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 ...

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8
votes
0answers
416 views

What is the origin of the formula for the Lie derivative along a Killing vector?

Background Let $(M,g)$ be an $n$-dimensioal riemannian manifold. A vector field $X$ on $M$ is said to be a Killing vector if the flow it generates is an isometry; that is, it preserves the metric ...
33
votes
13answers
2k views

Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation: What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit? I believe this to be a serious question because ...
6
votes
1answer
2k views

Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory

I am confused and don't get the big picture concerning the connection between Ito integral Stratonovich integral Standard results in probability theory concerning skewed distributions. Example: ...
16
votes
3answers
816 views

Natural setting for characteristic classes?

In my mind, algebraic topology is comprised of two components: Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks". ...
24
votes
1answer
1k views

Are there any “homotopical spaces” ?

This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense. [Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...
3
votes
5answers
1k views

Is it true that the only interesting topologies are metric topologies and weak topologies?

In "Infinite dimensional analysis, A hitchhikers guide" by Aliprantis and Border, they write that these 2 classes of topologies "by and large include everything of interest". @Pete Clarke: I was ...
12
votes
5answers
1k views

Are rings really more fundamental objects than semi-rings?

The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics. From then on, it seems ...
8
votes
7answers
2k views

Path integrals outside QFT

The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical ...
65
votes
6answers
8k views

How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...
10
votes
4answers
4k views

Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
25
votes
5answers
5k views

Mathematics of path integral: state of the art

I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called ...
26
votes
5answers
5k views

Doing geometry using Feynman Path Integral?

I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space. Coming from a background of studying Quantum Field Theory from the books like ...
7
votes
3answers
942 views

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 ...
21
votes
12answers
6k views

Why are tensors a generalization of scalars, vectors, and matrices?

Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear ...
15
votes
12answers
2k views

What are some fundamental “sources” for the appearance of pi in mathematics?

I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, ...
13
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 ...
26
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) ...
19
votes
7answers
2k 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
276 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 ...
82
votes
38answers
15k 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
366 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
532 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 ...
32
votes
6answers
5k 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
480 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 ...
8
votes
2answers
791 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): ...
102
votes
61answers
19k 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 ...
38
votes
7answers
4k 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. ...
7
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 ...
11
votes
4answers
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 ...
54
votes
33answers
8k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. Motivation I am aware about a few such cases and I think it will be useful to gather ...
32
votes
2answers
5k 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 ...
3
votes
1answer
413 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 ...
13
votes
2answers
859 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
206 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 ...
9
votes
2answers
719 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 ...
26
votes
14answers
2k 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 ...
19
votes
6answers
3k 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
6k 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
814 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 ...
16
votes
3answers
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 ...
29
votes
7answers
5k 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
434 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 ...
65
votes
11answers
12k 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
689 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 ...