use the book will have enough flexibility. The book

an

intense1

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.