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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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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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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Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?
From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...
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Axiom of choice as zero dimensionality
In the paper Quantifiers and Sheaves by Lawvere, at the bottom of the second page, the author writes:
"... the condition that every epi splits, which geometrically we would call 0-dimensionality ...
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What is the conceptual significance of supercommutativity?
A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is said to supercommute if $xy = (-1)^{|x| |y|} yx$; in other words, odd elements anticommute. Why is this the "right" definition of supercommutativity? (...
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Foundations of topology
I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here.
Also some time ago I read ...
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Monte Carlo integration
As probably many other people here, I learned integration, as an undergrad, from Rudin's books. I recently realized, however, that I don't quite use Lebesgue integration in my work, or at least I use ...
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A Sketch of "Esquisse d'un Programme"
I'm refering, of course, to Grothendieck's ambitious program available fully here: http://www.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf
The text is, as described in its title, a sketch,...
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Algebraic Geometry in Number Theory
It appears to me that there are two main ways by which algebraic geometry is applied to number theory. The first is by studying polynomials over fields of number-theoretic interest (which does not ...
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What is lambda calculus related to?
So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based.
I was wondering if anyone had a suggestion ...
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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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The advantage of asymmetric objects
We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of ...
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Kapranov's analogies
I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete ...
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Why the similarity between Hodge theory for compact Riemannian and complex manifolds?
I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic ...
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What do you use categorical glueing/sconing/Freyd covers for?
In the theory of programming languages and structural proof theory, one of the handiest techniques we have available is a method called "logical relations", in which you can prove properties of ...
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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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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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Is there any meaning to a "nice bijective proof?"
From Zeilberger's PCM article on enumerative combinatorics:
The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal ...
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What is the appropriate setting for Cauchy's Integral Formula?
For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:
$$
f(\zeta) = \...
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Justifying a theory by a seemingly unrelated example
Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...
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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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Is the ABC conjecture known to imply the Riemann hypothesis?
I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
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Why the search for ever larger primes?
I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the ...
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How are Modal Logic and Graph Theory related?
I am currently taking a graduate logic course on Modal Logic and I can't help notice that there are a certain class of graphs characterized by the modal axioms such as (4) $\Box p \rightarrow \Box \...
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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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What is the status of (universal) algebra in type theory?
With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...
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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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Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
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Why are Witten-Reshetikhin-Turaev invariants expected to be integral?
A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ ...
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What to expect from spectral algebraic geometry
So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
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What advanced area of mathematics can be delved into with only basic calculus and linear algebra
Hello Mathoverflow Community,
I would really appreciate some advice on this:
All I know is basic calculus and basic linear algebra,
I want to start learning more advanced material on my own while ...
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Generalized notions of solutions in various areas of mathematics
In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are ...
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What is the interface between functional analysis and algebraic geometry?
This is a very open ended curiosity of mine and I would be grateful to hear any comments in this direction. In particular I am interested in functional analysis/algebraic geometry books/papers ...
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What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?
Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several ...
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What is the applications of the dg-enhancements of derived categories of sheaves
Let $X$ be a scheme and let $D^b_{\text{coh}}(X)$ be the derived category of complexes of sheaves with bounded, coherent cohomologies.
We know that the category $D^b_{\text{coh}}(X)$ has some ...
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Can there be a polymath project for mathematical physics?
My hunch is that it might be possible to create something like https://polymathprojects.org/ for mathematical physics and I'd like to know whether MathOverflow users can recommend some appropriate ...
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Can the methods of classical algebraic geometry be made rigorous with a synthetic approach?
There are approaches to real analysis that use an axiomatization of nilpotent infinitesimals to enable rigorous synthetic reasoning about infinitesimals, which is arguably closer to the reasoning ...
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Proof by `universal receiver'
Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually ...
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Meaning of a quantum field given by an operator-valued distribution
I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me.
Let $\mathcal{H}$ be a Hilbert space in which ...
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What is about J. v. Neumann's "Continuous geometry"?
I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of ...
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Linear/Non-linear sigma model
This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model.
I gauge from some other interactions that it has considerable ...
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Hecke-algebras in your field of mathematics
(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important?
How would you define them and how do you think about them?
e.g. generators and relations, functions ...
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Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...
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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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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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Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
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Status of Beilinson conjectures?
(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...
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Comparing the existing formulations of universal algebra and their levels of generality
I am a newcomer to universal algebra and I just read this (very good, IMO) book on the topic:
Adámek, J., Rosický, J., & Vitale, E. M. (2010). Algebraic theories: a categorical introduction to ...
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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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Status of the Leopoldt conjecture ? [closed]
In 2009 Mihailescu published a proof of the famous Leopoldt conjecture on the arxiv. Later on, in 2011, he published a 'lightweight' version that proves the conjecture for CM fields. Also compare
...