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 ...
roger123's user avatar
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17 votes
8 answers
10k views

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 ...
Seva's user avatar
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17 votes
1 answer
6k views

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 ...
Joseph O'Rourke's user avatar
17 votes
1 answer
876 views

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 ...
Arrow's user avatar
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17 votes
2 answers
830 views

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? (...
Qiaochu Yuan's user avatar
17 votes
2 answers
2k views

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 ...
Bananeen's user avatar
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17 votes
2 answers
1k views

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 ...
Richard's user avatar
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17 votes
1 answer
2k views

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,...
Makhalan Duff's user avatar
17 votes
1 answer
4k views

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 ...
Anton Hilado's user avatar
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16 votes
7 answers
3k views

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 ...
Marcus Booster's user avatar
16 votes
2 answers
2k views

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^...
Anixx's user avatar
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16 votes
3 answers
2k views

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 ...
Veronica Phan's user avatar
16 votes
1 answer
4k views

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 ...
Thomas Riepe's user avatar
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16 votes
2 answers
1k views

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 ...
Akhil Mathew's user avatar
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16 votes
2 answers
1k views

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 ...
Neel Krishnaswami's user avatar
16 votes
1 answer
3k views

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 ...
Koushik's user avatar
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16 votes
3 answers
1k views

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 ...
16 votes
2 answers
1k views

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 ...
Harrison Brown's user avatar
16 votes
0 answers
1k views

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) = \...
Greg Zitelli's user avatar
15 votes
27 answers
3k views

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 ...
15 votes
5 answers
3k views

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 $T$". ...
kakaz's user avatar
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15 votes
5 answers
8k views

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 ...
James Weigandt's user avatar
15 votes
6 answers
2k views

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 ...
Tall Jeff's user avatar
  • 261
15 votes
4 answers
3k views

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 \...
Samuel Reid's user avatar
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15 votes
3 answers
992 views

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, ...
Joseph O'Rourke's user avatar
15 votes
2 answers
2k views

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 ...
Cory Knapp's user avatar
15 votes
2 answers
1k views

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 ...
Wolfgang's user avatar
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15 votes
1 answer
1k views

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. ...
Arrow's user avatar
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15 votes
1 answer
1k views

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 $\xi$ ...
Daniel Moskovich's user avatar
15 votes
0 answers
3k views

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 ...
Exit path's user avatar
  • 2,969
14 votes
12 answers
6k views

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 ...
14 votes
16 answers
1k views

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 ...
14 votes
4 answers
5k views

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 ...
gradstudent's user avatar
  • 2,136
14 votes
2 answers
2k views

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 ...
user17868's user avatar
  • 163
14 votes
1 answer
2k views

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 ...
Zhaoting Wei's user avatar
  • 8,657
14 votes
3 answers
1k views

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 ...
14 votes
2 answers
774 views

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 ...
Cameron Zwarich's user avatar
13 votes
6 answers
1k views

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 ...
13 votes
4 answers
2k views

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 ...
Jannik Pitt's user avatar
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13 votes
4 answers
2k views

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 ...
Thomas Riepe's user avatar
  • 10.7k
13 votes
3 answers
5k views

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 ...
Anirbit's user avatar
  • 3,453
13 votes
4 answers
2k views

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 ...
13 votes
0 answers
704 views

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_{...
Zhaoting Wei's user avatar
  • 8,657
13 votes
0 answers
330 views

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 ...
David Corfield's user avatar
12 votes
2 answers
3k views

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?
Thomas Riepe's user avatar
  • 10.7k
12 votes
3 answers
810 views

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 ...
jg1896's user avatar
  • 2,683
12 votes
2 answers
2k views

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 ...
12 votes
3 answers
592 views

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 ...
antoine_vm's user avatar
12 votes
1 answer
538 views

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 ...
Amit Sing Mukerjee's user avatar
12 votes
0 answers
2k views

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 ...
KBuck's user avatar
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