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.
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).
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 this Wikipedia listthis Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoebaamoeba 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. this blog post of minethis blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
(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.
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.
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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
(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.
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.
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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
(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.
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.
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 algebraic 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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
(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.
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.
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 algebraic 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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
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.
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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
(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.
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.
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 algebraic 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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,0,>)$$(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,0,>)$$(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
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.
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 algebraic 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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,0,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,0,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.
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.
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 algebraic 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).
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 this Wikipedia list, 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".
If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (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 amoeba 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. this blog post of mine). 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 $(<,=,>)$ trichotomy.
(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, 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.