**16**

votes

**3**answers

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

**7**answers

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

**2**answers

436 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

**1**answer

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 ...

**67**

votes

**11**answers

13k 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

**3**answers

708 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 ...

**33**

votes

**5**answers

5k 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 ...

**19**

votes

**4**answers

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 ...

**51**

votes

**16**answers

8k 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

**3**answers

385 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 ...

**118**

votes

**130**answers

28k 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

**3**answers

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 ...

**44**

votes

**11**answers

5k 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

**3**answers

334 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 ...

**7**

votes

**2**answers

470 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? ...

**30**

votes

**6**answers

2k 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

**1**answer

199 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, ...

**13**

votes

**2**answers

667 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 ...

**7**

votes

**1**answer

891 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

**5**answers

718 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

**7**answers

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 ...

**44**

votes

**50**answers

10k views

### Describe a topic in one sentence. [closed]

When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good ...

**14**

votes

**3**answers

2k views

### K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

**29**

votes

**15**answers

2k views

### What are examples of good toy models in mathematics?

This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...

**5**

votes

**7**answers

530 views

### Given a sequence defined on the positive integers, how should it be extended to be defined at zero?

This question is inspired by a lecture Bjorn Poonen gave at MIT last year. I have ideas of my own, but I'm interested in what other people have to say, so I'll make this community wiki and post my ...

**35**

votes

**8**answers

5k views

### Sheaf cohomology and injective resolutions

In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...

**39**

votes

**21**answers

6k views

### What's a groupoid? What's a good example of a groupoid?

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?

**43**

votes

**12**answers

5k views

### Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So ...

**10**

votes

**6**answers

1k views

### Why the search for ever larger primes?

I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the ...