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Godfrey Harold Hardy (G.H. Hardy has a famous saying: "... Mathematics is more of a young man's game than any other art or science. "Here, how correct is his understanding of" young man "?

To be sure, proving a mathematical theorem requires a lot of creativity, the ability to rethink problems, and thinking in ways that no one else has thought of.

But it also requires a lot of experience and knowledge. After all, if you don't understand a problem, you can't prove it. Many unproven conjectures are based on a mountain of ideas that often take years to reach.

Based on Wikipedia, I have laid out a timeline of mathematical development from 1501 to 2015, tracing 250 major mathematical events: new proofs of theorems, important work releases, or the germination of core mathematical concepts.

The following are some of the incidents:

In 1540, at the age of 18, Lodovico Ferrari solved a quartic equation.

In 1799, Carl Friedrich Gauss, aged 22, proved the fundamental theorem of algebra (every polynomial equation has a solution in complex numbers).

In 1925, Werner Heisenberg, at the age of 24, and Jordan and Born established the matrix representation of quantum mechanics.

Terence Tao and Ben Green proved the Green-Tao theorem in 2004 at the age of 29.

In 1522, Adam Ries, aged 30, explained the use of Arabic numerals and their superiority over Roman numerals.

In 2003, Gregory Perelman, 37, proved the Poincare conjecture.

In 1994, Andrew Wiles, 41, proved Fermat's Last Theorem by proving part of Taniyama's conjecture.

In 1929, Emmy Noether, 47, introduced the first universal representation theory of groups and algebras.

In 2013, 58-year-old Zhang Yitang proved that there are infinitely many pairs of prime numbers with finite gaps.

In 1618, John Napier, 68, first mentioned the base e of natural logarithm in a book on logarithm.

The author charts the ages of great mathematicians

Summary of data:

Their average age is 37 years;

The median age was slightly lower at 35 years.

25% of mathematicians in the Mathematics Chronicle have made important mathematical achievements under the age of 30;

42% of mathematicians made important achievements between the ages of 30 and 39;

33% of mathematicians have made important achievements at age 40 or older.

The youngest of them was 18 years old, and in 1540 Lodovico Ferrari derived the general solution of the quartic equation.

In 1825 Adrien-Marie Legendre and Peter Gustav Lejeune Dirichlet proved Fermat's Last Theorem for n=5.

From 20 to 70, I created an interactive chart called "The Age of Great Mathematicians," which readers can click here or below for detailed information (use the slider to survey the achievements of mathematicians of all ages). Some of the highlights are worth noting:

Between the ages of 20 and 29 In 1832, Évariste Galois, a French mathematician, proposed general conditions for the solvability of algebraic equations at the age of 21, thus basically establishing group theory and Galois theory, and he was also the first to use the mathematical term "group" to denote a set of permutations. But tragically and romantically, he was killed in a duel shortly after he advanced these theories.

In 1913, Srinivasa Ramanujan, a 26-year-old Indian mathematician, wrote Hardy a letter containing a long list of unproved theorems, in which Ramanujan also begged Hardy to help him escape poverty. Of course, Ramanujan's discovery in the letter must have occurred before he was 26 years old.

In 2008, at the age of 31, Iranian mathematician Marian Mirzahani proved a long-standing conjecture: William Thurston proposed that earthquake map flows on Teichmüller space are ergodic systems. Six years later, she was awarded the Fields Medal for "outstanding contributions to the dynamics and geometry of Riemann surfaces and their modular spaces." Mirzahani died of breast cancer on July 14, 2017, aged 40.

In 1837, the 32-year-old German mathematician Johann Peter Gustav Lejeune Dirichlet developed analytic number theory in a treatise on the existence of primes in a given arithmetic series. This is Dirichlet's fourth job on the mathematical timeline, the first three taking place when he was 20, 27 and 32.

1915. Albert Einstein, a 36-year-old theoretical physicist, published his general theory of relativity, ten years after his special theory was published while he was still a clerk at the Swiss patent office.

Between the ages of 40 and 49 In 1993, after years of secretly studying Fermat's Last Theorem, Andrew Wiles, 40, announced that he had proved Fermat's Last Theorem. It is well known that there was an error in the proof during the review process, but the error was corrected the following year. Wiles expanded his work at the age of 46, completing all of Taniyama Shimura's conjectures.

In 1918, Godfrey Harold Hardy (G.H. Hardy and Srinivasa Ramanujan developed an asymptotic formula for the partition function. Perhaps his collaboration with the young creative genius Ramanujan is one reason he laments that mathematics is "a game for young people."

Over 50 years old In 2013, Chinese-American mathematician Zhang Yitang proved for the first time that there are infinite pairs of prime numbers with finite gaps, thus making a qualitative breakthrough in the number theory problem of twin prime conjecture. Zhang Yitang did not get a recommendation letter from his tutor after obtaining his doctorate in 1991. His academic road was bumpy. He lived by doing odd jobs for a while. He worked as an accountant for several years and worked in a fast food restaurant in the subway. (Source)

In 1722, the French mathematician Abraham de Moivre linked complex numbers with trigonometry, proposing Moivre's formula; at the age of 66, he introduced a normal distribution to approximate a binomial distribution.

There is a humorous anecdote about the elderly mathematician:

It is often said that Domov, who has always been interested in series, predicted that he would need to sleep fifteen minutes more each day than he had the day before, and that he would die when he had slept twenty-four hours, on November 27, 1754, which was the time of his death. (Source)

Data and methods Most of the data are obtained from mathematical timelines on Wikipedia, starting with "modern"(16th century). It would be nice to start with Archimedes and Hypatia in ancient Greece, but the historical record is not accurate enough, so here we start with "modern times."

I have also expanded the data provided on Wikipedia's Timeline with other famous results, such as the work of mathematical physicists such as Einstein, Bohr, and Heisenberg.

Surprisingly, some of the most recent big events in mathematics have also had some data collection problems. I wanted to include Hao Huang's proof of his sensitivity conjecture for Boolean functions in 2019, but I couldn't find his date of birth. Extrapolating from his educational experience, he may be in his thirties.

Similarly, I would like to include Joan Taylor, an amateur mathematician who, along with Joshua Socolar, discovered the Socolar-Taylor tile collage and solved the problem posed by Roger Penrose. But I couldn't find her birth date, but on her website she said she started thinking about the problem in 1990, 20 years before the results were published in 2011.

A final thought statistically, though most of the remarkable achievements are made by mathematicians between the ages of 20 and 40, the age range is still wide. As mathematicians age, they tend to prefer books and synopsis-type work, although there are still examples of breakthrough proof by older mathematicians.

Every mathematician's story is different. There are prodigies who die young, prolific polymath who cover all fields, and people who stick to a problem for 20 years.

For those of us who are amateurs in math, it may not be that we want to change the world, but that we enjoy learning about it, knowing that our brains are always thinking, and that we can continue to enjoy the pleasures of math.

Original link:

https://www.cantorsparadise.com/mathematical-genius-youth-vs-experience-d6dadac4f6eb

The translation content only represents the author's point of view, not the position of the Institute of Physics, CAS

This article comes from Weixin Official Accounts: Institute of Physics, Chinese Academy of Sciences (ID: cas-iop), author: Russell Lim, translation: Nuor, revision: xux

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