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## A interesting proof on there exists irrational numbers a and b such that a^b is rational

I found a question on mathoverflow, it asks “is there  exists irrational numbers $a$ and $b$ such that $a^b$ is rational”.

Obviously, if you let $a=e$, $b=\log 3$, then $a^b=e^{\log 3}=3$ is rational.

But someone gave a very interesting proof.

You don’t have to know whether $\sqrt{2}^{\sqrt{2}}$ is rational or not, you can just get the answer. That is a clever trick.

## Proof without words

I found a quite interesting site http://memtropy.com/proofs-without-words/, if you can understand what it says, click it and see more.

## Advice to a Young Mathematician

• Always ask yourself questions. Atiyah says that one of the secrets of his success is to always be curious.
• Don’t get disheartened in your early years. Most mathematicians have a slow start. It’s important to keep on pushing and eventually a breakthrough will come. Indeed, Michael mentioned that when he was young he had a slow year, and so he sought advice from a great mathematician of the day. It turns out that this great mathematician had also had a disheartening year when he had started!
• Collaboration is important. There are benefits, mathematical and personal, when one collaborates in research with another. On the technical level, you get a second opinion and a fresh set of ideas to add to your own. On the personal level, you can make good friends while you work.
• Manage how much you get sidetracked. When you’re working on a big problem, you can often get sidetracked on a smaller problem. It’s important to realise that 9 out of 10 such sidetracks will lead nowhere, so it’s a good practice to know when to abandon your deviations. Of course, this management is tricky, as 1 out of 10 times you might just be on to something!
• Attending a bad lecture can be useful. There are many times when you go to a talk where the presenter will state a beautiful theorem. However, they make a complete mess of things when they go to the proof. If it looks like the talk is heading in this direction, have a go at understanding intuitively the ideas behind the proof. That is, try and nut out the proof yourself on a piece of paper, rather than continue to watch the presenter struggle.
• Keep a collection of grounding examples in your area. If you are working your way through a very theoretical area, it’s important to have at least one example of every theorem you come across. This helps to keep things both intuitive and connected, and makes for an easy catch-up if you ever have a few weeks away from work.
• The most important thing for your PhD is to have a good supervisor. In particular, try to get supervision by a world expert. Second rate mathematicians won’t be able to give you clear answers to high calibre questions. By having a sharp supervisor, you get a front row seat to the thinking that happens on the top floor of maths.
• Make the introduction to your papers readable to any mathematician. As it stands, about 99% of papers in mathematics are unreadable to the non-specialist. You should aim to make the introduction accessible to all mathematicians.
• A badly written paper is ignored. Writing a good paper is important. You should never rush it. A good practice is to start by first laying out the logical structure of the paper, before having a go at writing. Once you’re done, put the paper to the side for a few weeks before giving it a look over with a fresh set of eyes. Don’t be afraid to rewrite the entire paper, especially if it would improve it.
• Use rough ideas in your papers. Before you launch into your formal proof, give the reader a rough outline as to what is about to happen. You’d do this in a talk, right?

## William Thurston

For the most recently, I need some info about topology, here I collect  some introductions to William Thurston.

Bill Thurston is a topologist, though his work impinges on many other areas of mathematics. He has discovered unexpected links between topology, hyperbolic geometry, and complex analysis.

Highlights of his career include his classification of foliations of codimension greater than one, his classification of surface automorphisms, his hyperbolization theorem in three-dimensional topology, and the theories of automatic groups and confoliations. Thurston has also made fundamental contributions to the theory of symplectic and contact manifolds, dynamics of surface diffeomorphisms, and the combinatorics of rational maps.

His current research includes random 3-manifolds and relations of knot theory to computational complexity. His main interest remains his geometrization conjecture, a far-reaching proposed generalization of his hyperpolization theorem.

2. Wikipedia: http://en.wikipedia.org/wiki/William_Thurston
3. terrence tao: http://terrytao.wordpress.com/2012/08/22/bill-thurston/
5. The Mystery of 3-Manifolds: http://www.youtube.com/watch?v=4jdmkUQDWtQ

## Stein’s Functional Analysis photocopy released

After a long time waiting, Elias Stein’s Functional Analysis released the photocopy version in China this week. Luckily I registered the pre-selling notification at China-pub,  so I paid for it as soon as I received the SMS.

Functional Analysis:Introduction to Further Topics in Analysis (Princeton Lectures In Analysis) by Elias M.Stein  and Rami Shakarchi.

I paste the intro at china-pub here. http://product.china-pub.com/3020533

《泛函分析》是Stein的“Princeton Lectures In Analysis”四卷集中的最后一卷，这一个系列的教科书旨在全面剖析分析的核心，从泛函分析的基础开始，讲述巴纳赫空间、lp空间和分布理论，强调了 它们在调和分析中的核心地位。接着应用Baire范畴定理详解了一些重点，包括Besicovitch集合的存在性；本书的第二部分引导读者进入概率论和 布朗运动等分析的其他核心话题，以Dirichlet问题作为结束；最后几章讲述了多复变量和傅里叶分析中的振荡积分，并简述了在非线性色散方程中的计数 网格点问题中的应用。作者通篇紧紧围绕这个理论诸领域的核心思想，使得本课题的各个有机部分更加紧凑，层次分明，清晰易懂。

Low Dimensional Topology

Recent Progress and Open Problems

Gowers's Weblog

Mathematics related discussions

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Wir müssen wissen, wir werden wissen

The polymath blog

Massively collaborative mathematical projects

Chris Aldrich

Musings of a Modern Day Cyberneticist

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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao