# Tagged Questions

**-2**

votes

**2**answers

624 views

### Accidental, unplanned breakthroughs in Mathematics [closed]

In math/physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts. Moreover, there are some cases that originally having ...

**45**

votes

**18**answers

7k views

### Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of major theorems in mathematics whose proofs are ...

**0**

votes

**2**answers

176 views

### Intuition about covariant derivative/connections on real and complex manifolds

I was hoping to gain more intuition about the similarities and differences between the covariant derivative (of any connection, not necessarily the Levi Civita one if it exists) on real and complex ...

**33**

votes

**33**answers

4k views

### Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...

**15**

votes

**3**answers

965 views

### Research level applications of “row rank = column rank”?

No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...

**16**

votes

**3**answers

809 views

### Proofs that inspire and teach

I was just listening to the show "A Splendid Table" (which I'd recommend, if you're interested in food) in which they were discussing how to spot a good recipe: one which you can follow successfully ...

**1**

vote

**2**answers

1k views

### An undergraduate's guide to the foundational theorems of logic [closed]

How would you explain one of these theorems in the foundations of mathematics to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the ...

**6**

votes

**1**answer

1k views

### When are technical assumptions critical? [closed]

Apart from their technical statement and proof, a usual presentation of theorems is by leading up to them with a definite motivation or intuition, for example putting the results in the wider context ...

**18**

votes

**4**answers

1k views

### What information is contained in the Kazhdan-Lusztig polynomials?

The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations.
For example the character of a simple module over a Lie algebra with Weyl group $W$ ...

**38**

votes

**24**answers

6k views

### The concept of Duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.
...

**7**

votes

**1**answer

712 views

### What makes a theorem 'good'? [closed]

I have been pondering the issue of what makes a theorem noteworthy. There are many famous examples of 'outstanding' theorems, such as Roth's theorem in Diophantine approximation, Szemeredi's Theorem, ...

**43**

votes

**29**answers

9k views

### What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...

**58**

votes

**52**answers

18k views

### Theorems that are 'obvious' but hard to prove

There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would ...

**49**

votes

**13**answers

5k views

### What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...

**69**

votes

**16**answers

16k views

### What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...

**12**

votes

**4**answers

998 views

### How and how much do the notations and diagrams influence our understanding of mathematical concepts?

How and how much do the notations and diagrams influence
our understanding of mathematical
concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and ...

**5**

votes

**4**answers

866 views

### Hecke-algebras in your field of mathematics

(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important?
How would you define them and how do you think about them?
e.g. generators and relations, functions ...

**12**

votes

**16**answers

1k views

### Generalized notions of solutions in various areas of mathematics

In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are ...

**8**

votes

**4**answers

2k views

### Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...

**30**

votes

**9**answers

4k views

### Collection of equivalent forms of Riemann Hypothesis

This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...

**5**

votes

**2**answers

680 views

### Number of ways to construct mathematical objects

This question stems from this other one mentioning 7 ways of constructing smooth manifolds. I quote:
At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods ...

**22**

votes

**3**answers

2k views

### Online math history lectures

This question is somewhat similar to this: Best online mathematics videos?
I'm using the word "history" loosely here. What I'm looking for are those lectures that put various mathematical ...

**7**

votes

**2**answers

2k views

### Linear/Non-linear sigma model

This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model.
I gauge from some other interactions that it has considerable ...

**5**

votes

**3**answers

1k views

### Graphical representation of mathematical structures (in the spirit of unified modeling language)

In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are ...

**16**

votes

**7**answers

4k views

### What is Realistic Mathematics?

This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...

**64**

votes

**10**answers

7k 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 ...

**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 ...

**15**

votes

**12**answers

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, ...

**80**

votes

**37**answers

14k 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 ...

**100**

votes

**61**answers

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

**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 ...

**53**

votes

**33**answers

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 ...

**29**

votes

**21**answers

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 ...

**13**

votes

**27**answers

2k 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 ...

**113**

votes

**130**answers

26k 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 ...

**42**

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 ...