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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Recursion theory from the standoint of category theory

It is (I believe) a very easy exercise to prove that the general recursive functions over the natural number object $N$ form a category. But what sort of category is it? From the fact that one can ...
Thomas Benjamin'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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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 ...
24 votes
2 answers
2k views

Research directions in persistent homology

I am interested in what are the possible directions for new research in persistent homology (more of the mathematical theoretical aspects rather than the computer algorithm aspects). So far from ...
8 votes
1 answer
324 views

Existence of Randomized polynomial time algorithm and some arithmetic analog of $ACC^0$ circuits for Factoring of primitive polynomials before LLL?

Before LLL came along in $1982$ there was no deterministic polynomial (in degree and number of bits in coefficients) way to factor square free primitive polynomials in $\Bbb Z[x]$. However was there ...
user avatar
6 votes
1 answer
414 views

Dyson's invitation: Opportunities in juxtaposition of incompatibles

"Up to now, my examples of missed opportunities have been mathematical discoveries which actually occurred, although they could have occurred a long time earlier. In such cases one can be sure that an ...
3 votes
0 answers
367 views

On the precise concentration of permanent of $\pm1$ matrices

Obtain $M\in\{-1,+1\}^{n\times n}$ by unbiased coin flipping. What is known about the distribution of permanent $\mathsf{Perm}(M)$? It seems to be bimodal. Given a function $g(n)$ what is the ...
user avatar
3 votes
1 answer
601 views

Are the paradoxes of material or strict implication used anywhere to prove theorems in mathematics

In the Stanford Encyclopedia of Philosophy entry "Relevance Logic", the following inference is listed as classically valid: The moon is made of green cheese. Therefore, it is raining in Ecuador ...
Thomas Benjamin's user avatar
19 votes
2 answers
2k views

Locales as geometric objects

There is the following analogy: $$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes}\end{array}$$ Here, ...
HeinrichD's user avatar
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17 votes
3 answers
8k views

Why is the current math community not contributing to machine learning much? [closed]

This question was inspired from What advantage humans have over computers in mathematics? and the answer of Brendan McKay, part of which is quoted in the below: The day will come when not only will ...
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
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39 votes
4 answers
6k views

Linear algebra in terms of abstract nonsense?

The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think. I was wondering what portions of basic linear algebra (first couple of courses) fall ...
Arrow's user avatar
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5 votes
2 answers
484 views

Critical points in $ZF$ without Choice

Recall the definition of critical point for set theory: A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...
Thomas Benjamin's user avatar
2 votes
1 answer
730 views

Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?

Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$. Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\...
user avatar
10 votes
0 answers
2k views

How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry? What do I mean by complex geometry? ...
Seidarkman'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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7 votes
1 answer
254 views

Most natural equivalence between $C^*$-algebras in NCG

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
Melquíades Ochoa's user avatar
31 votes
4 answers
2k views

Expert, Intuitive, Organizing Analogies

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...
48 votes
2 answers
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Grothendieck says: points are not mere points, but carry Galois group actions

Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French). The following is an excerpt from ...
Arrow's user avatar
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42 votes
26 answers
8k views

Where can square roots come from when they are not distances?

In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...
62 votes
14 answers
8k views

What advantage humans have over computers in mathematics?

Now that AlphaGo has just beaten Lee Sedol in Go and Deep Blue has beaten Garry Kasparov in chess in 1997, I wonder what advantage humans have over computers in mathematics? More specifically, are ...
61 votes
3 answers
5k views

Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order to really understand this result a rather broad background is required: you need to know analysis (pseudodifferential ...
truebaran's user avatar
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1 vote
1 answer
437 views

A morphism-revealing category? [closed]

Categories of sets and functions can be considered as subcategories of Set but when considered as subcategories of the category SubSet, of pairs of sets with pairs $(X,S)$, $S\subseteq X$, as objects ...
Lehs's user avatar
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3 votes
1 answer
591 views

Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following: Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...
user86638's user avatar
17 votes
1 answer
877 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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4 votes
1 answer
641 views

What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
Arrow's user avatar
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26 votes
2 answers
3k views

how do you visualize characteristic class?

For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be ...
Ben's user avatar
  • 849
12 votes
1 answer
539 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
170 votes
36 answers
35k views

Proposals for polymath projects

Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
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
  • 10.3k
1 vote
0 answers
118 views

Programmatically computing dual Hopf algebras: state of the art

Given a graded Hopf algebra of finite type, we know the (graded) linear dual is also a graded Hopf algebra. For instance the dual Hopf algebra to the polynomial algebra on an even degree generator, $R[...
WMycroft's user avatar
  • 133
7 votes
1 answer
3k views

Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...
riu_ss's user avatar
  • 87
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
29 votes
7 answers
4k views

Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ...
6 votes
1 answer
3k views

Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list http://en.wikipedia.org/wiki/Statistical_distance the only ...
Jane Fulton's user avatar
1 vote
1 answer
3k views

How many of Ramanujan's discoveries have had a practical application? [closed]

I was reading about the Indian mathematician Srinivasa Ramanujan who, before dying at the age of 32, independently compiled nearly 3900 results (this is from Wikipedia). So based on this he seems to ...
x457812's user avatar
  • 167
2 votes
0 answers
409 views

Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given by $$H(\mathcal{A}...
Turbo's user avatar
  • 13.7k
5 votes
2 answers
814 views

What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...
5 votes
2 answers
551 views

Can one make a category concrete by "enlarging the universe"?

This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no ...
Qfwfq's user avatar
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6 votes
3 answers
2k views

The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical Logic,...
-3 votes
1 answer
165 views

Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem. Could any one give reference or a simple introduction about such result known in their ...
XL _At_Here_There's user avatar
41 votes
3 answers
4k views

Underlying idea for (automorphic) L-function?

Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things. Within the Langlands philosophy, L-...
Spencer Leslie's user avatar
23 votes
8 answers
3k views

Examples of intuition from fields other than Physics to solve math problems

This is a chaser for the examples of using physical intuition to solve math problems question. Physical intuition seems to be used relatively frequently for solving math problems as well as stating ...
2 votes
2 answers
640 views

Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...
Lehs's user avatar
  • 852
37 votes
13 answers
4k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful? I ...
Lehs's user avatar
  • 852
3 votes
1 answer
590 views

Computer Science applications of Roth's Theorem [closed]

I have been reading about Additive Combinatorics and in particular Roth's theorem which states any positive upper density set has infinitely many 3-step arithmetic progressions. Let $A \subset \...
john mangual's user avatar
  • 22.6k
18 votes
0 answers
462 views

What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....
Daniel Moskovich's user avatar
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
17 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
  • 2,076
70 votes
9 answers
6k views

How does one find out what's happening in contemporary mathematics research?

How does one find out what's happening in contemporary mathematics research? EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside ...

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