use the book will have enough flexibility. The book
summer REU (Research Experience for U
course I taught at Penn State in 2004. Some material
in the MASS (Mathematics Advanced Study Semeste
Penn State in 2000-2004 and at the Canada/USA Bi
ematical Camp Program in 2001. In the fall semeste
material will be used again for a MASS course in geo
A few words about the pedagogical philosophy of t
the reader without a solid mathematical basis of real a
ential geometry, topology, etc., will benefit from the
without saying, such knowledge would be helpful).
these fields are freely used when needed, and the rea
tensively rely on his mathematical common sense.
For example, the reader who does not feel comfo
notion of a smooth manifold should substitute a smo
space, the one who is not familiar with the general
differential form should use the one from the first c
lus ("an expression of the form..."), and the reader
yet know Fourier series should consider trigonometr
instead. Thus what I have in mind is the learning patt
ner attending an advanced research seminar: one take
to the frontier of current research, deferring a more
"linear" study of the foundations until later.
A specific feature of this book is a substantial nu
sions; they have their own titles and their ends are
Many of the digressions concern topics that even an
dergraduate student is not likely to encounter but, I
educated mathematician should be familiar with. Som
ics used to be part of the standard curriculum (for exa
and involutes, or configuration theorems of projective g
ers are scattered in textbooks (such as distribution o
various sequences, or a mathematical theory of rainb
vertex theorem), still others belong to advanced topics
theory, or Poincare recurrence theorem, or symplectic
1Six weeks, six hours a week.
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