New Math

[jordan]

I was just curious, since I have no idea what the most advanced math stuff is these days (I've never really understood anything higher than a calculus 1), I was wondering what the newest stuff to math is. What kinds of things are they still working on? What are some of the newest fields they've discover/invented? What's actualy left to work on? If I want to be a mathemetician, can I look forward to have anything to work on?

[matt grime]

Mathematics is never ending, don't worry.

 

Research Maths, for want of a better phrase, has approximately two extremes of style.

 

1. To answer a specific question. For instance, the following are open questions that don't look like being solved anytime soon:

 

Are there an infinite number of twin primes (that is prime numbers differing by 2, such as 3,5, or 11,13.

 

2. Can any even number be written as the sum of two primes?

 

3. Given n objects to be arranged in m ways satisfying p constraints is there any reasonably short way of finding the optimal arrangement?

 

4. Can you factorize numbers quickly?

 

5. What is the distribution of the primes?

 

These are reasonably easy to understand, since they do not involve anything very complicated. And that is precisely what makes them hard to solve. You know that Wiles proved Fermat's last theorem: there is no solution in integers to x^n+y^n=z^n for integer n greater than or equal to 3. Well, the proof has nothing to do with anything that you can think of. This leads us into style 2.

 

You take some set of rules, see what you can deduce, what happens if you add in more rules. Here you tend not to have any particular result in mind, you just play around. And there are a lot of unexplored things to play around with, and sometimes by doing this you get some odd results.

 

Most of us operate somewhere between the two extremes. That is we have some ideas of explicit problems in the background that motivate us to define abstract new systems to study to hope to get an idea how to solve the problems.

 

Wiles classified some objects called modular forms, these are functions of a complex variable satisfying some rules. He also showed that the data that classifies these things can be used to classify semi-stable elliptic curves, and this classification implies that there are no solutions to Fermat's equation in a very odd way. He was building on 50 years of collective work: that's how fast pure maths develops.

 

The problem is that in order to explain to you the pure parts I'd have to introduce so much more material. Applied is a little easier.

 

Essentially the equations that govern physical systems are too complicated to solve. So we make approximations, eg linearization. But we are continually trying to improve these approximations and find better ways of allowing more vairiables to be taken into consideration.

[jordan]

Thanks for the great reply. However, I noticed that the majority of your examples deal with prime numbers. Is that just coincidence or what the majority of the problems are today? And would a proof (or disproof) of the Reimann Hypothesis wrap that up for the most part?

 

4. Can you factorize numbers quickly?

I didn't understand this one.

[matt grime]

I used prime numbers since I thought there was a chance you had heard of them.

 

I can give you a list of 1001 open problems without any stretching of the imagination, however, I doubt that more than2 of them would mean anything to you right now (There is a derived equivalence between the princpal blocks of a group and the normalized of its sylow-p subgroup if the sylow group is abelian? there is a Krull-shcmidt type theorem for relatively stable categories? the constant coefficient of some L functions is non-zero iff there are infinitely many rational points....)

 

the factorize thing states that: given any integer, there is a guaranteed way to find its factors, sadly this takes more than a reasonable number of operations (universe would end before you'd factored any big numbers such as those in RSA encrpytion). So, can you find some tricks and short cuts that'd do it in a reasonable length of time.

[mezarashi]

I think you're question was generally, what's beyond Calculus. So let me try to fill you in some from my endeavors in Engineering. I know this isn't all the mathematics out there, but "researched mathematics" is beyond my scope.

 

Generally I can say that what is classified now as "advanced mathematics" is searching for general solutions to existing problems. For example, we still have much difficulty solving differential equations. It is still basically impossible for us to analytically solve alot of equations, which we end up using iterative/numerical methods like guessing the answer and checking and reguessing etc (now that we have the power of computers with us hehe ^^). Anyway, for the interesting theories you can explore beyond calculus. They are not necessarily ground-breaking new research, but they are "advanced":

 

Laplacian Mathematics, and the Laplace Transform

An interesting mathematics. It transforms our every calculus into a different domain, known as the Laplace domain or the S-domain, and all of a sudden, all the complex calculus jargon because algebraiac equatinos. Pretty neat eh.

 

Fourier Series/Transform

It's really hard to mathematically quantize certain signals, like human voices. If you've ever observed one through an oscilloscope. It looks like some crazy wave, but if you do a fourier series/transform on it, then it becomes pretty simple. It turns into a series of clean sine and cosine functions. Also pretty cool.

 

Differential Methods

Of course, as much as we still have difficulty solving them, there has been some proven methods to do simple things, like solving standard form differential equations. Ever heard something of the Bernoulli equation or homogeneous differentials? Yup, there are interesting methods that mathematicians have come up in the past to solve scary looking things to a calculus newbie.

 

Vector State-Space and Linear Algebra

Ever heard of eigenvectors and eigenspace, or about diagonalization? It's about stability analysis. Did you know that every set of differential equations has a characteristic solution? Yup, we're not putting plain numbers into matrices this time, we're putting differential equations in em. It's pretty interesting, but has a relatively daunting proof. Who would've known that matrix theory you studied in highschool has much more depth to it than you think. How does this apply in engineering? Given any physical system, a sattelite in orbit for example, there is only one particular axis in which it will safe to rotate without disturbing its equilibrium. Thats where this field of mathematics comes in.

 

Complex Analysis

So the imaginary numbers were mind-breaking in highschool, but this time you have much more to it. You're not analyzing complex number, but complex function, complex geometry, and complex planes. Gladly brought to you by Cauchy, Mr. Euler, and friends. I'm sure some of these names are familiar to you.

 

Advanced Statistics

You've probably heard of standard statistical models like the bell-shaped curve if you've done an introductory statistics class in highschool, but trust me, there are a billion more statistical models. The maxwell distribution used to govern gas molecule motion (like that in an ideal gas used in Thermodynamics), and Einstein came up with his own distribution with the help of Bose, known as Einstein-bose statistics used to govern his own physics governing particles at low temperatures (if I remember right). Guess what, there are many others being researched to accurately model our world, and methods in which to use these models to predict what will happen next, in economics and engineering.

 

Welp, thats all for now that I can remember. I'm glad I don't have to do anymore of that now though. One word for math, HATE! XD

[matt grime]

Vector State-Space and,,

 

is backwards. Eigenvalues are not the study of stablity, stability is studied using eigenvalues. It's a shame people don't see more linear algebra, really. It's the introduction to so much interesting mathematics, both pure and applied (crystallography, game theory. spectral analysis..)