# Areas of research in maths

1. Sep 15, 2008

### tgt

What are fields of research in maths that contain a large number of tricks? What are fields that contain the least number?

By not containing many tricks, I mean fields where each step can be deduced in a logical manner without huge jumps. Tricks will mean the opposite.

2. Sep 15, 2008

### cristo

Staff Emeritus
You ask the weirdest questions. I would say that all maths is logical, you just need to understand the steps. For example, by "huge jumps" I presume you mean the author of a certain piece of work you are reading has just missed out several steps.. this can happen in any field!

3. Sep 15, 2008

### tgt

I see your point but it is possible to distinguish a trick and something less so given the fact that mathematicians talk about something being a trick or not.

4. Sep 15, 2008

### HallsofIvy

So whether something is or is not a "trick" depends upon the individual mathematician.

My experience is that most mathematicians don't talk about "tricks" at all!

5. Sep 15, 2008

### CRGreathouse

Knuth did, but he might not count as a mathematician. He had some terminology here, something like 'tricks are things you use once, techniques are things you use repeatedly'. That's from the draft copy of one of his TAoCP series.

6. Sep 16, 2008

### tgt

Some mathematicians speculated that Grothendieck didn't want to prove the Weil conjectures because it contained a trick or something like that.

It seems to me that all maths competition problem contain tricks, no?

7. Sep 16, 2008

### mhill

perhaps he is referring a 'trick' in the sense that you can proof a math theorem in a small number of steps.

for example, if you consider the 'trick' $$\sum_{n= -\infty}^{\infty}e^{2i \pi n x} =\sum_{n= -\infty}^{\infty} \delta (x-n)$$

taking Mellin transform on both sides you can get an easy 'proof' of Riemann Functional equation.

another of my favourites is the expansion $$ex(-exp(x))= 1-exp(x)+exp(2x)-$$

used to proof 'Ramanujan master theorem' http://mathworld.wolfram.com/RamanujansMasterTheorem.html

i always dreamed about a similar kind of 'trick' to prove RH (Riemann Hypothesis)