\infty$, and transience is equivalent to $\sum a_n&lt;\infty$. I imagine this trichotomy for sequences appears in other places as well.</p> <p><strong>Edit:</strong> It's worth mentioning that this trichotomy is also true for random walks on directed graphs (weighted or unweighted) -- historically I believe this is where it was first studied and where the terminology came from, but as a dynamicist I more immediately think of the interpretation above. In this setting the interpretations are as follows:</p> <ul> <li>Positive recurrent -- with probability 1, a random walk returns to where it started, and the expected return time is finite.</li> <li>Null recurrent -- the walk returns to the starting position with probability 1, but the expected return time is infinite.</li> <li>Transient -- with probability 1, the walk never returns to its starting position.</li> </ul> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120629#120629 Answer by Alexandre Eremenko for Trichotomies in mathematics Alexandre Eremenko 2013-02-02T22:14:37Z 2013-02-02T22:14:37Z <p>Almost everything in mathematics, indeed, can be "hyperbolic", "parabolic" or "elliptic". Like PDE's, Riemann surfaces, or manifolds of higher dimension, fractional-linear transformations, fixed points of a map, etc.</p> <p>Not even mentioning the 3 kinds of the conic sections:-) </p> <p>Of course this can be traced back to the fundamental trichotomy "positive", "zero" and "negative".</p> <p>In differential geometry we have three great areas: "positive curvature", "negative curvature" and "zero curvature".</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120640#120640 Answer by Michael Joyce for Trichotomies in mathematics Michael Joyce 2013-02-03T00:52:53Z 2013-02-03T01:09:35Z <p>There are three types of subgroups of $PGL_2(\mathbb{C})$ that act on $\mathbb{P}^1$ non-transitively but with finitely many orbits:</p> <p>(1) Type $T$: a one-dimensional torus</p> <p>(2) Type $N$: the normalizer of a one-dimensional torus</p> <p>(3) Type $U$: containing a non-trivial one-dimensional unipotent subgroup</p> <p>This trichotomy plays a key role in the study of the geometry of spherical varieties, a class of algebraic varieties that includes grassmannians, flag varieties, toric varieties, algebraic monoids and symmetric spaces. It is particularly important in understanding the analogues of Schubert subvarieties (i.e. closures of orbits of a Borel subgroup) of a spherical variety.</p> <p>In this example, there is no "middle" case as there is no intrinsic order to the three types.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120642#120642 Answer by Harry Altman for Trichotomies in mathematics Harry Altman 2013-02-03T02:13:59Z 2013-02-06T01:04:13Z <p>A local domain can be pure characteristic $0$, pure positive characteristic, or mixed characteristic. (Meaning that the algebra has characteristic $0$ but its residue field has positive characteristic.)</p> <p>Of course, this being a trichotomy relies on the fact that it's a domain; otherwise you could also have characteristic $p^n$ (with the residue field having characteristic $p$). I don't really know how that case is classified (I guess it's a form of mixed characteristic)...</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120643#120643 Answer by Terry Tao for Trichotomies in mathematics Terry Tao 2013-02-03T03:03:59Z 2013-02-05T01:24:19Z <ol> <li><p>After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence of positive real numbers either (a) converges to a positive real; (b) diverges to $+\infty$; or (c) diverges to $0$. In nonstandard analysis, these trichotomies become those of being bounded, negative unbounded, or positive unbounded, or of being infinitesimal, unbounded, or neither (i.e. both bounded, and bounded away from zero). These are of course variants of the basic $(-,0,+)$ trichotomy.</p></li> <li><p>Up to isomorphism, there are only three types of (connected) one-dimensional algebraic groups over an algebraically closed field: the additive group of the field, the multiplicative group of that field, and the elliptic curves over that field. (This last family would definitely be the "middle column".) This is of course connected to many of the other trichotomies previously mentioned. On the Riemann surface side, it comes from the fact that all one-dimensional connected complex groups are isomorphic to ${\bf C}/\Gamma$ for some discrete subgroup $\Gamma$ of ${\bf C}$, which can have rank 0 (additive case), 1 (multiplicative case), or 2 (elliptic curve case).