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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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?
6 votes
0 answers
239 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 ...
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17 votes
3 answers
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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 ...
2 votes
1 answer
158 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, ...
4 votes
1 answer
302 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 ...
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0 votes
0 answers
201 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 ...
112 votes
8 answers
11k 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
2k 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
177 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} &...
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0 votes
1 answer
111 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 ...
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11 votes
2 answers
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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 ...
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23 votes
5 answers
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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
12 answers
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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
362 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 ...
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6 votes
3 answers
526 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 ...
7 votes
2 answers
584 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 ...
1 vote
0 answers
83 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 ...
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1 vote
0 answers
115 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 $n>1$...
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8 votes
1 answer
843 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 ...
11 votes
3 answers
468 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 ...
63 votes
2 answers
9k 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(\...
0 votes
0 answers
154 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 ...
0 votes
1 answer
131 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 ...
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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 ...
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2 votes
0 answers
103 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 ...
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1 vote
0 answers
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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 $2$...
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17 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^...
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5 votes
3 answers
405 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 ...
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6 votes
1 answer
238 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 ...
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11 votes
0 answers
337 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 (...
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11 votes
3 answers
745 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 ...
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76 votes
15 answers
12k 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 ...
157 votes
46 answers
28k 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
444 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
316 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, ...
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2 votes
0 answers
111 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 ...
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10 votes
2 answers
623 views

Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
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5 votes
0 answers
235 views

Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this?

If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system \begin{align*}L\psi&=\lambda\psi\\ \psi_t&=B\psi\end{align*} has as its integrability condition ...
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5 votes
3 answers
721 views

Update on "Hopf algebras: their status and pervasiveness" by Hazewinkel

Hazewinkel wrote this article in 2005. Perhaps it's time for an update. For example, updating item 34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of ...
5 votes
1 answer
269 views

The Idea of Kroneckerian geometry

Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$. Let's ...
8 votes
1 answer
965 views

Geometric intuition behind this chain homotopy

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion $$C_\bullet^\mathcal{...
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3 votes
2 answers
292 views

Theories requiring dual continuous and discrete constructs

Over the years there have been questions of a similar ilk on MO (e.g., Q1, Q2, Q3) concerning theories in which either continuous constructs or discrete constructs preceded the development of the ...
1 vote
1 answer
138 views

Understanding the reason for the particular formulation of the definition of a concrete reflector (as stated in The Joy of Cats)

This question is essentially a followup of this question. But before going into the question let me introduce the relevant definitions as given in The Joy of Cats. Definition 1. Let $\bf{X}$ be a ...
user avatar
3 votes
0 answers
253 views

What is the logical progression in algebraic tools for studying spaces (varieties -> schemes, sheaves, topos etc.)?

Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". ...
2 votes
2 answers
306 views

How should I think about concrete functors and in particular about concrete isomorphism?

All the definitions that follow is taken from The Joy of Cats. Definition 1. Let $\bf{X}$ be a category. A concrete category over $\bf{X}$ is a pair $({\bf{A}},U)$, where $\bf{A}$ is a category ...
user avatar
6 votes
0 answers
110 views

On earlier references for $P=BPP$ and Kolmogorov's possible view on modern breakthroughs involving randomness?

Kolmogorov and Uspenskii in this paper 'http://epubs.siam.org/doi/pdf/10.1137/1132060' speculate $P=BPP$ in $1986$. They do this without getting into circuit lower bounds and from a different view ...
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38 votes
3 answers
3k views

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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11 votes
0 answers
272 views

Hausdorff dimension and von Neumann dimension

There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
82 votes
15 answers
8k views

Theorems that impeded progress

It may be that certain theorems, when proved true, counterintuitively retard progress in certain domains. Lloyd Trefethen provides two examples: Faber's Theorem on polynomial interpolation: ...
3 votes
1 answer
1k views

Regarding learning Algebraic Topology [closed]

Recently, I read a little portion of homotopy theory from Bredon's 'Topology and Geometry' and found that I like it enough to want to continue reading material in Algebraic Topology. A little ...

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