**9**

votes

**6**answers

1k views

### Is the dual notion of a presheaf useful?

It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are ...

**0**

votes

**3**answers

1k views

### Intuitions/connections/examples for “eigen-*”

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...

**37**

votes

**8**answers

10k views

### Modern algebraic geometry vs. classical algebraic geometry

Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...

**11**

votes

**0**answers

1k views

### Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...

**23**

votes

**8**answers

5k views

### Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum
The Ito integral has due to the unbounded total variation ...

**9**

votes

**6**answers

1k views

### Defining variable, symbol, indeterminate and parameter

Are there precise definitions for what a variable, a symbol, a name, an indeterminate, a meta-variable, and a parameter are?
In informal mathematics, they are used in a variety of ways, and often in ...

**36**

votes

**5**answers

3k views

### What's wrong with the surreals?

Of all the constructions of the reals, the construction of the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...

**22**

votes

**8**answers

2k views

### Intuitive and/or philosophical explanation for set theory paradoxes

Every student of set theory knows that the early axiomatization of the theory
had to deal with spectacular paradoxes such as Russel's, Burali-Forti's etc.
This is why the (self-contradictory) ...

**5**

votes

**4**answers

2k views

### Geometric interpretation of the fundamental groupoid

Motivation
The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy ...

**18**

votes

**8**answers

4k views

### To what extent is it true that “number theory = mathematics”? [closed]

In a thought-provoking answer to this MO question, Kevin Buzzard
and several commentators have described a multitude of ways in which
number theory is related to other parts of mathematics. It seems ...

**84**

votes

**25**answers

33k views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

**11**

votes

**1**answer

971 views

### A Sketch of “Esquisse d'un Programme”

I'm refering, of course, to Grothendieck's ambitious program available fully here: http://www.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf
The text is, as described in its title, a ...

**12**

votes

**2**answers

489 views

### What do you use categorical glueing/sconing/Freyd covers for?

In the theory of programming languages and structural proof theory, one of the handiest techniques we have available is a method called "logical relations", in which you can prove properties of ...

**78**

votes

**11**answers

8k views

### Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...

**8**

votes

**0**answers

444 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 ...

**34**

votes

**13**answers

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

**1**answer

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: ...

**17**

votes

**3**answers

829 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

**1**answer

2k 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

**5**answers

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

**5**answers

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

**7**answers

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

**6**answers

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

**4**answers

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 ...

**26**

votes

**5**answers

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

**5**answers

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

**3**answers

973 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

**12**answers

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

**12**answers

3k 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, ...

**14**

votes

**2**answers

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

**2**answers

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

**7**answers

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

**5**answers

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

**0**answers

280 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 ...

**85**

votes

**41**answers

16k 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

**1**answer

382 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

**2**answers

552 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

**3**answers

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 ...

**33**

votes

**6**answers

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

**3**answers

486 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

**2**answers

811 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):
...

**119**

votes

**69**answers

21k 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

**11**answers

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

**7**answers

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.
...

**8**

votes

**5**answers

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

**5**answers

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 ...

**59**

votes

**34**answers

9k 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 ...

**36**

votes

**2**answers

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

**1**answer

433 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

**2**answers

875 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 ...