</p></li> <li><p>If one squints at it in just the right way, the classification of finite simple groups is a trichotomy: cyclic, Lie type (including Lie over F_1, i.e. alternating group), or sporadic. (Of course, it can be sliced in many other ways; counting the items in <a href="http://en.wikipedia.org/wiki/List_of_finite_simple_groups" rel="nofollow">this Wikipedia list</a>, for instance, would make it a tetratetracontachotomy.) Sometimes it is conceptually useful to split up the large Lie type groups into three regimes: large characteristic but bounded rank; large rank but bounded characteristic (including the alternating groups); and large characteristic and large rank. (Alternatively, one can partition into bounded rank, alternating, and unbounded rank.) One can debate as to which of these categories is the "middle column".</p></li> <li><p>If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the <em>Littlewood-Paley trichotomy</em>: (a) "high-low" interactions with $|\xi_1| \gg |\xi_2|$ and $|\xi_1| \sim |\xi_3|$; (b) "low-high" interactions with $|\xi_1| \ll |\xi_2|$ and $|\xi_2| \sim |\xi_3|$; and (c) "high-high" interactions with $|\xi_1| \sim |\xi_2|$ and $|\xi_1| \gg |\xi_3|$. (One has to carefully demarcate the boundaries between these three possibilities to ensure it is a true trichotomy.) To an algebraic geometer, this would reflect the Y-shaped nature of the <a href="http://en.wikipedia.org/wiki/Amoeba_%28mathematics%29" rel="nofollow">amoeba</a> of the set $\{ (\xi_1,\xi_2,\xi_3): \xi_3 = \xi_1 + \xi_2 \}$. This trichotomy is important in harmonic analysis and PDE, and in particular in the paradifferential calculus of products and paraproducts (see e.g. <a href="http://terrytao.wordpress.com/2010/08/20/spielman-meyer-nirenberg/" rel="nofollow">this blog post of mine</a>). Often, one of the three interactions will be the most dominant, reflecting either a high-to-low frequency cascade or a low-to-high frequency cascade, but it depends heavily on the situation. Note that this trichotomy is basically a variant of the $(&lt;,=,>)$ trichotomy.</p></li> <li><p>(ADDED LATER) Another variant of the $(&lt;,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE <em>problem</em>, such as an initial value problem in a certain function space) can be classified as subcritical, critical, or supercritical, depending on whether the nonlinear component of the PDE is "weaker than", "comparable to", or "stronger than" the linear component in a suitable asymptotic limit (usually the fine scale/high frequency limit, although for scattering theory the coarse scale/low frequency limit is the relevant one instead). This distinction (which can usually be made precise through a scaling analysis or dimensional analysis) is often decisive in determining the difficulty level of the PDE problem. For instance, the regularity problem for 3D Navier-Stokes is supercritical and thus considered close to intractable, but 2D Navier-Stokes is critical and was solved decades ago. The global analysis of Ricci flow (with surgery) was considered supercritical until Perelman discovered new monotone quantities that made it critical, which was absolutely necessary for Perelman to be able to execute the rest of Hamilton's program and solve the Poincare and geometrisation conjectures. In this trichotomy, the critical (or scale-invariant) case is generally viewed as the most interesting and delicate, with some very nice mathematical tools coming into play to control the interaction between different scales. Perhaps it should also be pointed out that this trichotomy is orthogonal to the elliptic/parabolic/hyperbolic trichotomy, which only concerns the linear component of the PDE and not the nonlinear component, and all nine combinations (critical elliptic PDE, supercritical parabolic PDE, etc.) are studied in the literature.