This is the title of a talk given at the CMU graduate student seminar on November 3, 2015. A video of the talk is on Youtube.
This talk will be about Saharon Shelah's first book: "Classification theory and the number of non-isomorphic models", for which he won the Leroy P. Steele prize in 2013.
The book discusses the following question: When does a class of mathematical structures (for example a class of groups or a class of fields) have a "structure theorem"? When can we fully understand the structure of the class? The main theorem of the book, the so-called "main gap" gives a resolution of this question for classes specified by a countable set of first-order axioms.
The main concept in Shelah's resolution is the notion of a dividing line. Roughly speaking, a dividing line is a property so that both it and its negation have strong consequences. Typically, a class falling on the bad side of the dividing line does not, in a strong sense, have a structure theorem, while a class falling on the good side is quite well-behaved and can be analyzed further (typically via more dividing lines).
We will try to give several examples and make some of the notions discussed above more precise. As we go along, we will explore what the main gap says and why some mathematicians find it interesting.
Saharon Shelah, Classification theory and the number of non-isomorphic models, 2nd ed. Studies in logic and the foundations of mathematics, vol. 92, North-Holland, 1990.
Saharon Shelah, Classification of first order theories which have a structure theorem, Bulletin of the American Mathematical Society, vol. 12, number 2 (1985) 227-232.