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My maths teacher always keeps telling me about this 'cutting edge maths' that is going on in the world, amazing maths research, etc.

A lot of the google searches I've done for 'Cutting Edge Mathematics' hasn't returned much useful information, so I've taken to mathematics stack exchange.

What kind of cutting edge maths research is going on in the world at the moment?

Don't we already know everything there is to know about numbers?

Considering maths is just numbers, I thought that we could apply maths to other subjects like physics and that was why maths was useful. However... I never realised that there could be more to find out about the numbers themselves...

Featherball
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    "Maths is just numbers" aghhhh – angryavian Dec 13 '16 at 07:24
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    Cutting-edge math, so to say, is all but _devoid of_ numbers. – Ivan Neretin Dec 13 '16 at 07:47
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    To follow on angryavian's comment and to hopefully correct a misconception you have, mathematics is about a great deal more than *just numbers*. Have you ever tried to pack a suitcase and get everything to fit? That's math. Have you ever tried to untangle a string of christmas lights? That's math. Have you ever looked at the pattern of a snowflake up close? That's math. More than just numbers, mathematics is our way of describing the world around us as well as purely hypothetical scenarios with a great deal of precision allowing us to recognize and describe patterns and properties. – JMoravitz Dec 13 '16 at 07:47
  • Further "*don't we already know everything there is to know about numbers*"... the (*not very*) surprising answer to that is *no*! Even in the field of number theory, there are several open questions. For example, we don't even have a proof yet whether or not $\pi\cdot e$ is an irrational number. – JMoravitz Dec 13 '16 at 07:50
  • Haha, well, I suppose maths is more than just numbers (and I knew that all along). I was just being a little generalistic when I wrote this answer. @JMoravitz – Featherball Dec 13 '16 at 15:20

3 Answers3

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This mathoverflow question contains a bunch of recent applications of math. Most of the topics on it are only learned in very math-intensive graduate programs.

Asinomás
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It is an interesting question. Mathematics meets the definition of a language. Although simpler in a way, english can act as a metaphor with the 26 letter latin alphabet representing 'numbers'. There certainly is a development of the alphabet itself going on, various accents and refinements, but it's better to look at words. And much better still to consider sentences.

Equations are like your sentences, they are a real unit of meaning. It's always better to translate sentences than individual words.

To come back to your question, the cutting edge is often in the refinement and well considered combination of equations, 'paragraphs' in this metaphor. Where the metaphor differs is that the english language allows for endless break down of the rules, such that hundreds of paragraphs can be written quickly, whereas a single mathematical paragraph may take a long time to be proven true.

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What is the best way to move a couch through a narrow turn in a hallway?

How many dots must be drawn so that no matter how we labeled them in one of two ways, there is always an interconnected group of $5$ of them?

Are there infinitely many pairs of prime numbers that differ by two?

Does a polynomial exist having $\pi+e$ as a root?

Do fluid flow equations always have nice solutions?

There is a ton more. Look for the Millenium Prize problems, the Goldbach Conjecture, the Travelling Salesman Problem, the normality of $\pi$, etc.

It might be worth it to look up 'recent' solved problems. There is the famous Fermat's Last Theorem, for example.

Mathematics is not just about solving problems, but also about developing new ways to frame the problems. At every university and research institute in the world, mathematicians are hard at work on hundreds of thousands of new ideas and methods of attack to solve unsolved problems or elucidate new knowledge.

Your question is good question. My students ask me this every semester. I enjoy answering it.

The Count
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