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It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of the proofs of Roths' theorem (or Szemeredi's theorem) seem to qualify, starting with Furstenberg's amazing realisation that this problem in combinatorial number theory was equivalent to one in ergodic theory, and that the structural theory of the latter could then be used to attack the former. Or the Ruzsa-Szemeredi observation (made somewhat implicitly at the time) that Roth's theorem follows from a result in graph theory (the triangle removal lemma) which, in some ways, was "easier" to prove than the result that it implied despite (or perhaps, because of) the fact that it "forgot" most of the structure of the problem. And in this regard, I can't resist mentioning Ben Green's brilliant observation (inspired, I believe, by some earlier work of Ramare and Ruzsa) that for the purposes of finding arithmetic progressions, that the primes should not be studied directly, but instead should be viewed primarily [pun not intended] as a generic dense subset of a larger set of almost primes, for which much more is known, thanks to sieve theory...

Another problem that seems to generate radically new thinking every few years is the Kakeya problem. Originally a problem in geometric measure theory, the work of Bourgain and Wolff in the early 90s showed that the combinatorial incidence geometry viewpoint could lead to substantial progress. When this stalled, Bourgain (inspired by your own work) introduced the additive combinatorics viewpoint, re-interpreting line segments as arithmetic progressions. Meanwhile, Wolff created the finite field model of the Kakeya problem, which among other things lead to the sum-product theorem and many further developments that would not have been possible without this viewpoint. In particular, this finite field version enabled Dvir to introduce the polynomial method which had been applied to some other combinatorial problems, but whose application to the finite field Kakeya problem was hugely shocking. (Actually, Dvir's argument is a great example of "new thinking" being the key stumbling block. Five years earlier, Gerd Mockenhaupt and I managed to stumble upon half of Dvir's argument, showing that a Kakeya set in finite fields could not be contained in a low-degree algebraic variety. If we had known enough about the polynomial method to make the realisation that the exact same argument also showed that a Kakeya set could not have been contained in a high-degree algebraic variety either, we would have come extremely close to recovering Dvir's result; but our thinking was not primed in this direction.) Meanwhile, Carbery, Bennet, and I discovered that heat flow methods, of all things, could be applied to solve a variant of the Euclidean Kakeya problem (though this method did appear in literature on other analytic problems, and we viewed it as the continuous version of the discrete induction-on-scales strategy of Bourgain and Wolff.) Most recently is the work of Guth, who broke through the conventional wisdom that Dvir's polynomial argument was not generalisable to the Euclidean case by making the crucial observation that algebraic topology (such as the ham sandwich theorem) served as the continuous generalisation of the discrete polynomial method, leading among other things to the recent result of Guth and Katz you mentioned earlier.

EDIT: Another example is the recent establishment of universality for eigenvalue spacings for Wigner matrices. Prior to this work, most of the rigorous literature on eigenvalue spacings relied crucially on explicit formulae for the joint eigenvalue distribution, which were only tractable in the case of highly invariant ensembles such as GUE, although there was a key paper of Johansson extending this analysis to a significantly wider class of ensembles, namely the sum of GUE with an arbitrary independent random (or deterministic) matrix. To make progress, one had to go beyond the explicit formula paradigm and find some way to compare the distribution of a general ensemble with that of a special ensemble such as GUE. We now have two basic ways to do this, the local relaxation flow method of Erdos, Schlein, Yau, and the four moment theorem method of Van Vu and myself, both based on deforming a general ensemble into a special ensemble and controlling the effect on the spectral statistics via this deformation (though the two deformations we use are very different, and in fact complement each other nicely). Again, both arguments have precedents in earlier literature (for instance, our argument was heavily inspired by Lindeberg's classic proof of the central limit theorem) but as far as I know it had not been thought to apply them to the universality problem before.

It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of the proofs of Roth's theorem (or Szemerédi's theorem) seem to qualify, starting with Furstenberg's amazing realisation that this problem in combinatorial number theory was equivalent to one in ergodic theory, and that the structural theory of the latter could then be used to attack the former. Or the Ruzsa–Szemerédi observation (made somewhat implicitly at the time) that Roth's theorem follows from a result in graph theory (the triangle removal lemma) which, in some ways, was "easier" to prove than the result that it implied despite (or perhaps, because of) the fact that it "forgot" most of the structure of the problem. And in this regard, I can't resist mentioning Ben Green's brilliant observation (inspired, I believe, by some earlier work of Ramare and Ruzsa) that for the purposes of finding arithmetic progressions, that the primes should not be studied directly, but instead should be viewed primarily [pun not intended] as a generic dense subset of a larger set of almost primes, for which much more is known, thanks to sieve theory….

