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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Which revolutions in topology and geometry can we expect in the next 20 years? [closed]

In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
27 votes
5 answers
9k views

Why does mathematics seem to have a polarity bias?

Why does mathematics seem to have a polarity bias, i.e., why are products more common than coproducts, algebras more common than coalgebras, limits more common than colimits, monads more common than ...
Cameron Zwarich's user avatar
78 votes
9 answers
11k views

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 ...
0 votes
1 answer
342 views

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
user1248224's user avatar
2 votes
0 answers
72 views

Clique-coclique and uncertainty

The clique-coclique inequality states that for a graph $G$ on $n$ vertices that is either distance-regular or vertex-transitive, the independence number $\alpha(G)$ and the clique number $\omega(G)$ ...
Seva's user avatar
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10 votes
1 answer
362 views

Can $N!$ be computed in less than $\mathcal{O}(N)$ operations?

The standard algorithm to compute the factorial function $N!$ via repeated multiplications has complexity $\mathcal{O}(N)$, in the model in which each operation costs 1, no matter how many digits the ...
user6873235's user avatar
1 vote
0 answers
188 views

Can this kind of formal Mereological based interpretation of Set Theory be significant in understanding Set Theory?

To Atomic General Extensional Mereology $\sf AGEM$ add primitives of non-directional linking, which is a ternary relation symbol, and the primitive binary relation "is connected to". The ...
Zuhair Al-Johar's user avatar
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
  • 1,103
19 votes
7 answers
4k views

Do empirical studies have a place in contemporary mathematics research?

I was thinking about the Collatz conjecture a while back (I know, not the healthiest thing to think about). It occurred to me that while I might not be able to prove it true for all positive integers, ...
52 votes
11 answers
6k views

What is an important mathematical question?

$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste....
6 votes
1 answer
255 views

A geometric interpretation of the fractional Fourier transform

I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18): Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
Luke's user avatar
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6 votes
3 answers
338 views

Intuition behind Kleene's “second algebra” $\mathcal{K}_2$

The “second Kleene algebra” $\mathcal{K}_2$ is defined, e.g. here on nLab, or in section 1.4.3 of van Oosten's book Realizability: an Introduction to its Categorical Side (2008), or as example 3.4 of ...
Gro-Tsen's user avatar
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8 votes
0 answers
361 views

What is the relationship (if any) between constructivism, finitism and predicativism?

The terms “constructivism”, “finitism” and “predicativism” refer to ideas / currents in the philosophy of mathematics (or loosely defined conditions on a system of logic) that I think I understand ...
Gro-Tsen's user avatar
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4 votes
1 answer
253 views

Comparing Selberg and Eichler-Selberg trace formulas

The trace formula of Selberg gives an equality between trace of Hecke operators (a spectral sum) on spaces of Maass forms and sums over closed geodesics mostly. The Eichler-Selberg trace formula, ...
Lyer Lier's user avatar
  • 141
1 vote
0 answers
71 views

Any concrete survey on infinite and finite injury method?

I hope to have a historic outline of infinite and finite injury method and their main technical introdution.Any concrete survey on infinite and finite injury method recommended?
XL _At_Here_There's user avatar
6 votes
0 answers
300 views

Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?

If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
Anixx's user avatar
  • 9,316
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
2 votes
1 answer
252 views

Sieve theory through variational principles

Disclaimer: I'm just starting to read Sieve Methods by Halberstam and Richert, so my present knowledge of the subject is close to zero, but it made me wonder if some connection to physics could exist, ...
Sylvain JULIEN's user avatar
4 votes
1 answer
347 views

Examples of rich theories that started out as an infinite-dimensional inquiry

It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it ...
liuyao's user avatar
  • 485
0 votes
0 answers
209 views

Stories where a different definition lead to an inaccurate conclusion/a misunderstanding/etc

The overall question: What are some good examples where a different understanding of terminology or notation caused you to misinterpret a result in a way that was inaccurate? The intent here is of ...
120 votes
9 answers
13k views

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 ...
18 votes
4 answers
3k views

What are the "hot" topics in mathematical QFT at the time?

I am currently finishing my Master's studies in mathematical physics. One topic which always interested me a lot were modern mathematical approaches to Quantum Field Theory (QFT) as well as the ...
-5 votes
1 answer
196 views

Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]

It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance, $0^ \left( \begin{array}{cc} \frac{1}{2} &...
Anixx's user avatar
  • 9,316
0 votes
1 answer
124 views

In what circumstances do we typically encounter expressions like $(c/2+1/2)^n \pm(c/2-1/2)^n$?