</p></li> <li><p>(ADDED YET LATER) In analysis, there are basically three scenarios that prevent a weakly convergent sequence $f_n$ of functions in some function space from being strongly convergent in that space: (a) escape to "horizontal infinity" (basically, the support of the function runs off to spatial infinity, i.e. moving bump type examples); (b) escape to "vertical infinity" (the peaks of the function go to infinity, e.g. a sequence of approximations to the identity converging weakly but not strongly to a delta function); and (c) escape to "frequency infinity" (the functions become increasingly oscillatory). If one can shut down all three modes of escape then one can recover strong convergence, and thus also strong (pre)compactness, cf. the Arzela-Ascoli theorem which has three hypotheses (compact domain, pointwise boundedness, equicontinuity) to shut down (a), (b), and (c) respectively. In Section 2.9 of Lieb-Loss, these three scenarios are called "wanders off to infinity", "goes up the spout", and "oscillates to death" respectively.</p></li> </ol> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120646#120646 Answer by Koushik for Trichotomies in mathematics Koushik 2013-02-03T03:44:12Z 2013-02-03T04:14:52Z <p>The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: the $sphere$; the connected sum of $g$ $tori$, ; the connected sum of $k$ $real$ $projective$ $planes$. this is a simple example of the trichotomy.sphere can be taken as $0$ tori. so $SPHERE$ serves the middle column. </p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120650#120650 Answer by Sam Lewallen for Trichotomies in mathematics Sam Lewallen 2013-02-03T05:31:43Z 2013-02-03T05:47:00Z <p>There are exactly three finite-dimensional, associative division algebras over $\mathbb R$: $$\mathbb R, ~~ \mathbb C,~~\mathbb H$$</p> <p>This was also mentioned in Paul Reynolds' answer. </p> <p>Arnol'd thought this was a particularly fundamental trichotomy, and he has a fascinating table of "related" trichotomies in his book "Arnol'd's Problems" (sorry for the apostrophe catastrophe). Here's a picture I uploaded a few years ago:</p> <p><a href="http://concretenonsense.files.wordpress.com/2008/11/arnoldtable.jpg" rel="nofollow">Arnol'd's table</a></p> <p>I wonder if these triples are in the same "family" as $(-1 ,0 , 1)$. At first they seem to be different because there is no distinguished element, analogous to $0$ - but perhaps $\mathbb C$ plays this role? I think the proof that there are three division algebras might actually use the $(-1,0,1)$ trichotomy, but I don't have a moment to think about it right now - maybe someone can leave a comment confirming this? (<strong>Edit</strong>: I could also imagine $\mathbb R$ playing the role of $0$, because it is the identity element for the tensor product, for example - any opinions either way?)</p> <p>Another question I find interesting, which I'd hoped to think about at some point for fun, asks not just for (related) occurrences of 3-element sets in mathematical classifications, but more generally, for related occurrences of $n$-element sets for small $n$. I'd always dreamed of taking the number $5$ of regular platonic solids, for example, and trying to deduce from this as many other "small, finite number classifications" as possible. </p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120665#120665 Answer by Joe Silverman for Trichotomies in mathematics Joe Silverman 2013-02-03T13:21:22Z 2013-02-03T13:21:22Z <p>The reduction (special fiber) $E_{\mathfrak p}$ of (the Neron model of) an elliptic curve $E$ modulo a prime ${\mathfrak p}$ is one of:</p> <ol> <li>good reduction = stable reduction = $E_{\mathfrak p}$ is non-singular</li> <li>multiplicative reduction = semi-stable reduction = $E_{\mathfrak p}$ is a product of the multiplicative group times a finite group</li> <li>additive reduction = unstable reduction = $E_{\mathfrak p}$ is a product of the additive group times a finite group</li> </ol> <p>Of course, this trichotomy is a reflection of the fact that there are only three sorts of connected one-dimensional Lie groups, namely the additive group, the multiplicative group, and the compact case (elliptic curves).</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120667#120667 Answer by Andrea Mori for Trichotomies in mathematics Andrea Mori 2013-02-03T13:49:45Z 2013-02-03T13:49:45Z <p>Did anybody mention yet the different arithmetic-wise behaviour of algebraic curves of genus $0$, $1$, or $\geq2$?