He was nervous, active, competitive, critical, and very articulate he tended to dominate every meeting he attended. His memory was so phenomenal, he could correctly quote an enormous number of poems and limericks, the entire Bible, tables of mathematical functions, and almost anything he had learned about astronomy.

His students later remembered him as a man whose thinking was three times faster than just about anybody else’s. When Henry Norris Russell graduated from Princeton University, his work had been so brilliant that the faculty decided to create a new level of honors degree beyond “summa cum laude” for him. This investigation, and a similar independent study in 1911 by Danish astronomer Ejnar Hertzsprung, led to the extremely important discovery that the temperature and luminosity of stars are related ( Figure 18.13). In 1913, American astronomer Henry Norris Russell plotted the luminosities of stars against their spectral classes (a way of denoting their surface temperatures). Or you might have a very tall, skinny fashion model with great height but relatively small weight, who would be found near the upper right.Ī similar diagram has been found extremely useful for understanding the lives of stars. You occasionally see a short human who is very overweight and would thus be more to the bottom left of our diagram than the average sequence of people. And, of course, there are some dramatic exceptions. It’s not mathematically exact-there is a wide range of variation-but it’s not a bad overall rule. Typically, if we have bigger bones, we have more flesh to fill out our larger frame. This makes sense if you are familiar with the structure of human beings. Generally speaking, taller human beings weigh more, whereas shorter ones weigh less. We can conclude from this graph that human height and weight are related. Most points lie along a “main sequence” representing most people, but there are a few exceptions. The plot of the heights and weights of a representative group of human beings. Even when this does not happen, it is often possible to get useful information from a spectral sequence by various tricks.Figure 18.12 Height versus Weight. The easiest cases to deal with are those in which the spectral sequence eventually collapses, meaning that going out further in the sequence produces no new information. This information is usually contained in a rank three lattice of abelian groups or modules. Unfortunately, because of the large amount of information carried in spectral sequences, they are difficult to grasp. This is true even when many of the terms of the spectral sequence are incalculable. While their theoretical importance has decreased since the introduction of derived categories, they are still the most effective computational tool available. Spectral sequences were found in diverse situations, and they gave intricate relationships among homology and cohomology groups coming from geometric situations such as fibrations and from algebraic situations involving derived functors. It was soon realized that Leray's computational technique was an example of a more general phenomenon. The limit of this infinite process was essentially the same as the cohomology groups of the original sheaf. The cohomology of the cohomology again formed a chain complex, and its cohomology formed a chain complex, and so on. This was still not the cohomology of the original sheaf, but it was one step closer in a sense. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. The relation involved an infinite process. This gave a relation between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray ( 1946a, 1946b), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations.

Main sequence spectral class