Timeline for The probabilistic method outside of discrete mathematics
Current License: CC BY-SA 4.0
29 events
| when toggle format | what | by | license | comment | |
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| S Jun 2, 2023 at 8:38 | history | bounty ended | Pulcinella | ||
| S Jun 2, 2023 at 8:38 | history | notice removed | Pulcinella | ||
| May 26, 2023 at 5:10 | answer | added | PseudoNeo | timeline score: 3 | |
| S May 25, 2023 at 23:38 | history | bounty started | Pulcinella | ||
| S May 25, 2023 at 23:38 | history | notice added | Pulcinella | Canonical answer required | |
| Mar 24, 2023 at 15:23 | answer | added | John Dawkins | timeline score: 9 | |
| Mar 6, 2023 at 3:04 | comment | added | Pulcinella | @YemonChoi Not really (though maybe I feel less able to tell if an analysis application is "morally" really an instance of the probabilistic method). | |
| Mar 6, 2023 at 0:19 | comment | added | Yemon Choi | @PseudoNeo I'm not aware of something that exactly fits; I mainly learned of this in the context of Banach space theory by reading various comments in articles/books by people like Pisier and Szarek. There are particular results such as Kashin's decomposition theorem, the Johnson-Lindenstrauss lemma, Milman's proof of Dvoretzky's theorem ... | |
| Mar 5, 2023 at 17:27 | comment | added | PseudoNeo | @YemonChoi: do you recommend a book or a survey article about applications of the probabilistic method in analysis? I know they abound, but is there a tour guide out there? | |
| Feb 28, 2023 at 20:32 | comment | added | Yemon Choi | Sorry to be an analyst, but: is there a reason analysis wasn't mentioned in the OP's question? | |
| Feb 28, 2023 at 20:30 | answer | added | Will Sawin | timeline score: 26 | |
| Feb 27, 2023 at 2:10 | comment | added | Benjamin Steinberg | I think "generic" style arguments are one of the main analogues of the probabilistic method in math since they follow the same idea that to prove things exist you prove the set of things without this property is small. | |
| Feb 26, 2023 at 23:57 | comment | added | Dustin G. Mixon | One way to formulate the probabilistic method is that every random variable $X\colon\Omega\to\mathbb{R}$ satisfies $\sup_{\omega\in\Omega}X(\omega)\geq\mathbb{E}X$. The fact that the supremum is at least the average generalizes the pigeonhole principle and is under the hood of a lot of analysis, e.g., the proof of $\|x\|_2\leq\sqrt{n}\|x\|_\infty$ for $x\in\mathbb{R}^n$, the proof that the spectral norm of a positive semidefinite matrix is its top eigenvalue, and the proof of the mean value theorem (for definite integrals). | |
| Feb 26, 2023 at 22:33 | comment | added | Dustin G. Mixon | @BenjaminSteinberg - Non-computable numbers are an even more extreme example. en.wikipedia.org/wiki/Computable_number | |
| Feb 26, 2023 at 4:11 | comment | added | Benjamin Steinberg | What about things like the existence of transcendental numbers? A simple counting argument shows that most numbers are transcendental but it requires more work to find one. | |
| Feb 26, 2023 at 0:41 | history | became hot network question | |||
| Feb 25, 2023 at 21:30 | comment | added | Michael Greinecker | The empirical method of Maurey might count. | |
| Feb 25, 2023 at 21:14 | comment | added | Jason Gaitonde | Would Bourgain's example for the tightness of Pisier's inequality count? He uses the probabilistic method to construct Banach spaces that achieve the upper bound given in Pisier's inequality for the operator norm of the Rademacher projection. I'm not sure if there are known, explicit examples. | |
| Feb 25, 2023 at 20:55 | history | made wiki | Post Made Community Wiki by David Roberts♦ | ||
| Feb 25, 2023 at 18:43 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Feb 25, 2023 at 17:56 | comment | added | Benjamin Steinberg | Some of the examples I had in mind are of a similar type to Dustin's. For instance where the bad instances are a finite union of proper closed subsets of an irreducible variety and hence not all instances are bad | |
| Feb 25, 2023 at 17:45 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Feb 25, 2023 at 17:33 | comment | added | Pulcinella | @DustinG.Mixon I'd prefer to avoid examples of this form. | |
| Feb 25, 2023 at 17:32 | comment | added | Pulcinella | @BenjaminSteinberg That sounds good, I don't want to be too prescriptive. | |
| Feb 25, 2023 at 17:31 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Feb 25, 2023 at 16:58 | comment | added | mathworker21 | The most basic uses of the probabilistic method are just interchanging summations / double counting, which are used all over mathematics. | |
| Feb 25, 2023 at 16:53 | comment | added | Dustin G. Mixon | I know of several applications in which a point of interest is only known to exist because it resides in the complement of a (semi) algebraic set of low dimension. Does this count? | |
| Feb 25, 2023 at 16:51 | comment | added | Benjamin Steinberg | Are you willing to replace positive probability by things like set of measure nonzero or proving the set of examples is dense without an explicit construction? | |
| Feb 25, 2023 at 16:40 | history | asked | Pulcinella | CC BY-SA 4.0 |