**Colloquium Publications**

Volume: 5;
1918;
286 pp;
Softcover

MSC: Primary 00;
Secondary 45; 55

Print ISBN: 978-0-8218-4642-1

Product Code: COLL/5

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

**Electronic ISBN: 978-1-4704-3155-6
Product Code: COLL/5.E**

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

# The Cambridge Colloquium

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*Griffith Conrad Evans; Oswald Veblen*

The 1916 colloquium of the American Mathematical Society was held as
part of the summer meeting that took place in Boston. Two sets of
lectures were presented: Functionals and their
Applications. Selected Topics, including Integral Equations, by
G. C. Evans, and Analysis Situs, by Oswald Veblen.

The lectures by Evans are devoted to functionals and their
applications. By a functional the author means a function on
an infinite-dimensional space, usually a space of functions, or of
curves on the plane or in 3-space, etc. The first lecture deals with
general considerations of functionals (continuity, derivatives,
variational equations, etc.). The main topic of the second lecture is
the study of complex-valued functionals, such as integrals of complex
functions in several variables. The third lecture is devoted to the
study of what is called implicit functional equations. This study
requires, in particular, the development of the notion of a
Fréchét differential, which is also discussed in this
lecture. The fourth lecture contains generalizations of the
Bôcher approach to the treatment of the Laplace equation, where
a harmonic function is characterized as a function with no flux
(Evans' terminology) through every circle on the plane. Finally, the
fifth lecture gives an account of various generalizations of the
theory of integral equations.

Analysis situs is the name used by Poincaré when he
was creating, at the end of the 19th century, the area of mathematics
known today as topology. Veblen's lectures, forming the second part of
the book, contain what is probably the first text where
Poincaré's results and ideas were summarized, and an attempt to
systematically present this difficult new area of mathematics was
made.

This is how S. Lefschetz had described, in his 1924 review of the book,
the experience of “a beginner attracted by the fascinating and difficult
field of analysis situs”:

“Difficult reasonings beset him at every step, an unfriendly
notation did not help matters, to all of which must be added, most
baffling of all, the breakdown of geometric intuition precisely when most
needed. No royal road can be created through this dense forest, but a good
and thoroughgoing treatment of fundamentals, notation, terminology, may
smooth the path somewhat. And this and much more we find supplied by
Veblen's Lectures.”

Of the two streams of topology existing at that time, point set topology
and combinatorial topology, it is the latter to which Veblen's book is
almost totally devoted. The first four chapters present, in detail, the
notion and properties (introduced by Poincaré) of the incidence matrix
of a cell decomposition of a manifold. The main goal of the author is to
show how to reproduce main topological invariants of a manifold and their
relations in terms of the incidence matrix.

The (last) fifth chapter contains what Lefschetz called “an excellent
summary of several important questions: homotopy and isotopy, theory of
the indicatrix, a fairly ample treatment of the group of a manifold,
finally a bird's eye view of what is known and not known (mostly the
latter) on three dimensional manifolds.”

#### Readership

Graduate students and research mathematicians interested in functionals.

#### Table of Contents

# Table of Contents

## The Cambridge Colloquium

- Cover Cover11
- Title page A-i2
- Griffith Conrad Evans. Functionals and Their Applications. Selected Topics, Including Integral Equations. A-iii4
- Dedication A-iii4
- Preface A-v6
- Author’s Preface A-vii8
- Contents A-ix10
- Errata Corrige. A-xiii14
- Lecture I. Functionals, derivatives, variational equations A-118
- Lecture II. Complex functionals A-3047
- Lecture III. Implicit functional equations A-5269
- Lecture IV. Integro-differential equtions of Bôcher type A-7390
- Lecture V. Direct generalizations of the theory of integral equations A-101118
- Oswald Veblen. Analysis Situs. B-i154
- Author’s Preface B-iii156
- Contents B-v158
- Chapter I. Linear graphs B-1162
- Chapter II. Two-dimensional complexes and manifolds B-34195
- Chapter III. Complexes and manifolds of 𝑛 dimensions B-73234
- Chapter IV. Orientable manifolds B-100261
- Chapter V. The fundamental group and certain unsolved problems B-125286
- Back Cover Back Cover1312