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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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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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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Proofs that inspire and teach
I was just listening to the show "A Splendid Table" (which I'd recommend, if you're interested in food) in which they were discussing how to spot a good recipe: one which you can follow successfully ...
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Which mathematical questions are objectively true or false? [closed]
Which parts of mathematics are objectively true or false like finite arithmetic and which are only true, false or undecidable relative to a particular axiom system such as Euclidean geometry?
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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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Finite subgroups of the unimodular group
This is related to this MO question (and others as well).
Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:
1) The problem of classifying ...
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Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?
Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?
I have long been intrigued by the observation that much of mathematics ...
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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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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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Intrinsic vs. Extrinsic [closed]
Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, differential geometry (abstract manifolds vs. ...
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An undergraduate's guide to the foundational theorems of logic [closed]
How would you explain one of these theorems in the foundations of mathematics to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the ...
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What are other applications of difference equations in other branches of mathematics ?
What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?
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Characterizing specific "concrete" mathematical objects by abstract general properties
In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a function space, ...
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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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ubiquitous modulicity?
On the one hand, as mentioned here, basically "everything" in algebraic geometry could be seen in the context of "moduli problems" - on the other hand, Grothendieck's few remarks on a possible "tame ...
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What information is contained in the Kazhdan-Lusztig polynomials?
The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations.
For example the character of a simple module over a Lie algebra with Weyl group $W$ ...
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Hilbert's 3rd problem,number theory, motives, cyclic homology,...
This talk by Jinhyun Park connects a lot of interesting themes, making me curious to read more about that. Do you know where?
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More on "Transalgebraic Theories" (a 19th century yoga)?
Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful '...
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Usefulness of symbolic devices
Each mathematician knows that good notation or symbolism – which seems to be irrelevant from a purely logical point of view – makes theorems more plausible and motivates results which would ...
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Thom's Principle: rich structures are more numerous in low dimension
Marcel Berger states Thom's Principle as:
"rich structures are more numerous in low dimension,
and poor structures are more numerous in high dimension."
This is in
Geometry II
(Springer-Verlag, ...
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Interesting mathematical topics arising from biology
I've heard that there's a relatively new field of science called mathematical biology.
It will certainly apply well known and less known mathematical techniques to the understanding of some biological ...
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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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How are mathematical objects defined from an ultrafinitist perspective?
I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
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Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?
Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...
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Did Joseph Doob prove that random sequences don't exist?
In the book "The Mathematical Experience" it says:
"An infinite [binary] sequence $x_1, x_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x_{n_1}, x_{n_2}, \ldots$...
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What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?
I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a ...
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On what kind of objects do the Galois groups act?
I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...
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Various flavours of infinitesimals
I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for ...
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What is an algebraic object? [closed]
The question is in the title. In order to give a bit more backround about the question, one knows that their are several different notions of an algebraic object. One approach is that of Lavere and ...
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Why the triangle inequality?
[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y) = d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and ...
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Categorical Invariants
I apologize in advance if this question seems too vague.
In many topology courses, concepts like the fundamental group and homology groups are introduced as a means of distinguishing non-...
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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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The first eigenvalue of a graph - what does it reflect?
A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the ...
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Why is the Arthur trace formula so powerful?
Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all ...
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Does mathematical fecundity ever deviate from its applicability?
We are all familiar with Wigner's "unreasonable effectiveness
of mathematics" thesis (1), and of Hardy's opinion
that "the great bulk of higher mathematics is useless" (2).
I am wondering if there are ...
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Logic in mathematics and philosophy
What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second ...
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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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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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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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Surprising and Useful Physical Intuition for Mathematical Objects
I believe I.M. Gelfand said that when beginning to learn a new subject, one should learn it like a physicist.
In this spirit, what are some helpful and surprising physical intuitions accompanying ...
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What do whitehead towers have to do with physics?
First let me say something that I don't completely understand, since I do not know enough physics. If I say anything wrong, someone please tell me:
For the spinning particle, there is a sigma-model, ...
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(Non?)-linearity of the consistency strength ordering in ZF
Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...
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Why chain homotopy when there is no topology in the background?
Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow D_{n+...
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Knowledge base about topology [closed]
We are studying topology. There are a lot of definitions and theorems. I wonder if there somewhere knowledge base about topology and reasoning system exists. So I expect some tool that systematizes ...
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Canonical Time Evolution for Type $II_{1}$-Factors?
This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (...
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Counting Lattice Points in Real Polytopes
Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...
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Analogue to Serre spectral sequence for cofiber sequences and homotopy
(This is a follow-up question to this one).
As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well ...
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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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Why does homotopy behave well with respect to fibrations and homology with respect to cofibrations?
(I apologize that this is a vague question).
I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups ...
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Why semigroups could be important?
There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
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