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.
369
questions
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 ...
10
votes
2
answers
843
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 ...
6
votes
0
answers
308
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 ...
5
votes
3
answers
795
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
287
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
1k
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
votes
2
answers
340
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
145
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 ...
3
votes
0
answers
279
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
359
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 ...
6
votes
0
answers
115
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 ...
40
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 ...
11
votes
0
answers
314
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
9k
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
2k
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 ...
1
vote
0
answers
129
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 ...
31
votes
2
answers
3k
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 ...
20
votes
1
answer
1k
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 ...
6
votes
0
answers
520
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 ...
4
votes
0
answers
108
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.
...
2
votes
0
answers
325
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 ...
66
votes
4
answers
5k
views
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 ...
9
votes
1
answer
631
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' ...
11
votes
1
answer
634
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 ...
-1
votes
1
answer
940
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 ...
11
votes
0
answers
509
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 ...
3
votes
2
answers
212
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 ...
6
votes
3
answers
419
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 ...
15
votes
0
answers
3k
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 ...
0
votes
0
answers
74
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 ...
2
votes
0
answers
142
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 ...
18
votes
8
answers
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 ...
14
votes
2
answers
776
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 ...
8
votes
1
answer
753
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
votes
1
answer
200
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 ...
41
votes
12
answers
6k
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
votes
1
answer
314
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 ...
2
votes
1
answer
161
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
votes
0
answers
144
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 ...
8
votes
1
answer
925
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 currently are ...
25
votes
3
answers
2k
views
Interpretations and models 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 ...
18
votes
2
answers
1k
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 ...
6
votes
0
answers
126
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 (...
3
votes
4
answers
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 ...
9
votes
2
answers
501
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 ...
40
votes
4
answers
5k
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
votes
2
answers
311
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 ...
31
votes
2
answers
3k
views
Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme
(Disclaimer: I'm no expert in homotopy theory nor in higher categories!) If I understand it correctly, Grothendieck's homotopy hypothesis states that there should be an equivalence (of $(n+1)$-...
4
votes
2
answers
368
views
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 ...
13
votes
0
answers
704
views
Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...