Questions tagged [big-list]
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
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books well-motivated with explicit examples
It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
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Blackbox Theorems [closed]
By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite ...
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Proof synopsis collection
I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself!
Definition (Fraleigh): A proof synopsis ...
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What could be some potentially useful mathematical databases?
This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a ...
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Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
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Examples of algorithms requiring deep mathematics to prove correctness
I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.
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Contest problems with connections to deeper mathematics
I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay.
We all know that problems from, for ...
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What programming language should a professional mathematician know? [closed]
More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable?
If I am right,...
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How would you have answered Richard Feynman's challenge?
Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students ...
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Nontrivially fillable gaps in published proofs of major theorems
Prelude: In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem, as presented in Nash's well-known paper "The Imbedding ...
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Books you would like to see translated into English
I have recently been told of a proposal to produce an English translation
of Landau's Handbuch der Lehre von der Verteilung der Primzahlen, and
this prompts me to ask a more general question:
...
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Are there proofs that you feel you did not "understand" for a long time?
Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...
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More open problems [closed]
Open Problem Garden and Wikipedia are good resources for more or less famous open problems. But many mathematicians will be happy with more specialized problems. They may want to find a research theme,...
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Algebraic geometry examples
What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that ...
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What would you want on a Lie theory cheat poster?
For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
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Breakthroughs in mathematics in 2023
At the end of 2021, Johnny Cage asked about breakthroughs in 2021 in different mathematical disciplines. A similar question has been asked at the end of 2022, so it looks like Johnny Cage originated a ...
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Examples of interesting false proofs
According to Wikipedia False proof
For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...
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Sophisticated treatments of topics in school mathematics
Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
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Mathematical conjectures on which applications depend
What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
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What practical applications does set theory have?
I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the ...
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Results that are widely accepted but no proof has appeared
The background of this question is the talk given by Kevin Buzzard.
I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here.
...
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Each mathematician has only a few tricks
The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
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Applications of mathematics
All of us have probably been exposed to questions such as: "What are the applications of group theory...".
This is not the subject of this MO question.
Here is a little newspaper article that I found ...
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Good papers/books/essays about the thought process behind mathematical research
Papers in mathematics are generally written as if the major insights suddenly appeared, unbidden, in a notebook on the researcher's desk and then were fleshed out into the final paper.
While this is ...
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Dimension leaps
Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" ...
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Atlas-like websites on specific areas of mathematics
In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples:
GroupNames: https://...
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Free open-access peer-reviewed math journals
Is there any free (as in free beer, i.e., no publication fees or other fees whatsoever), open-access (free and open access to everyone) and peer-reviewed mathematics journal?
I am interested in a ...
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Least collaborative mathematician
The recent question about the most prolific collaboration interested me. How about this question in the opposite direction, then: can anyone beat, amongst contemporary mathematicians, the example of ...
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Math Annotate Platform?
Suppose most mathematical research papers were freely accessible online.
Suppose a well-organized platform existed where responsible users could write comments on any paper (linking to its doi, ...
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Free, high quality mathematical writing online? [closed]
I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable ...
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What are some deep theorems, and why are they considered deep?
All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
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How should one present curl and divergence in an undergraduate multivariable calculus class?
I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and ...
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What precisely Is "Categorification"?
(And what's it good for.)
Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
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Counterexamples in PDE
Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...
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An example of a beautiful proof that would be accessible at the high school level?
The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
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Proofs where higher dimension or cardinality actually enabled much simpler proof?
I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...
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Facts from algebraic geometry that are useful to non-algebraic geometers
A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...
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Essays and thoughts on mathematics
Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written ...
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(Preferably rare) Audio/Video recordings of famous mathematicians?
Terence Tao's homepage has a link to a collection of quotes, and one among them was Hilbert's famous "We must know, we will know" quote. This quote also had an audio link to it. Now although I'm not ...
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Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?
What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...
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What are some examples of ingenious, unexpected constructions?
Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by ...
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Mathematical research published in the form of poems
The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the ...
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Still Difficult After All These Years
I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
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The use of computers leading to major mathematical advances II
I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances.
This is a continuation of a question ...
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Surprisingly short or elegant proofs using Lie theory
Today, I was listening to someone give an exhausting proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory:
A finite extension $K$ of $\mathbb{C}$...
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Is there a nice application of category theory to functional/complex/harmonic analysis?
[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in most ...
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What programming languages do mathematicians use? [closed]
I understand this might be a slightly subjective question, but I am honestly curious what programming languages are used by the mathematics community.
I would imagine that there is a group of ...
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Elementary / Interesting proofs of the Nullstellensatz
Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques?
One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...
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PhD dissertations that solve an established open problem
I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor).
In my question I search for every possible ...
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Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...