</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120668#120668 Answer by Asaf Karagila for Trichotomies in mathematics Asaf Karagila 2013-02-03T14:32:09Z 2013-02-03T14:32:09Z <p>Finite, countable, and uncountable.</p> <p>Those are the three distinctions we have for cardinalities. For most people uncountable would usually mean $2^{\aleph_0}$. But even for set theorists, given a model of ZFC, the finite sets are finite, the countable sets are countable, and the rest is madness$^*$.</p> <p>Replacing cardinality by topological-measure theoretic properties of subsets of an ordinal $\kappa$, there are non-stationary sets (small); stationary sets (big, but not too big); and clubs (which is practically everything).</p> <hr> <p>$^*$ <em>Good</em> madness! Like a smile without a cat.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120673#120673 Answer by Lee Mosher for Trichotomies in mathematics Lee Mosher 2013-02-03T15:12:25Z 2013-02-03T15:12:25Z <p>Every isometry of a proper $CAT(-1)$ space is either elliptic, parabolic, or hyperbolic: elliptic means fixes a finite point; hyperbolic means fixes two infinite points connected by a translation axis, equivalently translation distance bounded away from zero; parabolic means translation distance limiting to zero but no fixed point. There are versions for proper Gromov hyperbolic spaces, and even for the nonproper case, if you are willing to "quasify" the statements of the cases, and if you are willing to let the trichotomy degenerate to a dichotomy.</p> <p>Every isometry of Teichmuller space is either elliptic, parabolic, or hyperbolic. This is Bers' form of Thurston's trichotomy for mapping classes: finite order, reducible, pseudo-Anosov. This trichotomy also has an interpretation in terms of the action of the mapping class group on the curve complex which is a nonproper Gromov hyperbolic space by a theorem of Masur and Minsky. </p> <p>For elements of $Out(F_n)$, the outer automorphism group of a rank $n$ free group, there are related trichotomies and other -otomies coming from the work of Bestvina, Feighn, and Handel on relative train track theory. The simplest one is that every element of $Out(F_n)$ is either of finite order, or of polynomial growth, or of exponential growth. </p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120737#120737 Answer by GH for Trichotomies in mathematics GH 2013-02-04T08:03:30Z 2013-02-04T08:03:30Z <p>Each set in a topological space partitions the space into three parts: interior, boundary, exterior.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120750#120750 Answer by David Corwin for Trichotomies in mathematics David Corwin 2013-02-04T10:13:03Z 2013-02-04T10:13:03Z <p>The endomorphism ring of an elliptic curve is either $\mathbb{Z}$, an order in a quadratic field, or an order in a quaternion algebra (ranks $1,2$, and $4$, respectively).</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120752#120752 Answer by A. Bellmunt for Trichotomies in mathematics A. Bellmunt 2013-02-04T10:16:26Z 2013-02-04T10:16:26Z <p>Knots are either torus, satellite or hyperbolic.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120754#120754 Answer by Guy R. for Trichotomies in mathematics Guy R. 2013-02-04T10:33:16Z 2013-02-04T10:33:16Z <p>In number theory, there is the trichotomy $\infty$, odd $p$, and $2$ (the oddest prime).</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120763#120763 Answer by unknown (yahoo) for Trichotomies in mathematics unknown (yahoo) 2013-02-04T12:29:05Z 2013-02-04T12:54:11Z <p>Previous MO questions:</p> <ul> <li><p>rational/trigonometric/elliptic trichotomy <a href="http://mathoverflow.net/questions/58040/groups-quantum-groups-and-fill-in-the-blank" rel="nofollow">http://mathoverflow.net/questions/58040/groups-quantum-groups-and-fill-in-the-blank</a></p></li> <li><p>trichotomy of interrelated model structures: h-model, q-model, m-model <a href="http://mathoverflow.net/questions/86942/is-the-category-of-metric-spaces-and-continuous-maps-quillen-equivalent-to-top" rel="nofollow">http://mathoverflow.net/questions/86942/is-the-category-of-metric-spaces-and-continuous-maps-quillen-equivalent-to-top</a></p></li> <li><p>"such "log-exp functions" are either eventually positive, eventually zero, or eventually negative. ... It guarantees that the germs at infinity of such functions do indeed form a field K." <a href="http://mathoverflow.net/questions/45284/examples-of-sequences-whose-asymptotics-cant-be-described-by-elementary-function" rel="nofollow">http://mathoverflow.net/questions/45284/examples-of-sequences-whose-asymptotics-cant-be-described-by-elementary-function</a></p></li> <li><p>a function of a complex variable with an algebraic addition theorem must be: 1) A rational function, 2) A rational function of e^px, or 3) A rational function of the Weierstrass elliptic function and its derivative. <a href="http://mathoverflow.net/questions/96452/trig-functions-based-on-convex-curves" rel="nofollow">http://mathoverflow.net/questions/96452/trig-functions-based-on-convex-curves</a></p></li> <li><p>"Every finitely generated infinite profinite group has a just infinite quotient. There is a trichotomy due to Wilson (and refined by Grigorchuk) describing what they can look like." <a href="http://mathoverflow.net/questions/49591/what-is-the-virtue-of-profinite-groups-as-mathematical-objects/68895" rel="nofollow">http://mathoverflow.net/questions/49591/what-is-the-virtue-of-profinite-groups-as-mathematical-objects/68895</a></p></li> <li><p>"there is a trichotomy of curves given by g=0, g=1, and g≥2. If you look at topological, geometric, arithmetic properties of these curves, their properties align very strongly with these classes." <a href="http://mathoverflow.net/questions/56011/why-should-i-believe-the-mordell-conjecture" rel="nofollow">http://mathoverflow.net/questions/56011/why-should-i-believe-the-mordell-conjecture</a></p></li> <li><p>Kodaira dimension. κ(Y)&lt;0, κ(Y)=0, κ(Y)=dimY. <a href="http://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle" rel="nofollow">http://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle</a></p></li> <li><p>"Rank and period of primes in the Fibonacci sequence; a trichotomy," Fib. Quart., 45 (No. 1, 2007), 56-63). <a href="http://mathoverflow.net/questions/84797/can-the-difference-of-two-distinct-fibonacci-numbers-be-a-square-infinitely-often" rel="nofollow">http://mathoverflow.net/questions/84797/can-the-difference-of-two-distinct-fibonacci-numbers-be-a-square-infinitely-often</a> </p></li> </ul> <p>M.SE:</p> <ul> <li><p>"The set-theoretic setup of Categories for the working mathematician is somewhat subtle. ... There is therefore a trichotomy of small sets, large sets, and proper classes. This is not the usual practice: we normally think of all sets as being small." <a href="http://math.stackexchange.com/questions/201062/confusion-over-the-use-of-universes-in-category-theory" rel="nofollow">http://math.stackexchange.com/questions/201062/confusion-over-the-use-of-universes-in-category-theory</a></p></li> <li><p>"There are three distinct aspects of schemes that each have their own purpose" <a href="http://math.stackexchange.com/questions/99605/why-study-schemes/99615" rel="nofollow">http://math.stackexchange.com/questions/99605/why-study-schemes/99615</a></p></li> </ul> <p>TCS.SE:</p> <ul> <li>"one of the most amazing facts about logic is that consistency strength boils down to the question "what is the fastest-growing function you can prove total in this logic?" As a result, the consistency of many classes of logics can be linearly ordered! If you have an ordinal notation capable of describing the fastest growing functions your two logics can show total, then you know by trichotomy that either one can prove the consistency of the other, or they are equiconsistent." <a href="http://cstheory.stackexchange.com/questions/4816/axioms-necessary-for-theoretical-computer-science/4821" rel="nofollow">http://cstheory.stackexchange.com/questions/4816/axioms-necessary-for-theoretical-computer-science/4821</a></li> </ul> <p><hr> A frequently cited paper: "A trichotomy theorem in natural models of AD+", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120812#120812 Answer by André Henriques for Trichotomies in mathematics André Henriques 2013-02-05T00:08:40Z 2013-02-05T00:08:40Z <p>In algebraic topology, the sequence $$\text{ordinary cohomolgy},\qquad K\text{-theory},\qquad \text{elliptic cohomology}$$ closely parallels the trichotomy in the classification of algebraic groups ($\mathbb G_a$, $\mathbb G_m$, elliptic curves).