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The Cambridge Colloquium
 
The Cambridge Colloquium
Softcover ISBN:  978-0-8218-4642-1
Product Code:  COLL/5
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-3155-6
Product Code:  COLL/5.E
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
Softcover ISBN:  978-0-8218-4642-1
eBook: ISBN:  978-1-4704-3155-6
Product Code:  COLL/5.B
List Price: $188.00 $143.50
MAA Member Price: $169.20 $129.15
AMS Member Price: $150.40 $114.80
The Cambridge Colloquium
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The Cambridge Colloquium
Softcover ISBN:  978-0-8218-4642-1
Product Code:  COLL/5
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-3155-6
Product Code:  COLL/5.E
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
Softcover ISBN:  978-0-8218-4642-1
eBook ISBN:  978-1-4704-3155-6
Product Code:  COLL/5.B
List Price: $188.00 $143.50
MAA Member Price: $169.20 $129.15
AMS Member Price: $150.40 $114.80
  • Book Details
     
     
    Colloquium Publications
    Volume: 51918; 286 pp
    MSC: Primary 00; Secondary 45; 55;

    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
     
     
    • Griffith Conrad Evans. Functionals and Their Applications. Selected Topics, Including Integral Equations.
    • Lecture I. Functionals, derivatives, variational equations
    • Lecture II. Complex functionals
    • Lecture III. Implicit functional equations
    • Lecture IV. Integro-differential equtions of Bôcher type
    • Lecture V. Direct generalizations of the theory of integral equations
    • Oswald Veblen. Analysis Situs.
    • Chapter I. Linear graphs
    • Chapter II. Two-dimensional complexes and manifolds
    • Chapter III. Complexes and manifolds of $n$ dimensions
    • Chapter IV. Orientable manifolds
    • Chapter V. The fundamental group and certain unsolved problems
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 51918; 286 pp
MSC: Primary 00; Secondary 45; 55;

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.

  • Griffith Conrad Evans. Functionals and Their Applications. Selected Topics, Including Integral Equations.
  • Lecture I. Functionals, derivatives, variational equations
  • Lecture II. Complex functionals
  • Lecture III. Implicit functional equations
  • Lecture IV. Integro-differential equtions of Bôcher type
  • Lecture V. Direct generalizations of the theory of integral equations
  • Oswald Veblen. Analysis Situs.
  • Chapter I. Linear graphs
  • Chapter II. Two-dimensional complexes and manifolds
  • Chapter III. Complexes and manifolds of $n$ dimensions
  • Chapter IV. Orientable manifolds
  • Chapter V. The fundamental group and certain unsolved problems
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.