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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1answer
347 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. ...
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1answer
97 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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102 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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2answers
383 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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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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3answers
596 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
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1answer
222 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 ...
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1answer
548 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{...
3
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2answers
229 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 ...
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0answers
46 views

Daniell integral of “generalized (of some sort)” functions?

Let $E$ be a (Dedekind $\sigma$-complete) Riesz space and $H\subseteq E$ a subspace. A Daniell integral $I\colon H\to\mathbb R$ is defined to be a positive linear functional which is continuous with ...
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1answer
130 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 ...
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231 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". ...
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2answers
220 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 ...
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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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3answers
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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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205 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 ${\rm II_1}$ ...
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15answers
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: ...
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1answer
531 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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0answers
117 views

Arithmetic that corresponds to combinatorial rectangles and cylinder intersections?

Definable subsets of $\mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets. In communication complexity the interpretation is more on intersection and union of ...
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2answers
1k views

Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
16
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1answer
876 views

Axiom of Choice versus V=L in opposition to large cardinals

Consider the following two observations: The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals. The axiom of Choice is incompatible with ...
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335 views

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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0answers
100 views

Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory: never perform quotients, add structure instead; never require subobjects, take fibres instead. ...
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239 views

Fundamental groupoid and fibration

In this post, it is said that a functor from the fundamental groupoid of a space $X$ (denoted by $\Pi(X)$) to the category $\mathrm{Vect}$ of vector spaces gives a flat vector bundle over $X$. But I ...
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4answers
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Why did Voevodsky consider categories “posets in the next dimension”, and groupoids the correct generalisation of sets?

Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...
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1answer
310 views

Three theorems on the number of nonzero coefficients of a polynomial

The number of positive real roots of a polynomial with real coefficients is strictly smaller than the number of nonzero coefficients of the polynomial. This is an immediate corollary of Descartes' ...
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1answer
495 views

Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness

Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question: Is it true that one can find a manifold $M$ which is homotopy ...
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1answer
719 views

What is the big picture of algebraic geometry? [closed]

I am trying to understand a big-picture for Algebraic Geometry: Given a category of commutative rings $\mathrm{CRing}$, we can create objects that locally look like these objects called schemes. This ...
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0answers
435 views

Floer cohomology from mapping spaces of $\infty$ categories

There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...
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2answers
194 views

Viewing parts of $\mathbb{V}$ 'from the top down' or 'from the bottom up'

I am curious about instances where we can glean nontrivial information about a certain piece of the universe by viewing it as being 'built over' a smaller part of the universe, or alternatively ...
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0answers
136 views

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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0answers
1k 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 ...
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0answers
66 views

Transformation or correspondence between language and real number

As we know, formal language can be regarded as a set of strings of alphabet, and real number can be regarded as sequence generated by set of integers, for example, denominators of the simple continued ...
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0answers
97 views

Any proved connection between Roth theorem and hartmanis stearns conjecture?

Roth theorem classifies numbers into two classes, one is rational and transcendental, another is irrational algebraic numbers, by the number of solutions to the inequality (finite or infinite), and ...
17
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8answers
2k views

Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?

What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found. A good example is Lovász's proof of ...
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2answers
594 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 ...
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1answer
434 views

What are the uses of coefficient systems for arithmetic cohomology theories?

In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local ...
3
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1answer
170 views

Is there a Fourier Analytic way to approximate volume?

Suppose a convex compact room in $3$-dimensions is given and source and microphones recorders are provided in the room that can locate echo timings there are works in literature which can give you the ...
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11answers
4k views

Why is the definition of the higher homotopy groups the “right one”?

If someone asked me the question for the fundamental group, I would answer as follows: The connection to classification of covering spaces. The fundamental group of many spaces is an object of ...
4
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1answer
294 views

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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1answer
143 views

On Shannon information theoretic capacity to coding distance metric translation

Shannon theory says that given a channel source variable $X$ and received variable $Y$ and channel $Y/X$ there is a capacity associated with this channel. The notion of maximum likelihood leads from ...
3
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0answers
135 views

How much can analogy between $\Bbb Z$ and $\Bbb F_q[t]$ work out to give better distance measures in information theory?

Let $x$ be transmitted symbol and $y$ be received symbol and $n$ be noise Given $y=x+n$ where symbols $x,y,n$ are in $\Bbb K$. If $\Bbb K=\Bbb Z$ then we take $|n|$ to be the magnitude of noise while ...
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1answer
609 views

Steps in Geometric Complexity Theory

GCT purports to provide a program to show that $NP \not \subset P/poly$. At the high level what are the steps involved in the program and what stage is each step in? What difficulties ...
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2answers
1k views

Interpretations of permanent

The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix ...
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2answers
938 views

Measuring a presheaf's failure to be a sheaf?

Apologies for the vagueness of question. Background this thread has some nice examples of presheaves failing to be sheaves. Question Is there a generic way to measure "how badly" a presheaf fails ...
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0answers
112 views

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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4answers
1k views

Is there a physically realizable inductive turing machine that can solve Hilbert's $10$th problem and can it overcome Church-Turing Hypothesis?

There is a claim on https://en.wikipedia.org/wiki/Super-recursive_algorithm#Inductive_Turing_machines that 'Simple inductive Turing machines are equivalent to other models of computation such as ...
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2answers
457 views

categorification of q-series

In his talk, S. Gukov asked two questions: What is the categorification of a $q$-serie ? How to associate to a 3-manifold a $q$-serie ? As far as I understand, he was looking for a bigarded ...
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4answers
4k views

Is algebraic geometry constructive?

Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an ...
5
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2answers
224 views

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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