Another problem that seems to generate radically new thinking every few years is the Kakeya problem. Originally a problem in geometric measure theory, the work of Bourgain and Wolff in the early 90s showed that the combinatorial incidence geometry viewpoint could lead to substantial progress. When this stalled, Bourgain (inspired by your own work) introduced the additive combinatorics viewpoint, re-interpreting line segments as arithmetic progressions. Meanwhile, Wolff created the finite field model of the Kakeya problem, which among other things lead to the sum-product theorem and many further developments that would not have been possible without this viewpoint. In particular, this finite field version enabled Dvir to introduce the polynomial method which had been applied to some other combinatorial problems, but whose application to the finite field Kakeya problem was hugely shocking. (Actually, Dvir's argument is a great example of "new thinking" being the key stumbling block. Five years earlier, Gerd Mockenhaupt and I, in "Restriction and Kakeya phenomena for finite fields", managed to stumble upon half of Dvir's argument, showing that a Kakeya set in a finite field could not be contained in a low-degree algebraic variety. If we had known enough about the polynomial method to make the realisation that the exact same argument also showed that a Kakeya set could not have been contained in a high-degree algebraic variety either, we would have come extremely close to recovering Dvir's result; but our thinking was not primed in this direction.) Meanwhile, Carbery, Bennet, and I discovered that heat flow methods, of all things, could be applied to solve a variant of the Euclidean Kakeya problem (though this method did appear in literature on other analytic problems, and we viewed it as the continuous version of the discrete induction-on-scales strategy of Bourgain and Wolff.) Most recent is the work of Guth, who broke through the conventional wisdom that Dvir's polynomial argument was not generalisable to the Euclidean case by making the crucial observation that algebraic topology (such as the ham sandwich theorem) served as the continuous generalisation of the discrete polynomial method, leading among other things to the recent result of Guth and Katz you mentioned earlier.

EDIT: Another example is the recent establishment of universality for eigenvalue spacings for Wigner matrices. Prior to this work, most of the rigorous literature on eigenvalue spacings relied crucially on explicit formulae for the joint eigenvalue distribution, which were only tractable in the case of highly invariant ensembles such as GUE, although there was a key paper of Johansson extending this analysis to a significantly wider class of ensembles, namely the sum of GUE with an arbitrary independent random (or deterministic) matrix. To make progress, one had to go beyond the explicit formula paradigm and find some way to compare the distribution of a general ensemble with that of a special ensemble such as GUE. We now have two basic ways to do this, the local relaxation flow method of Erdős, Schlein, Yau, and the four moment theorem method of Van Vu and myself, both based on deforming a general ensemble into a special ensemble and controlling the effect on the spectral statistics via this deformation (though the two deformations we use are very different, and in fact complement each other nicely). Again, both arguments have precedents in earlier literature (for instance, our argument was heavily inspired by Lindeberg's classic proof of the central limit theorem) but as far as I know it had not been thought to apply them to the universality problem before.

It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of the proofs of Roths' theorem (or Szemeredi's theorem) seem to qualify, starting with Furstenberg's amazing realisation that this problem in combinatorial number theory was equivalent to one in ergodic theory, and that the structural theory of the latter could then be used to attack the former. Or the Ruzsa-Szemeredi observation (made somewhat implicitly at the time) that Roth's theorem follows from a result in graph theory (the triangle removal lemma) which, in some ways, was "easier" to prove than the result that it implied despite (or perhaps, because of) the fact that it "forgot" most of the structure of the problem. And in this regard, I can't resist mentioning Ben Green's brilliant observation (inspired, I believe, by some earlier work of Ramare and Ruzsa) that for the purposes of finding arithmetic progressions, that the primes should not be studied directly, but instead should be viewed primarily [pun not intended] as a generic dense subset of a larger set of almost primes, for which much more is known, thanks to sieve theory...

Another problem that seems to generate radically new thinking every few years is the Kakeya problem. Originally a problem in geometric measure theory, the work of Bourgain and Wolff in the early 90s showed that the combinatorial incidence geometry viewpoint could lead to substantial progress. When this stalled, Bourgain (inspired by your own work) introduced the additive combinatorics viewpoint, re-interpreting line segments as arithmetic progressions. Meanwhile, Wolff created the finite field model of the Kakeya problem, which among other things lead to the sum-product theorem and many further developments that would not have been possible without this viewpoint. In particular, this finite field version enabled Dvir to introduce the polynomial method which had been applied to some other combinatorial problems, but whose application to the finite field Kakeya problem was hugely shocking. (Actually, Dvir's argument is a great example of "new thinking" being the key stumbling block. Five years earlier, Gerd Mockenhaupt and I managed to stumble upon half of Dvir's argument, showing that a Kakeya set in finite fields could not be contained in a low-degree algebraic variety. If we had known enough about the polynomial method to make the realisation that the exact same argument also showed that a Kakeya set could not have been contained in a high-degree algebraic variety either, we would have come extremely close to recovering Dvir's result; but our thinking was not primed in this direction.) Meanwhile, Carbery, Bennet, and I discovered that heat flow methods, of all things, could be applied to solve a variant of the Euclidean Kakeya problem (though this method did appear in literature on other analytic problems, and we viewed it as the continuous version of the discrete induction-on-scales strategy of Bourgain and Wolff.) Most recently is the work of Guth, who broke through the conventional wisdom that Dvir's polynomial argument was not generalisable to the Euclidean case by making the crucial observation that algebraic topology (such as the ham sandwich theorem) served as the continuous generalisation of the discrete polynomial method, leading among other things to the recent result of Guth and Katz you mentioned earlier.

