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Paulo Ney de Souza
Jorge-Nuno Silva

1'

Berkeley Problems
in Mathematics

Springer

Preface

In 1977 the Mathematics Department at the University of California, Rerkeley, instituted a written examination as one of the first major requirements
toward the Ph.D. degree in Mathernalics. This examination replaced a
system of standardized Qualifying Exams. 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.
Historically, any one examination is passed by approximately half of the
students taking it and students are allowed three attempts. Since its incep
tion, the exam has become a major hurdle to overcome in the pursuit of
the degree and, therefore, a measure of the minimum requirements to successful completion of the program at Berkeley. Even though students are
allowed three attempts, most would agree that the ideal time to complete
the requirement is during the first month of the program rather than in the
middle or end of the first year. This book was conceived on this premise,
and its intent is to publicize the material and aid in the preparation for the
examination during the undergraduate years, when one is deeply involved
with the material that it covers.
The examination is now offered twice a year in the second week of each
semester, and consists of 6 hours of written work given over a 2-day period
with 9 problems each (10 before 1988). Students select 6 of the 9 problems
(7 of 10 before 1988). Most of thc examination covers material, mainly in
analysis and algebra, that should be a part of a well-prepared mathematics
student’s undergraduate training. This book is a compilation of the almost
1000 problems which have appeared on the Prelims during the last 20

b

viii

Preface

years and currcntly makc up a collection which is a dclightful field to plow
through, and solutioiis to most of them.
When Berkeley was on the Quarter system, cxams werc given three times
a year: Spring, Summer, and Fall. Since 1986, the exams have been given
twice a year, in January and September.
From the first cxamination through Fall 1981, the policy was: two attempts allowed; each examination 6 hours; total 14/20 problems. From
Winter 1982 through Spring 1988, the policy was: two attempts allowed;
each examination 8 hours; total 14/20 problems. Starting Fall 1088, the
policy was: three attempts allowed; each examination 6 hours; total 12/18
p...
Paulo
Ney de
Souza
Jorge-Nuno
Silva
1'
Berkeley
Problems
in
Mathematics
Springer
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