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,...

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17
votes
2answers
1k views

Big Picture: What is the connection of Malliavin calculus with differential geometry?

I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of ...
32
votes
2answers
2k views

How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)

What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me) ...
22
votes
7answers
3k views

Why is it useful to classify the vector bundles of a space?

It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
7
votes
5answers
1k views

Is it necessary that model of theory is a set?

From Model Theory article from wikipedia : "A theory is satisfiable if it has a model $ M\models T$ i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set $T$". ...
4
votes
0answers
296 views

Q-construction and Gabriel-Zisman Localization

It might be a stupid question. When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...
112
votes
45answers
21k views

Examples of eventual counterexamples

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make. where '...
2
votes
1answer
410 views

“$\kappa$ 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 ...
5
votes
2answers
621 views

visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...
3
votes
3answers
1k views

Why is 2 so odd? [duplicate]

Possible Duplicate: Is there a high-concept explanation for why characteristic 2 is special? There are so many results on primes that either fail for $p=2$ or are not known to be true for $p=2.$ ...
35
votes
6answers
6k views

The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics

In this question, Orbicular made the following comment to Feb7 and my own answers; Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" ...
2
votes
3answers
490 views

Given is “model”. How many theories may it be a model?

Usually we have axiomatic theory and the we look for model for it - this is book picture. Of course in real math usual one has a "model" that is given structure and looks for proper axiomatizing of ...
9
votes
2answers
941 views

Integrable dynamical system - relation to elliptic curves

From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system): ...
149
votes
77answers
25k views

Your favorite surprising connections in Mathematics

There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
6
votes
11answers
2k views

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 ...
41
votes
7answers
5k views

“Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. ...
11
votes
5answers
3k views

Analogies between analogies

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between ...
16
votes
5answers
2k views

Category theory and model theory as “natural” counterparts

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory (...
85
votes
42answers
12k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples ...
38
votes
2answers
6k views

current status of crystalline cohomology?

The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously (...
4
votes
1answer
510 views

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 ...
14
votes
2answers
901 views

Why the similarity between Hodge theory for compact Riemannian and complex manifolds?

I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic ...
1
vote
0answers
211 views

Classification of properties of structures

Is there a sensible classification of the properties of structures with a given signature $\sigma$, e.g. graphs with $\sigma = \lbrace R \rbrace$? For example like this: properties defined by first-...
10
votes
2answers
827 views

What are important examples of filtered/graded rings in physics?

Hi, what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics? Of ...
28
votes
14answers
3k views

What are interesting families of subsets of a given set?

Motivation The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$. Indeed, one defines a topology on $S$ to be a family of subsets ...
20
votes
6answers
4k views

Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform? Moreover, could someone recommend a concise introduction to the subject?
19
votes
6answers
2k views

Generalizations of “standard” calculus

We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...
29
votes
21answers
7k views

What are some slogans that express mathematical tricks?

Many "tricks" that we use to solve mathematical problems don't correspond nicely to theorems or lemmas, or anything close to that rigorous. Instead they take the form of analogies, or general methods ...
6
votes
1answer
889 views

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 ...
13
votes
27answers
3k views

Justifying a theory by a seemingly unrelated example

Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...
19
votes
2answers
2k views

Why is the decomposition theorem awesome?

I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...
31
votes
7answers
6k views

Why is it useful to study vector bundles?

I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just ...
2
votes
2answers
454 views

How much faith should I put in numerics? [closed]

Edit: Let me summarize what this question was meant to ask. Is there a quantitative theory of "approximate" soundness? Arguments are usually either sound or unsound. This is binary. If we don't ...
10
votes
1answer
2k views

Kapranov's analogies

I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete ...
91
votes
12answers
18k views

What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...
7
votes
3answers
775 views

“Right” Way of Introducing Modular Forms to Undergraduate Audience?

I am giving an extremely short talk (~12 minutes) in a few weeks in which I need to be able to introduce and motivate the idea of a classical modular form to an audience of undergraduates in as short ...
42
votes
5answers
6k views

What is a cohomology theory (seriously)?

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"? I know that there exist generalized cohomology theories, Weil ...
22
votes
4answers
2k views

Visualizing how Cech cohomology detects holes

I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space. How can we directly visualize how and in what sense the Cech cohomology of a cover does this? In ...
63
votes
16answers
10k views

Why is it a good idea to study a ring by studying its modules?

This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
5
votes
3answers
400 views

Abstract Relation between Presehaves and Simplicial Sets

Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...
144
votes
136answers
31k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
9
votes
3answers
2k views

What is the relationship between algebraic geometry and quantum mechanics?

The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...
48
votes
11answers
6k views

What precisely Is “Categorification”?

(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
8
votes
3answers
351 views

How does one identify properties of objects with good “inheritance”?

When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the ...
11
votes
2answers
518 views

What is the conceptual significance of supercommutativity?

A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is said to supercommute if $xy = (-1)^{|x| |y|} yx$; in other words, odd elements anticommute. Why is this the "right" definition of supercommutativity? (...
31
votes
6answers
3k views

Universal definition of tangent spaces (for schemes and manifolds)

Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...
3
votes
1answer
213 views

limits of algebraic varieties

I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject). For the kind of example I have in mind, ...
14
votes
2answers
737 views

Is there any meaning to a “nice bijective proof?”

From Zeilberger's PCM article on enumerative combinatorics: The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal ...
6
votes
1answer
919 views

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.
5
votes
5answers
770 views

Visual representation of mathematical research interrelationships

I remember seeing a visualization in the form of a 2d (nodal) graph of all areas of academia, with math, physics and engineering over in one section, connecting in an arc to the central area of ...
4
votes
7answers
1k views

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 ...