Riemann

When learning calculus for the first time, I think you must know Riemann before Newton. In addition to the Riemann integral that everyone knows, the Riemann \(\zeta\) function, Riemann-Hilbert problem and Cauchy-Riemann equation will also occur in advanced mathematics. Then what kind of person is Riemann? I’ll introduce him to you in this article…

 


Info of Riemann

Georg Friedrich Bernhard Riemann
Born:1826-09-17
Died:1866-07-20
Nationality:German

Hilbert’s portrait
(Image Source:Wikipedia

Early life

Riemann’s father, Friedrich Bernhard Riemann, is a Lutheran minister. Friedrich acted as teacher to his children and plays a important role in Riemann’s early education. It was not until Riemann was 14 or 15 years old that he moved to Hanover to live with his grandmother, and entered high school. At that time, he met a great teacher who taught him to read the classic of science.Riemann showed outstanding skills on math when he was a child, and he also showed strong interests in math. Although he entered the theology faculty due to his father’s encouragement at the age of 20, he still listened to several math lectures (at that time, Gauss was also at the University of Göttingen). He realized that he still loved mathematics. Since Riemann was very closed to his family and he would never changed courses without his father’s permission. Fortunately, he got approval from his father and then took courses in mathematics from Gauss.

Go to the Berlin University

At the University of Göttingen, Riemann found that most of the professors only taught basic concepts because of the law level of mathematics. Everyone was in awe of the professors’ authority. But in such an environment, it’s difficult to discuss with the professor. Riemann heard that the atmosphere of Berlin University was relatively free and open(democratic trend was influencing). Jacobi and Dirichlet, the two great mathematicians, even taught students about the topics they were still thinking about. This is attractive to Riemann to transfer to Berlin. He stayed in Berlin for two years and returned to Göttingen in 1849.

Those who influenced Riemann

The instructor of Riemann’s doctoral thesis was Gauss, and the mathematics lecture he was inspired by was also taught by Gauss. During this period, not only Gauss had a significant influence on Riemann, but Jacobi and Dirichlet discussed mathematics very eagerly with Riemann. The result of Dirichlet’s research also had a profound impact on Riemann’s thesis. The well-known physicist, Weber returned to Göttingen as the chair of physics during that time, and Riemann was his assistant for 18 months. Through Weber and Listing ( Listing had been appointed as a proffessor in 1849), Riemann gained a strong background and important ideas in topology.

Some Riemann’s “stuff”

I’ll talk about some related terms of Riemann’s achievement. The people who has taken advanced mathematics courses would be familiar, I think.

•Riemann integral

Let me start with a term that most people have heard of XD Those who have take calculus courses must be familiar with the concept of “integrable”. Both Cauchy and Dirichlet had discussed integrability before, and Riemann proposed a different definition of it, which is the Riemann integral that we are now familiar with. Until now, Riemann integral was extended to Lebesgue integral, which has played an important role in many fields of mathematics.

•Riemann zeta function

Suppose there is a complex number \(s\) whose real part is greater than \(1\) (\(Re(s)>1\)), we define the Riemann \(\zeta\) function as:

\(\displaystyle{ \zeta (s) = \sum_{n=1}^{\infty}{\frac{1}{n^{s}}}}\) 

The Riemann \(\zeta\) function is considered to play a pivotal role in number theory (The super pure mathematics XD) and also has applications in physics, probability theory, and applied statistics. You can see related information on Wikipedia – Riemann zeta function

•Riemann surface

According to Wikipedia’s explanation:In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different……

Hey, are you still awake XD Never mind, I don’t understand it, either. Others such as Riemann curvature tensor, Riemann sphere, Riemann-XXX theorem, etc., also developed Riemann geometry, and provided Einstein a mathematical basis to general relativity. All of these can be seen that Riemann made great contribution to modern mathematics. There are many concepts from Riemann have not been completed yet, they also expanded new fields of mathematics.

The Holy Grail of Mathematics – Riemann hypothesis

Have you seen a report on “Riemann hypothesis” a few years ago? Based on the definition of Riemann \(\zeta\) function above,

\(\displaystyle{ \zeta(s) = \frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+…}\)

the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part \(\frac{1}{2}\)

This hypothesis has confused many mathematicians for many years. It is also one of the Millennium Prize Problems. A hundred years ago, none of mathematician can come up with a rigorous proof. There are two jokes about Riemann hypothesis. One of them is: If Riemann is resurrected 500 years later, the first thing he does when he gets up is to ask, “Has the Riemann hypothesis been proved?” And the other of them is: If the devil and a mathematician trade souls for a proof of a proposition, then the mathematician will choose Riemann’s hypothesis.

The reason why I mentioned the report at the beginning was because a mathematician named Michael Atiyah claimed that he had proved the Riemann hypothesis. This caused an uproar. Of course, it’s still called Riemann’s “hypothesis” so it means that Atiyah didn’t prove it. Although the old Atiyah has become a stubborn old man in mathematics in some respects, anyway, this old man has won the Fields Medal. It’s not enough for him to affect his reputation.

Die young

Compared with other mathematicians, the information about Riemann himself is less. He got married at the age of 36. He lived a healthy life very early due to his lung disease, but it get worse at the age of 40 and he died in Italy. Although Riemann died young, his foresight made the mathematicians expand many new fields and still have impact to this day.

Add some ingredient

Bernhard Riemann
Mactutor-Georg Friedrich Bernhard Riemann

Seasoned with Chinese

◊ Topology – 拓樸學
◊ Riemann integral – 黎曼積分
◊ Riemann surface – 黎曼曲面
◊ Riemann hypothesis – 黎曼猜想
◊ general ralativity、general theory of relativity – 廣義相對論
◊ Millennium Prize Problems – 千禧年大獎難題

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