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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,...

5
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207 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
0answers
89 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
0answers
173 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 ...
0
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0answers
78 views

How well does convolution model $\mathbb Z$ multiplication?

In $\mathbb K[x]$ or $\mathbb K$ where $\mathbb K$ is a ring we can think of multiplication of polynomials as convolution. Over $\mathbb Z$ this line of thought has led to fast integer multiplication ...
54
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4answers
3k 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
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1answer
209 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
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1answer
432 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 ...
-2
votes
1answer
507 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 ...
9
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0answers
376 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 ...
2
votes
2answers
179 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 ...
3
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0answers
124 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 ...
14
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0answers
862 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
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0answers
65 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
0answers
78 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 ...
14
votes
2answers
530 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 ...
0
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0answers
135 views

On a permanent related rank?

Given a matrix $M\in\Bbb Z^{n\times n}$ let $r(m,M)$ be the number of $m\times m$ submatrices of $M$ whose permanent is $\neq0$. Given a matrix $M$ what is the typical behavior of the function $r(m,M)...
7
votes
1answer
304 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
1answer
159 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 ...
33
votes
11answers
3k 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
1answer
279 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 ...
1
vote
1answer
134 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
134 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 ...
7
votes
1answer
471 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 ...
21
votes
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 ...
15
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2answers
793 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
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0answers
99 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 (...
2
votes
4answers
871 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 ...
54
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19answers
8k views

When has discrete understanding preceded continuous?

From my limited perspective, it appears that the understanding of a mathematical phenomenon has usually been achieved, historically, in a continuous setting before it was fully explored in a discrete ...
8
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2answers
419 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 ...
35
votes
3answers
3k 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
2answers
203 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 ...
23
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2answers
2k 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
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2answers
168 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
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0answers
321 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_{...
6
votes
1answer
407 views

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 ...
12
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1answer
1k 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 ...
12
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2answers
663 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 ...
8
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1answer
289 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 ...
6
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1answer
328 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
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0answers
266 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 ...
4
votes
1answer
421 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 ...
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2answers
678 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, ...
11
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3answers
5k 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 ...
8
votes
2answers
1k 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 ...
32
votes
3answers
3k 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 ...
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2answers
326 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 ...
2
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1answer
338 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 $\...
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0answers
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? ...
15
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1answer
978 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 ...