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viii Preface to the Second Edition

There are three new appendices, one by Stefan Theisen on the role of Calabi–
Yau manifolds in string theory and one by Otto Forster on the use of elliptic curves
in computing theory and coding theory. In the third appendix we discuss the role of
elliptic curves in homotopy theory. In these three introductions the reader can get a
clue to the far-reaching implications of the theory of elliptic curves in mathematical
sciences.
During the final production of this edition, the ICM 2002 manuscript of Mike
Hopkins became available. This report outlines the role of elliptic curves in homo-
topy theory. Elliptic curves appear in the form of the Weierstasse equation and its
related changes of variable. The equations and the changes of variable are coded in
an algebraic structure called a Hopf algebroid, and this Hopf algebroid is related to
a cohomology theory called topological modular forms. Hopkins and his coworkers
have used this theory in several directions, one being the explanation of elements
in stable homotopy up to degree 60. In the third appendix we explain how what we
described in Chapter 3 leads to the Weierstrass Hopf algebroid making a link with
Hopkins’ paper.

Max-Planck-Institut f¨ ur Mathematik Dale Husem¨ oller
Bonn, Germany