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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Does the presence of cocycle conditions indicate the existence of an underlying cohomology theory?
Motivation: We have two examples:
(Abelian) Kummer theory (resp. Artin-Schreier theory) has a hidden cohomology theory given by Galois cohomology. The cocycle conditions become clear when you look ...
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Given a sequence defined on the positive integers, how should it be extended to be defined at zero?
This question is inspired by a lecture Bjorn Poonen gave at MIT last year. I have ideas of my own, but I'm interested in what other people have to say, so I'll make this community wiki and post my ...
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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 ...
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Is there a deep reason for the fecundity of involutions?
You might have come across the book of involutions in your travels. A colleague of mine asked whether there is a natural global reason (versus ad-hoc trickery) for considering involutions in ...
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Elementary questions in arithmetic geometry
In many theories there is a rough divide between elementary problems that can be solved with "one's hands", and "deep results that require powerful tools". For example, I am told that Hodge theory is ...
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Duality of eta product identities: a new idea?
Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs:
let's call two eta product identities $\sum\limits_{i=1}^r a_iP_i=...
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Under which constraints are there only finite numbers of irreducible eta product identities?
For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n})$, let for brevity $e_k:=\eta(q^k)$.
An eta product identity (or eta identity for ...
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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 ...
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Does mathematical fecundity ever deviate from its applicability?
We are all familiar with Wigner's "unreasonable effectiveness
of mathematics" thesis (1), and of Hardy's opinion
that "the great bulk of higher mathematics is useless" (2).
I am wondering if there are ...
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What can't be described by categories?
I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that ...
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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 ...
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Various concepts of "closure" or "completion" in mathematics
Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them ...
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Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
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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,...
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Number of ways to construct mathematical objects
This question stems from this other one mentioning 7 ways of constructing smooth manifolds. I quote:
At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods ...
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"$\kappa$ strongly inaccessible" = "every function $f:V_\kappa\to V_\kappa$ can be self-applied"?
Strongly inaccessible cardinals are usually introduced either as (a) cardinalities of models of ZFC or (b) cardinals which are not the power set of a smaller cardinal nor the supremum of a set with ...
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Clean introduction to toric varieties for an undergraduate audience
I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need ...
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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 ...
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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 ...
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How are mathematical objects defined from an ultrafinitist perspective?
I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
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What is Drinfeld's manuscript "Best Dream" (in Russian!) about?
I would like to know what Drinfeld's scanned manuscript "Best Dream" is about: the title makes me curious.
It's in Russian.
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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 ...
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Which constants are ambivalent and why?
This question is possibly a bit more philosophical $-$ compatible with the Christmas season, which is an appropriate moment to look at the world from a more universal angle... My last question with a ...
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Graphical representation of mathematical structures (in the spirit of unified modeling language)
In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are ...
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What notions of universe does predicative type theory admit?
Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in ...
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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 ...
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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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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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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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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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Grothendieck letter to Jun-Ichi Yamashita on tame topology
I am looking for Grothendieck writings on tame topology:
a manuscript on tame topology mentioned by Scharlau; a letter to Jun-Ichi Yamashita; a letter to Z.Mebkhout.
I am also interested in ...
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How to decompose cuspidal representations?
Let $\mathbb{G}$ be a connected reductive group over $\mathbb{F}_q$. Let $R_{T}^{\theta}$ be a Deligne--Lusztig representation of $\mathbb{G}(\mathbb{F}_q)$. Assume that $R_{T}^{\theta}$ is cuspidal (...
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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 ...
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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 ...
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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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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 ...
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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 ...
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Convex hull with genus information
Are there convexity generalizations that admit genus information?
For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can ...
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Background reading for proving irrationality of real numbers
I'm almost finishing my PhD in applied mathematics, but I'm planning soon (after doing a post-doc) to start seriously doing research on problems about proving irrationality of real numbers. Whenever I ...
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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 ...
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Differential Galois number theory
Following https://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...
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Algebraic number theory: building and simplifying
This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.
As we all know, ...
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Intrinsic vs. Extrinsic [closed]
Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, differential geometry (abstract manifolds vs. ...
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About the cone being unique up to non-unique isomorphism
In an answer to this MO question [link] Fernando Muro sais:
the mapping cone of a morphism in a triangulated category is unique up
to non-unique isomorphism. This fact has originated a lot of ...
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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 ...
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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, ...
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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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Maximality without Zorn
When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.
I am interested in instances of proving the existence of ...
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Toric ideal of slice of a polytope?
Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...
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What do orbital integrals have to do with reciprocity?
Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program.
In an attempt to sort through the articles ...