The Poincare Conjecture by Donal O’Shea is very enjoyable.
Because it covers sophisticated mathematical material in an understandable manner, and also because it spends time on the various people (such as Gauss, Riemann, Klein, Hilbert) who have created such beautiful mathematics.
For instance, I hope to use this line (from Grigory Perelman) in some future talk😏:
I am not good at talking linearly, so I intend to sacrifice clarity for liveliness.
About Bernhard Riemann’s Probationary Lecture (with his advisor Carl Friedrich Gauss in the audience), O’Shea writes:
The speech completely recast three thousand years of geometry, and did so in plain German with almost no mathematical notation….Riemann first distinguished the notion of space from a geometry…argued that distance was even more fundamental than Euclid’s primitive notions and had to be specified independently….Curvature is not one number, but a whole collection, one for each pair of directions at a point…..and fundamentally altered the way geometry and topology would develop. Recognizably modern mathematics begins with Riemann.
On Felix Klein:
By 1880, Klein had the world by the tail. A virtuoso teacher and a masterful lecturer, he had attracted a large number of talented students to Leipzig.
About Henri Poincare, “the true intellectual heir to Riemann”:
:..was a “monster of mathematics.” He never seemed to take notes, had a virtually photographic memory. Poincare was even-tempered and good natured. “He was very balanced. In his judgements of others, he avoided all exaggeration,” his sister wrote.
On David Hilbert:
He was a minimalist and sought the shortest way into a subject…was utterly direct. Klein was an impresario, and Hilbert a stunningly capable, no-nonsense mathematician. Together, the two were formidable.
And of course (I have always wondered: without Einstein would we really care about all these people as much?🤷🏽♂️):
Einstein expressed gravity as the curvature of space-time. The Einstein equation describes how the [Riemann] curvature tensor varies in the presence of matter. Matter curves space-time. “To my great joy,” he wrote, “I completely succeeded in convincing Hilbert and Klein.” Einstein’s work became the rage. He would become a world celebrity in 1919 when a British scientific expedition….
Which brings me to our most recent paper, The Topology of Mutated Driver Pathways.
We open with a quote from Poincare:
Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.
Our key point is that while the search for cancer pathways is important, and we are ourselves also doing so in a parallel paper using novel formulations and algorithms, it is akin to adding to the heap of stones.
Our Conclusion section reads as follows:
The main goal of this paper was to suggest a study of the space of cancer pathways, using the natural language of algebraic topology. We hope that the consideration of the pathways collectively, that is, as a topological space, helps in revealing novel relations between these pathways. Indeed, we have seen that the homology in the case of AML indicates that the mutation data has a shape of a sphere. However, in the case of GBM, the final set of pathways has the topology of a double torus (or more technically a genus-2 surface). This intriguing observation raises the question of whether these facts translate into a new biological understanding about cancer. Studying the space of pathways of other cancers will be illuminating as well, if they also show similar structures, and we can classify cancers by the topology of their mutated driver pathways. This is an example of the new type of hypotheses one can now formulate about the data. Eventually, our goal (recalling Poincare) is to help build a house through revealing patterns among the stones.
We close our paper with a quote from The Emperor of All Maladies (by Siddhartha Mukherjee):
The third, and arguably most complex, new direction for cancer medicine is to integrate our understanding of aberrant genes and pathways to explain the behavior of cancer as a whole, thereby renewing the cycle of knowledge, discovery and therapeutic intervention.