Academician Zhang Jiping Talks about Group Theory

On the morning of September 18, 2020, Professor Zhang Jiping from Peking University, an academician of Chinese Academy of Sciences was invited to the Peiyang Maths Forum and gave the first math class to the freshmen from the School of Mathematics on group theory.

“The greatest joy of teach-student interaction lies in the mutual understanding and learning of each other.” Professor Zhang’s opening remarks immediately struck a chord with the audience. He further elaborated that the temperament of a mathematician often complemented the characteristics of his research direction. He hoped that by discussing group theory, students could learn something deeper and more profound than the theory itself.

Zhang Jiping started his lecture talking about the origin of the Chinese character "群" (indicating group) and its related ancient writings. He said that many people thought of mathematics research as a solitary meditation. However, mathematics today cannot be defined by old concepts. “It is more of a group work that advocates unity and teamwork as are required by the group theory and are perceivable in the personal characters of many group theorists,” said Prof. Zhang.

As for the strict definition of “group” in mathematics, Prof. Zhang wrote down the equations on blackboard and explained them in detail. A group is a set with an operation, which satisfies the associative law, and has a unit element, each of which has an inverse element. The origin of “group” concept is mainly derived from the famous genius mathematician Galois who considered solving questions with the root of an equation in unary higher order algebra. Galois creatively related questions to the groups consisting of all permutations of the root set of the equation.

If G is a finite group and H Ì G is a subgroup, then according to Lagrange's theorem, we have |H|||G|. So naturally, the notion of a subgroup is associated with the factors of integers and the quotients of integers induce the so-called quotient groups.

Analogous to prime numbers, there are so-called simple groups. Prof. Zhang Jiping emphasized that finding all prime numbers was a very difficult and unsolved problem. However, group theorists have obtained the famous classification theorem of finite simple groups through cooperation. Zhang said that the result was completed by more than 500 group theorists in over 2000 papers between 1954 and 1983. “The second simplified version of the theorem is currently progressing well and is expected to be completed in 2023. This is an epoch-making result in honor of Feit and Thompson 60 years later, that is, any odd-order group is a solvable group.”

This conclusion was published in the Pacific Journal of Mathematics in 1963, with a total of 255 pages, and was reported by The Times as the greatest mathematical theorem in contemporary times. Later, many people and even mathematics master Atiyah tried to simplify the theorem but failed. Although the simplification theorem failed, Atiyah's article left many important mathematical ideas to the younger generation. Professor Zhang told students that the ideas, the simplicity and beauty of the truly great mathematical theorems were worthy of lifelong efforts to understand, appreciate, and gain inspiration.

Prof. Zhang Jiping also put forward Thompson's conjecture, sowing the seeds for mathematical savants. Then he shared his views on the Monster Group and offered suggestions on learning abstract algebra. He emphasized that the development of mathematics nowadays was becoming more and more comprehensive. Although Algebra, analysis, and geometry are of unique characteristics，students should not focus on a certain subject with great passion. Mathematics should be regarded as a whole at the university level. For example, finding the roots of equations is an algebraic problem, solving the roots of a bunch of equations is a geometric problem, and approximating arbitrary functions with polynomials is analytical problem. Zhang drew an analogy between the shaolin Monks' martial arts and the basic mathematical skills training to illustrate the importance of diligence. He said that students must change their learning attitudes and stand no excuse forslacking off in basic skill training. “As we know, there is no natural-born genius,” Prof. Zhang noted.

“For a mathematician, nothing can be more cheerful than holding a piece of chalk and turning a flash of inspiration into a jumble of mathematical symbols at one’s fingertips. Each and every one of you here is about to embark on a journey of mathematics research.”

By the School of Mathematics

Editor: Eva Yin