Calculus without limits

# Wenjun Wu, 1919-2017

 According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year. I encountered some mathematics developed by Wu when working on Wu characteristic. See the Slides and the paper on multi-linear valuations. There is an other paper on this in preparation, especially dealing with the cohomology belonging to Wu characteristics. Just as a reminder, the Wu characteristic of a simplicial complex G is the sum $\sum_{x \sim y} \omega(x) \omega(y)$, where $\omega(x)=(-1)^{{\rm dim}(x)}$ and $x \sim y$ means that the sets $x$ and $y$ intersect. The Euler characteristic $\sum_{x} \omega(x)$ is the linear version. There are higher order versions like the cubic $\sum_{x,y,z} \omega(x) \omega(y) \omega(z)$ where the sum is over all triplets of sets which simultaneously intersect. Each of these numbers are combinatorial invariants meaning that they don’t change when taking a Barycentric refinement of the complex. Each of these invariants also has an associated calculus and cohomology, Gauss-Bonnet, and Brouwer-Lefschetz fixed point theorems. They naturally extend the calculus we know. The school calculus we teach and which has been built by Betti and Poincare finalized by de Rham and Cartan is the calculus which belongs to Euler characteristic.

 But first about Wenjun Wu. His life must have been very interesting. One can get a glimpse on this in the introduction of the “Selecta” which Wu wrote in 2007: the research in the first stage from 1947-1965 was in the field of pure mathematics, in particular in algebraic topology and algebraic geometry. In the second stage, influenced by the culttural revolution, where Wu was sent to a computer manufacture company to learn and work with laborers, he began using computers to study mathematics, especially the method to prove geometric theorems computer assisted. He worked so on computational algebraic geometry, graph theory or mechanics. One paper from 1987 does a mechanical derivation of the Newton equations from Kepler’s law. An other paper from 1991 deals with computer assisted proofs in differential geometry. Of course, much of this work is nowadays extended and absorbed by powerful computer algebra systems which are specialized in various fields like algebraic geometry, algebra, number theory or group theory.

Here are two pages from the back of his “Selecta” which show Wu as a student in France, with Family, with students and with Shiin-Shen Chern.

And here are more pictures from the back of the “Selecta”: