Vaughan Jones, The Geat Mathematician, Has Died

The contemporary mathematics master Vaughan Jones (Jones) died on September 6, 2020 at the age of 67, from the complications caused by a heavy ear infection.

In 1984, when New Zealand mathematician Jones specialized in operator algebra, he made a historic discovery in a completely unexpected direction - he found a new kink invariant: Jones polynomial. It can be said that the emergence of Jones polynomials immediately gave new connotations to the kink theory, a branch that only appeared in this century, and led to the solution of many classic knot theory problems, which aroused the attention of mathematicians to low-dimensional topology and inspired later the emergence of research methods and technologies.

Jones made the kink theory one of the focuses of attention in the mathematics circle at that time. Numerous mathematicians have developed research interest, which has led to a series of important advances and opened up channels of contact with other branches of mathematics and physics.

Jones polynomials and kink theory have now been widely used in low-dimensional topology, statistical mechanics, quantum field theory, string theory, quantum group, ergodic theory, representation theory, etc.

He himself was awarded the Fields Medal in 1990, one of the two highest honors in mathematics. (If it is not an accidental death, he will definitely get another highest award in the future - the Abel Prize, which is equivalent to the lifetime achievement award). 

Brief Talk on Kink Theory

For people who loves fishing, you should be familiar with concepts such as sailor knot and double bar knot. They are essentially kink and the subject of chain-link theory. It's just that for convenience, mathematicians generally require that the knotted rope be a closed loop connected end to end.

In the time of Lord Kelvin, physicists generally believed that there was a space medium called ether—where light travels through the universe from.

Kelvin believes that the motion of the ether is vortex-like, and the axis of the vortex can be knotted. Different knots correspond to different chemical elements. Therefore, scientists at the time were encouraged to figure out the structure of various knots. This gave birth to the discipline of classifying kinks.

Mathematicians want to know whether there are really different kinks with different appearances. In other words, whether a knot can be transformed into another knot through the deformation rules allowed by nature. According to different specific rules, we call it homeomorphism, isomorphism or homologous.

Here is Jones' achievement: He established an important and profound relationship between objects that were originally topological and algebraic objects—polynomials. That is, if you know the polynomial of a knot, you can understand the topological properties of the knot through algebraic operations and analyzing the properties of the polynomial.

What's more ingenious is that the theory of Jones polynomials can essentially be expressed in very simple mathematics!