接下来我想用一分钟的时间向你们介绍我的演出 M³，在数学当中M³等于M乘以M再乘以M。第一个M指Music，音乐，我知道你们一定都喜欢它。第二个M，我确定你们不喜欢它，它代表数学—Mathematics， 但是在我演讲结束后你会喜欢上它。第三个M是一个德语单词，Malerei，翻译成英语就是绘画的意思。绘画用中文怎么说？ Huihua，这是我学会的第五个中文单词了。
我们看到这样的连续函数，我们有一条(函数的）曲线 ，f (0)比0大 ，f (1)比0小，中间有某个时刻的值是0，对应到桌子的情况，这个点就是桌子稳定的时刻，因为其他桌子腿都挨着地，现在4号桌腿的高度也是0，所以桌子不再摇晃。
Freude, schöner Götterfunken, Tochter aus Elysium.
Ni Hao! This is not my only Chinese word.But it is the only one I will use in this performance.
So I would like to explain you in one minute the idea of my program M³. So in mathematics "cubed" means M times M times M，and the first M is music. You all like that, I'm sure. The second M, I'm sure you don't like it. It's mathematics, but you will like it after the show.And the third M is a German word. It is called Malerei, and in English Painting.What is the Chinese word for painting? Huihua! It's my fifth Chinese word.
So if you expect that mathematics will explain the music or painting, you are wrong. The idea is that these three arts stand for themselves. So when you listen to the music, only concentrate on the music, forget the mathematics. That might be easy. If you listen to mathematics, forget the music. That's probably not so easy. And then there will be paintings, the paintings are combined with the mathematics. So there you are allowed to think at the mathematics. The paintings will actually be cartoons; you will see them.
So concentrate on the music, concentrate on the mathematics, and then the paintings come,
concentrate on both paintings and mathematics.
I promised to say one minute, I still have ten seconds. I have here a wonderful watch, but I don't use this 10secondsI play instead more music. I play a Sonata by AntonioVivaldi, the Italian composer.He's famous for one piece Four Seasons. I don't play this. I play a Sonata for Cello, uh, the Sonata number three, two movements.
So for me, that was the hard part, the mathematics part is easy for me.For you, the mathematics part might be a little bit harder, but I make it easy for you. If you think at Germans, you probably think thatwe sit in a beer garden, twenty-four hours a day and drink beer. We like that very much. Not twenty-four hours.
And on this painting, you see me in such a beer garden. And I ordered a beer, the beer comes.I’m very glad that I have the beer.And now the table is wobbling and I’m extremely angry. Then all beer is poured out. I hate that.
So I complain to the owner of the place.I throw the table and then I say, “What table do you have?”And he says, “No, no, no. The table is completely alright. The problem is the ground on which the table stand. The ground is not flat, and that's why the table wobbles.”Ok, I excuse him. It's not the table, it's the ground and the beer garden.
Of course, the ground cannot be completely flat. So we all have experienced such a situation and we know what to do. So I wonder what to do. And next video, a little boy comes and tells me,“Very simple. Namely, take a sheet of paper, put it under the leg and the table is okay.”But you know that all only okay for a while. Because of the paper, after a while, it's a little bit compressed and unstablity again.
Mathematicians hate unstability. They don't accept that, and you should also hate it. And I show you something in the future. When you have a wobbling table,you will always apply and you will have never a wobbling table again.You will think this is not possible. And the answer is very simple and can all the time applied.
Namely, we do a quarter turn of the table. We turn it up to a quarter, we turn it forward and backwards.And you see, now we watch at some moment.Now it's completely stable. Try that out at home or wherever you are. If you have a wobbling table, turn it up to a quarter and it will be completely fixed.
And this is not by chance. It's a mathematics talk. It's not a beer garden talk although It's hard to define what is a beer garden talk. It's a mathematics talk. And there is a mathematical proof that this works. And this proof we want to work out together.And for this, we first have to translate.And that's an important part. Also in mathematical research.The problem of reality into a mathematical problem, we have to make a so called mathematical model.
And that we do now together.And we look at the next video We enumerate the four legs, one, two, three, four,and we fix leg two, three, and four on the ground.So that one leg is above the ground.
Yeah, if it would not wobble, then all four would be on the ground.So that wobble means you see leg one.is above the ground.And that's why it wobbles, and now comes an important step.
We should see the next slide.So you see now the points on the ground where the four legs are, leg two, three, and four are fixed on the ground.And leg one is above the ground.If I push back leg one down, then leg three is above the ground.So it wobbles.
And now please next slide.So we keep the height of the leg one, we measure that.And that's a certain number, say, one centimeter.And this number is positive because of it’s one centimeter above the ground.
And now we turn the table.And while we turn it we keep leg two, three, and four on the floor.two, three, and four on the floor, and we turn it and keep these three on the floor.What will happen with leg one? For a while, it will be above the floor.But if we are there now, we fix these three legs, before this was above the ground.If we make this below the ground, then this will be under the ground.
