Questions tagged [big-picture]
Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
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Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
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Examples of eventual counterexamples
Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
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Explaining Mukai-Fourier transforms physically
A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).
The basic algorithm is ...
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Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
Consider the operator $\frac D{e^D-1}$ which we will call "shadow":
$$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
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What is an integrable system?
What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "...
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Experimental mathematics leading to major advances
I would like to ask about examples where experimentation by computers has led to major mathematical advances.
A new look
Now as the question is five years old and there are certainly more examples of ...
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The concept of duality
I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come....
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Your favorite surprising connections in mathematics
There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
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Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
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What recent discoveries have amateur mathematicians made?
E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
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How do you decide whether a question in abstract algebra is worth studying?
Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
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Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
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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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Mathematics of path integral: state of the art
I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called ...
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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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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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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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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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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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What is the significance of non-commutative geometry in mathematics?
This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more ...
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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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Why is Set, and not Rel, so ubiquitous in mathematics?
The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling ...
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What is the high-concept explanation on why real numbers are useful in number theory?
The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...
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What advantage humans have over computers in mathematics?
Now that AlphaGo has just beaten Lee Sedol in Go and Deep Blue has beaten Garry Kasparov in chess in 1997, I wonder what advantage humans have over computers in mathematics?
More specifically, are ...
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How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)
What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
...
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Are there any "homotopical spaces"?
This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.
[Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...
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Heuristic behind the Fourier-Mukai transform
What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?
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Is it necessary that model of theory is a set?
From Model Theory article from wikipedia : "A theory is satisfiable if it has a model $ M\models T$ i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set $T$". ...
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Analytic tools in algebraic geometry
This is not a very precise question, but I hope it will get some good answers.
As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ...
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What makes four dimensions special?
Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...
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Why are modular forms interesting?
Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
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Breakthroughs in mathematics in 2021
This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important ...
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Examples of major theorems with very hard proofs that have not dramatically improved over time
This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of
Major theorems in mathematics whose proofs are ...
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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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What is a cumulant really?
A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants ...
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Grothendieck's manuscript on topology
Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...
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A map of non-pathological topology?
I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the ...
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"Softness" vs "rigidity" in Geometry
According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually,...
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Coefficients of Weil Cohomology Theories
A Weil Cohomology theory is a functor from the category of smooth projective varieties (over some fixed field $k$) to graded $K$-algebras (for some fixed field $K$) satisfying various axioms. For ...
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Why are finiteness conditions important (and how to recognize them)?
I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a ...
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Using higher-order Bring radicals to solve arbitrary polynomials
It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...
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3D generalizations of permutations, RSK correspondence, contingency tables, etc.
I want to gather facts and questions related to 3D generalizations
of permutations, RSK correspondence, contingency tables,
etc. One reason I am interested in this is because it is potentially
related ...
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Differences in philosophy between Lie Groups and Differential Galois Theory
As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the ...
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Why are some q-analogues more canonical than others?
It is striking that some q-analogs of functions, operators, identities and especially whole theorems seem quite "canonical", e.g.
the factorial and the q-Gamma function
the basic hypergeometric ...
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On Bailey–Borwein–Plouffe formula for irrational numbers
A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...
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Differentiation of functions over graphs
In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary.
Definitions.
Let $G=(V,E)$ be a directed ...
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Advice on choosing an area of specialization [closed]
I'm not sure if this is an appropriate question for MO, but I figured it couldn't hurt to ask. I'm a second year graduate student, currently gearing up to construct a committee and syllabus for my ...
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Duality of eta product identities: a new idea?
Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs:
let's call two eta product identities $\sum\limits_{i=1}^r a_iP_i=...
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Differential Galois number theory
Following https://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...