It attracted my attention that in many areas of mathematics we sometimes encounter expressions of the form $(c/2+1/2)^n \pm(c/2-1/2)^n$, where $c$ is some kind of a known constant. Split-complex ...
Anixx's user avatar
  • 9,316
11 votes
2 answers
2k views

Motivation for birational geometry

I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
roymend's user avatar
  • 221
23 votes
5 answers
7k views

Why do we have two theorems when one implies the other?

Why do we have two theorems one for the density of $C^{\infty}_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ and one for the density of $C^{\infty}_c(\Omega)$ in $L^p(\Omega)$? with $\Omega$ an open subset ...
72 votes
13 answers
10k views

The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question ...
2 votes
1 answer
430 views

Where do these divergent integrals appear in mathematics and physics?

I have already asked a similar question, albeit far more extensive, but it was criticized and closed for being too extensive and promotional. So, here is a greatly truncated and focused version. Since ...
Anixx's user avatar
  • 9,316
6 votes
3 answers
548 views

Anomalous phenomena [closed]

What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern? This is most interesting when ...
9 votes
2 answers
1k views

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 ...
Matthieu Latapy's user avatar
1 vote
0 answers
98 views

What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?

I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
Anixx's user avatar
  • 9,316
2 votes
0 answers
126 views

On primes of specified length and bit pattern

Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$...
Turbo's user avatar
  • 13.6k
8 votes
1 answer
1k views

Motivation for $C^*$-algebras

I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
Emiel Lanckriet's user avatar
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
67 votes
3 answers
11k views

Nonconvexity and discretization

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
Peter Scholze's user avatar
0 votes
0 answers
179 views

Morphism of non-commutative algebras

Disclaimer: this question is a "big picture" one that comes from my personal thoughts in physics. If it doesn't fit this site, please tell me. While having a walk, I thought a bit about what ...
Sylvain JULIEN's user avatar
0 votes
1 answer
172 views

Sober spaces vs. spatial frames-a big picture

For any topological space $X$ one can consider the so called frame of all open subsets of $X$ to be denoted by $\mathcal{O}(X)$. If $f:X \to Y$ is continuous taking the inverse image we get the ...
truebaran's user avatar
  • 9,140
26 votes
3 answers
3k views

What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?

First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that ...
exfret's user avatar
  • 479
2 votes
0 answers
108 views

Evidence of optimality of sieve algorithms

Sieve techniques apply to integer factoring and discrete logarithm to provide $2^{O(((\log n)(\log\log n)^2)^{1/3})}$ complexity for $n$ bit factoring and $n$ bit prime discrete logarithm. The state ...
Turbo's user avatar
  • 13.6k
1 vote
0 answers
108 views

Is there a bipartite graph whose determinant corresponds to number of perfect matchings?

Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$...
Turbo's user avatar
  • 13.6k
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
  • 9,316
5 votes
3 answers
476 views

Is there a quantum analog of Kolmogorov Complexity?

Kolmogorov Complexity (interpreted in terms of shortest program computing a string) and Shannon Entropy are quite similar. Since there is a quantum entropy is it reasonable to ask if there is quantum ...
Turbo's user avatar
  • 13.6k
6 votes
1 answer
272 views

Geometric intuition for $R[x,y]/ (x^2,y^2)$, kinematic second tangent bundle, and Wraith axiom

This is a sort of continuation of this question. In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that ...
Arrow's user avatar
  • 10.3k
11 votes
0 answers
402 views

What is the motivation for a Frobenius manifold?

A Frobenius manifold is a type of manifolds with extra structure. The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities (...
Pulcinella's user avatar
  • 5,506
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
77 votes
15 answers
13k views

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 ...
163 votes
46 answers
31k views

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 ...
0 votes
1 answer
467 views

Mathematics based only on real numbers [closed]

I'm aware that >90% will outright reject this, so feel free to ignore it. I'd much appreciate those trying to figure out in which way this question (or rather its eventual answer) would make sense. ...
user avatar
-1 votes
1 answer
528 views

What are some interesting relationships between pi and phi? [closed]

Phi is the golden mean solution to the 1/x=1+x and pi the transcendental number relating the radius of the circle to its area. A side note: while there are really interesting series converging to pi, ...
Kugutsu-o's user avatar
  • 147
2 votes
0 answers
118 views

Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
VS.'s user avatar
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