It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of the proofs of Roths' theorem (or Szemeredi's theorem) seem to qualify, starting with Furstenberg's amazing realisation that this problem in combinatorial number theory was equivalent to one in ergodic theory, and that the structural theory of the latter could then be used to attack the former. Or the Ruzsa-Szemeredi observation (made somewhat implicitly at the time) that Roth's theorem follows from a result in graph theory (the triangle removal lemma) which, in some ways, was "easier" to prove than the result that it implied despite (or perhaps, because of) the fact that it "forgot" most of the structure of the problem. And in this regard, I can't resist mentioning Ben Green's brilliant observation (inspired, I believe, by some earlier work of Ramare and Ruzsa) that for the purposes of finding arithmetic progressions, that the primes should not be studied directly, but instead should be viewed primarily [pun not intended] as a generic dense subset of a larger set of almost primes, for which much more is known, thanks to sieve theory...

Another problem that seems to generate radically new thinking every few years is the Kakeya problem. Originally a problem in geometric measure theory, the work of Bourgain and Wolff in the early 90s showed that the combinatorial incidence geometry viewpoint could lead to substantial progress. When this stalled, Bourgain (inspired by your own work) introduced the additive combinatorics viewpoint, re-interpreting line segments as arithmetic progressions. Meanwhile, Wolff created the finite field model of the Kakeya problem, which among other things lead to the sum-product theorem and many further developments that would not have been possible without this viewpoint. In particular, this finite field version enabled Dvir to introduce the polynomial method which had been applied to some other combinatorial problems, but whose application to the finite field Kakeya problem was hugely shocking. (Actually, Dvir's argument is a great example of "new thinking" being the key stumbling block. Five years earlier, Gerd Mockenhaupt and I managed to stumble upon half of Dvir's argument, showing that a Kakeya set in finite fields could not be contained in a low-degree algebraic variety. If we had known enough about the polynomial method to make the realisation that the exact same argument also showed that a Kakeya set could not have been contained in a high-degree algebraic variety either, we would have come extremely close to recovering Dvir's result; but our thinking was not primed in this direction.) Meanwhile, Carbery, Bennet, and I discovered that heat flow methods, of all things, could be applied to solve a variant of the Euclidean Kakeya problem (though this method did appear in literature on other analytic problems, and we viewed it as the continuous version of the discrete induction-on-scales strategy of Bourgain and Wolff.) Most recently is the work of Guth, who broke through the conventional wisdom that Dvir's polynomial argument was not generalisable to the Euclidean case by making the crucial observation that algebraic topology (such as the ham sandwich theorem) served as the continuous generalisation of the discrete polynomial method, leading among other things to the recent result of Guth and Katz you mentioned earlier.

EDIT: Another example is the recent establishment of universality for eigenvalue spacings for Wigner matrices. Prior to this work, most of the rigorous literature on eigenvalue spaces relied crucially on explicit formulae for the joint eigenvalue distribution, which were only tractable in the case of highly invariant ensembles such as GUE, although there was a key paper of Johansson extending this analysis to a significantly wider class of ensembles, namely the sum of GUE with an arbitrary independent random (or deterministic) matrix. To make progress, one had to go beyond the explicit formula paradigm and find some way to compare the distribution of a general ensemble with that of a special ensemble such as GUE. We now have two basic ways to do this, the local relaxation flow method of Erdos, Schlein, Yau, and the four moment theorem method of Van Vu and myself, both based on deforming a general ensemble into a special ensemble and controlling the effect on the spectral statistics via this deformation (though the two deformations we use are very different, and in fact complement each other nicely). Again, both arguments have precedents in earlier literature (for instance, our argument was heavily inspired by Lindeberg's classic proof of the central limit theorem) but as far as I know it had not been thought to apply them to the universality problem before.

EDIT: Another example is the recent establishment of universality for eigenvalue spacings for Wigner matrices. Prior to this work, most of the rigorous literature on eigenvalue spacings relied crucially on explicit formulae for the joint eigenvalue distribution, which were only tractable in the case of highly invariant ensembles such as GUE, although there was a key paper of Johansson extending this analysis to a significantly wider class of ensembles, namely the sum of GUE with an arbitrary independent random (or deterministic) matrix. To make progress, one had to go beyond the explicit formula paradigm and find some way to compare the distribution of a general ensemble with that of a special ensemble such as GUE. We now have two basic ways to do this, the local relaxation flow method of Erdos, Schlein, Yau, and the four moment theorem method of Van Vu and myself, both based on deforming a general ensemble into a special ensemble and controlling the effect on the spectral statistics via this deformation (though the two deformations we use are very different, and in fact complement each other nicely). Again, both arguments have precedents in earlier literature (for instance, our argument was heavily inspired by Lindeberg's classic proof of the central limit theorem) but as far as I know it had not been thought to apply them to the universality problem before.