</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121001#121001 Answer by Todd Trimble for Trichotomies in mathematics Todd Trimble 2013-02-06T18:39:29Z 2013-02-06T18:39:29Z <p>Here is a cluster of examples with a common theme, based partly on comments here and in an <a href="http://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree" rel="nofollow">MO thread</a> on whether an empty space should be considered connected, and partly on an article in the nLab, <a href="http://ncatlab.org/nlab/show/too+simple+to+be+simple" rel="nofollow">"too simple to be simple"</a>. </p> <p>One should note that some of these trichotomies were once-upon-a-time considered dichotomies; for example, for many people in the past, 1 was a prime number. Also one should notice that there are a number of cross-connections (homomorphisms, if you will) between these examples, and that list is by no means complete (this is CW, so feel free to add more!). </p> <ul> <li><p>A module can be reducible, irreducible, or zero. </p></li> <li><p>A filter in a Boolean algebra can be "submaximal" (<em>faute de mieux!</em>), a maximal filter = ultrafilter, or an improper filter. </p></li> <li><p>An element in a p.i.d. is composite, prime, or a unit. </p></li> <li><p>A topological space (or a graph) can be disconnected, connected, or <a href="http://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree/120548#120548" rel="nofollow">"unconnected"</a> (empty). </p></li> <li><p>As for elements satisfying a predicate, one can have a multiplicity, unique existence, or nonexistence. </p></li> </ul> <p>The last trichotomy is often considered a dichotomy: nonuniqueness vs. uniqueness. (I.e., nonexistence falls under the scope of uniqueness = "at most one".) But experience in mathematics, e.g. in category theory and its focus on universal properties, shows that unique-existence deserves to be considered in a category of its own. </p> <ul> <li>Theories can be incomplete but consistent, complete, or inconsistent. </li> </ul> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121003#121003 Answer by Lee Mosher for Trichotomies in mathematics Lee Mosher 2013-02-06T18:46:18Z 2013-02-06T18:46:18Z <p>Every finitely generated group is either of polynomial growth, intermediate growth, or exponential growth. </p> <p>As a statement, there is not much to this, the only mathematical content is that the growth function of every finitely generated group has an exponential upper bound. </p> <p>But as a method of classifying finitely generated groups, it has been very fruitful: Gromov's theorem on groups of polynomial growth; the incredibly rich theory that arose from Grigorchuk's original construction of an intermediate growth group; and the emergence of rich classes of exponential growth groups such as word hyperbolic groups.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121020#121020 Answer by Axel Boldt for Trichotomies in mathematics Axel Boldt 2013-02-06T21:05:20Z 2013-02-06T21:05:20Z <p>Every system of linear equations over the reals say has either no solution, or one solution, or infinitely many solutions.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121281#121281 Answer by Franz Lemmermeyer for Trichotomies in mathematics Franz Lemmermeyer 2013-02-09T08:01:17Z 2013-02-09T08:01:17Z <p>Everything has an end, except for a sausage, which has two. </p> <p>good-bye.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121505#121505 Answer by Alexander Chervov for Trichotomies in mathematics Alexander Chervov 2013-02-11T17:07:44Z 2013-02-11T17:07:44Z <p>Let me point out that Vladimir Arnold was quite interested in similar question. He called subj. "mathematical trinities", see e.g. his paper <a href="http://www.maths.ed.ac.uk/~aar/papers/arnold4.pdf" rel="nofollow">"Symplectization, Complexification and Mathematical Trinities"</a>. As far as I remember from his lectures, his ideas were that many of these "trinities" are actually related to each other; and he also considered subj. as a tool to invent to theories: see question marks at already cited "Arnold's table": <a href="http://concretenonsense.files.wordpress.com/2008/11/arnoldtable.jpg" rel="nofollow">jpg</a>.