And you'll see that in the next slide.Now, the position of leg one after the quarter turn.That's. I admit this is something very difficult.And if you don't understand it, ask me after the lecture again.So this is now under the ground.So we turn in the beginning above the ground, we keep these three on the ground. Now these three are on the ground.This means this is under the ground.So that's very important information.And with this in mind, we can now make our mathematical model.
So see the next video here.We see that's a very nice thing.You see how leg one goes under the ground.Can you see can we show this video again? That would be nice to see this important step again.So now we translate into mathematics, you know, in mathematics, you know, from school,we draw certain curves and study them.
And we draw now such a curve.Namely, we measure the height of leg one at every time while we turn,and we turn in time. At the beginning of the time is zero, at the end the time is one.So at time zero, it's above.And at time one, it is below.And now we measure the height at any moment.And we get a curve. Here.
You see now I draw the curve,this measure, measure measure measure, at this time measure measure measure.It goes up again, goes down, and at the end of the time, it's below.I hope you'll see what I did.I take the table, measure the height of this while I turn.This gives such a curve.And now, we are very close to an interesting mathematical theorem.
So we look at the next slide.First we do it again.It's good.We see leg two, three, four, always zero on the ground, but leg one goes up and down.Ok, now comes the next slide with a mathematical theorem.In mathematics, we have theorems.And it's a very important theorem called Intermediate Value Theorem.And students learn it in the first year with all details and proofs.Students take very long to learn it.You learn it in five minutes from me.You are better than the students.
So we have such a continuous function.This is such a curve.You have a curve with f at times zero, greater than zero, f at time one smaller than zero.Then there is some time in the middle where it is zero.And that's in the case of the table, the point where the table is stable because of all others have height zero.And now also leg four has height zero, nothing wobles.
So this important theorem, you could say, I believe it.Now we prove it.And now we come to the proof, and we do the proof by hunting.So imagine you have a field and there is some animal in the field.You want to hunt it.What do you do?We are not in the U.S, we don’t shoot it.No, no, no. Shooting is not allowed.We half the field in two halves.And we look where the animal is.We forget the other half.We half that again.We look where the animal is.Half it, make a fence and so on.And the end, we have the animal.This is how we hunt.
And we hunt here by dividing our time into half.Look where the table height is at half.And then there are two possibilities.You see at time zero, it was positive.At time one it was negative Here at time one half, it is negative.Then we throw this half away, where it has the same sign there.They are both are negative.That should be thrown away because of we hunt, the zero on the left side, this zero we want to hunt.So next slide, we show again the same curve.Now you see the curve on the right.I have thrown away.So this is like our animal we want to hunt.Now, the zero is between zero and one half.
Ok.We do the same again, we half again, this time, the zero is on the right side, we throw the left side away and remain with this part and next slide, we do it again.And you see if you do it again and again and again,the interval will be smaller and smaller.And of course, we perhaps have to it infinitely many times.That's something you learn in mathematics.What infinity is, that's tricky,but for you, no problem.We will see the infinity, we get closer and closer.And in the end, we hunt the zero.
And of course, that this hunting works,in the theorem it is said that the curve has to be continuous.And continuous means the curve can be drawn by a pen without a jump.If you have jumps, then of course, you can go from plus to minus without hitting the zero.So continuous is roughly draw without jumps.Mathematically, the tricky definition is precisely that.What we do here, that you can hunt the zero.
So I'm very proud. I'm very proud because of I managed to solve the problem.You'll see the peoples carry me away, but I hate that because of I want to drink the beer first.So I drink the beer and I disappearand.
I hope you have learned two things.First of all, mathematicians can make jokes.Very important, very important thing.Secondly, mathematicians can solve real problems, like the wobbling table.Yeah, we are sometimes useful,certainly most of the time we work theoretical.The translation into mathematical problem is for us the beginning of thinking,
if the problem comes from physics, chemistry, whatever, or inner mathematics.Most time, I'm a theoretical mathematician.I sit at the table, and when somebody comes in, the person says he's not working at all.Because I just sit and people think I sleep.But I think, and to prove that I think is that,from time to time, I publish a paper in the famous journal.So that's the way mathematicians work.
And you should take at home, mathematicians make jokes.They can solve the real problem, like the wobbling table.And they can really make proofs, which is something normal people don't know any more.You have seen a proof.And so I thank you extremely much for your audience after so many hours of listening. I am deeply impressed by the Chinese audience.
So at the end there is, of course, music again.And I count on you.I want to perform some music together with you.I will play now two songs, which I hope most of you know, and you sing with me, please.
So this was an English song.The next song is a German song.I'm sorry, but there exists some not so bad German composers, as you know, and I play one of them, you sing with me.
Thank you.I have one minute and forty-one seconds left, and I use them to thank you for the most wonderful concert I played in my life, to play with such an educated Chinese crowd, who knows my German colleague Beethoven. This is very impressive for me.
Do you know the text of this?
Freude, schöner Götterfunken, Tochter aus Elysium.
very strange, but it is the song of joy and the song of freedomand I think in this world we need both, and peace. Thank you again.