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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Top specialized journals
In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA.
What journals ...
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Are there any good websites for hosting discussions of mathematical papers?
I was wondering if there are any websites out there which
systematically provide space for the discussion of mathematics articles (particularly those on the arXiv, though not necessarily just those),...
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What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
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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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Examples of seemingly elementary problems that are hard to solve?
I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
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Results true in a dimension and false for higher dimensions
Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
A positive polynomial not reaching its infimum. ...
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What are some mathematical sculptures?
Either intentionally or unintentionally.
Include location and sculptor, if known.
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Approaches to Riemann hypothesis using methods outside number theory [closed]
Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The ...
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Fundamental Examples
It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...
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Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
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How To Present Mathematics To Non-Mathematicians?
(Added an epilogue)
I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching).
In the last ...
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Nonequivalent definitions in Mathematics
I would like to ask if anyone could share any specific experiences of
discovering nonequivalent definitions in their field of mathematical research.
By that I mean discovering that in different ...
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What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...
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Fields of mathematics that were dormant for a long time until someone revitalized them
I thought that the closed question here could be modified to a very interesting question (at least as far as big-list type questions go).
Can people name examples of fields of mathematics that were ...
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Proofs of the uncountability of the reals
Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem (Wayback Machine). At first, I was excited to see a variant proof (as it did not ...
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Have you solved problems in your sleep?
I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending through ...
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Theorems that are 'obvious' but hard to prove
There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would ...
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Short papers for undergraduate course on reading scholarly math
(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)
Today, I was reminded of the existence of this ...
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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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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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Unnecessary uses of the axiom of choice
What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?
I'm interested in standard proofs that use the axiom of choice, but where
...
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Prominent non-mathematical work of mathematicians
First of all, sorry if this post is not appropriate for this forum.
I have a habit that every time I read a beautiful article I look at the author's homepage and often find amazing things.
Recently I ...
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Theorems for nothing (and the proofs for free) [closed]
Some theorems give far more than you feel they ought to: a weak hypothesis is enough to prove a strong result. Of course, there's almost always a lot of machinery hidden below the waterline. Such ...
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Contemporary philosophy of mathematics
Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in ...
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How to write math well?
Let's learn about writing good mathematical texts.
For some people it could be especially interesting to answer about writing texts on Math Overflow, though I personally feel like I've already ...
35
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What's your favorite equation, formula, identity or inequality? [closed]
Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule. What's your favorite equation, formula, identity or inequality?
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Non-rigorous reasoning in rigorous mathematics
I was wondering what role non-rigorous, heuristic type arguments play in rigorous math. Are there examples of rigorous, formal proofs in which a non-rigorous reasoning still plays a central part?
...
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What are the applications of operator algebras to other areas?
Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
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Noteworthy achievements in and around 2010?
The goal of this question is to compile a list of noteworthy mathematical achievements from about 2010 (so somewhat but not too far in the past).
In particular, this is meant to include (but not ...
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3
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What are the current breakthroughs of Geometric Complexity Theory?
I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods.
That program seems ...
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A search for theorems which appear to have very few, if any hypotheses [closed]
I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.
By "few hypotheses" I ...
232
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16
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What elementary problems can you solve with schemes?
I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...
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Video lectures of mathematics courses available online for free
It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
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Have any long-suspected irrational numbers turned out to be rational?
The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
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Proofs that require fundamentally new ways of thinking
I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
191
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Books you would like to read (if somebody would just write them…)
I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different ...
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What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
189
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Which math paper maximizes the ratio (importance)/(length)?
My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
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Suggestions for good notation
I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:
Iverson introduced the notation [X] to mean 1 if X is ...
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Magic trick based on deep mathematics
I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...
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Old books you would like to have reprinted with high-quality typesetting
There are some questions on mathoverflow such as
What out-of-print books would you like to see re-printed?
Old books still used
with answers that tell us things such as:
Mathematicians prefer to use ...
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Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...
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Extremely messy proofs
Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...
148
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31
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What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
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Occurrences of (co)homology in other disciplines and/or nature
I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
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Collection of equivalent forms of Riemann Hypothesis
This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...
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Tweetable Mathematics
Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.
I am looking for ...
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What is entropy, really?
I first saw the term "entropy" in a chemistry course while studying thermodynamics.
During my graduate studies I encountered the term in many different areas of mathematics.
Can anyone explain why ...
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Why do Bernoulli numbers arise everywhere?
I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...
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Theorems with unexpected conclusions
I am interested in theorems with unexpected conclusions. I don't mean
an unintuitive result (like the existence of a space-filling curve), but
rather a result whose conclusion seems disconnected from ...