</p> <hr> <p>Let me also mention some "trinities" which occur in my own research related to <a href="http://en.wikipedia.org/wiki/Capelli%27s_identity" rel="nofollow">Capelli identities</a> (which are some non-commutative analogs of det(AB)=det(A)det(B) ).</p> <p>Matrix trinity - a) generic b) symmetric c) antisymmetric</p> <p>Here how it goes in Capelli (and related Cayley) identities: </p> <p>a) generic matrices - original Capelli identity has been discovered by Capelli in 19-th century - it is for "generic matrices" $A=x_{ij}$ $B = \partial_{ji}$ </p> <p>b) <a href="http://en.wikipedia.org/wiki/Capelli%27s_identity#Turnbull.27s_identity_for_symmetric_matrices" rel="nofollow">symmetric matrices - analog of the Capelli identity</a> has been discovered by Turnbull around 1940-ies - here $A=(x_{ij}+x_{ji})$ $B= \partial_{ji} + \partial_{ij}$.</p> <p>c) <a href="http://en.wikipedia.org/wiki/Capelli%27s_identity#The_Howe.E2.80.93Umeda.E2.80.93Kostant.E2.80.93Sahi_identity_for_antisymmetric_matrices" rel="nofollow">antisymmetric matrices - analog</a> has been found by Howe-Umeda and Kostant-Sahi around 1990, here $A=(x_{ij}-x_{ji})$ $B= \partial_{ji} - \partial_{ij}$.</p> <p>Similar generalization were found for Cayley identity respectively: a) attributed to Cayley b) Garding 1948 c) Shimura 1984 - see <a href="http://arxiv.org/abs/1105.6270" rel="nofollow">arXiv:1105.6270</a> for quite a complete information.</p> <p><strong>My question</strong>: is it really trinity ? Or you can propose some analogs of Cayley-Capelli for some other matrices, say "symplectic" ? </p> <hr> <p>It is might be strange, but other trinities like R,C,H also appears in the Capelli story - and they give different identities. Moreover trinities can be combined and we might get trinity^trinity^trinity...</p> <p>Actually H-analog of the Capelli identity is not fully known for the momemnt - only analog for 1x1 matrices has been discovered quite recently by student of R. Borcherds, <a href="http://arxiv.org/abs/1102.2657" rel="nofollow">An Huang</a>. Looking at this example I proposed some C-analogs of <a href="http://arxiv.org/abs/1203.5759" rel="nofollow">Capelli identities</a>. Actually all generic/symmetric/antisymmetric can be complexified, hopefully there should exist quaternionic analogs and thus we might have trinity^trinity. Some partial results of trinity^trinity spirit for Cayley identity contained in loc. cit.</p> <p>There are certain analogs of <a href="http://arxiv.org/abs/q-alg/9712021" rel="nofollow">Capelli identities for classical Lie algebras</a>: this can seen as gl/so/su trinity, well probably it is not the trinity in some strict sense. I have no idea can we have something like trinity^trinity^trinity ...</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121679#121679 Answer by LeBlanc for Trichotomies in mathematics LeBlanc 2013-02-13T06:47:18Z 2013-02-13T06:47:18Z <p>Every prime of a field either ramifies, splits or is inert in a field extension. </p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121681#121681 Answer by unknown (google) for Trichotomies in mathematics unknown (google) 2013-02-13T07:00:39Z 2013-02-13T07:00:39Z <p>NP-hard, Intermediate(conjectural), Polynomial time classes will separate $P$ from $NP$.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121694#121694 Answer by shane.orourke for Trichotomies in mathematics shane.orourke 2013-02-13T09:42:01Z 2013-02-13T09:42:01Z <p>Suppose that a group $G$ has an action on a tree with no inversions and no global fixed point. Then either (a) $G$ is expressible as a (non-trivial) amalgamated free product; (b) $G$ is indicable (i.e. it maps onto an infinite cyclic group); or (c) $G$ is expressible as the union of a strictly ascending sequence of subgroups.</p> <p>The negation of this property (every action of $G$ on a tree without inversions has a fixed point) is called FA by (J.-P.) Serre.</p> <p>A more mundane, but related, example: a single automorphism of a tree either fixes a point, inverts an edge, or has an invariant (doubly-infinite) line contained in the tree on which the automorphism acts by translations.</p>