Berkeley problems in mathematics (Q5906261)

This is a collection of problems that were offered on written exams to Ph.D. mathematics students at University of California, Berkeley, from 1977 through 1998. The authors explain that its purpose was to determine whether ``first-year students in the Ph.D. program had mastered basic mathematics well enough to continue in the program with a reasonable chance of success''. The 443-page book includes numerous problems of varying degree of difficulty, their solutions and appendices. So what are the topics Berkeley believes need to be mastered? The chapter titles tell the story: Real Analysis, Multivariable Calculus, Differential Equations, Metric Spaces, Complex Analysis, Algebra, and Linear Algebra. Ought basic Ph.D. program mathematics be reduced to analysis -- with a piece of algebra? Where are such foundations of mathematics as Set Theory, Mathematical Logic, History of Mathematics, Number Theory? Only Number Theory from this list can be found -- it is a mere last section in chapter Algebra. What about that Erdősian mathematics -- Combinatorics, Graph Theory, Ramsey Theory, etc.? Problem 6.13.1 reads: ''Prove that if six people are riding together in an Evans Hall elevator, there is either a three-person subset of mutual friends (each knows the other two) or a three-person subset of mutual strangers (each knows neither of the other two).'' Does this one high school contest problem from the birth of Ramsey Theory constitute a sufficient knowledge of Combinatorics for a Ph.D. student? And what does this problem have to do with Number Theory? Yet, it opens Number Theory section! This is a fine problem book, which would give students and instructors another resource in Analysis and Algebra. In the opinion of this reviewer, however, it is sad when a flagship of mathematics - Berkeley - preaches an obsolete idea that mathematics is more or less synonymous with Analysis.