**Colloquium Publications**

Volume: 4;
1914;
238 pp;
Softcover

MSC: Primary 11; 32;
**Print ISBN: 978-0-8218-4598-1
Product Code: COLL/4**

List Price: $58.00

Individual Member Price: $46.40

# The Madison Colloquium

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*Leonard Eugene Dickson; William Fogg Osgood*

Following the tradition of the American Mathematical Society, the seventh
colloquium of the Society was held as part of the summer meeting that took
place at the University of Wisconsin, in Madison. Two sets of lectures
were presented: On Invariants and the Theory of Numbers, by L. E.
Dickson, and Functions of Several Complex Variables, by W. F.
Osgood.

Dickson considers invariants of quadratic forms, with a special
emphasis on invariants of forms defined in characteristic
\(p\), also called modular invariants, which have
number-theoretic consequences. He is able to find a fundamental set of
invariants for both settings. For binary forms, Dickson introduces
semi-invariants in the modular case, and again finds a fundamental
set. These studies naturally lead to the important study of invariants
of the standard action of the modular group. The lectures conclude
with a study of “modular geometry”, which is now known as
geometry over \(\mathbf{F}_p\).

The lectures by Osgood review the state of the art of several complex
variables. At this time, the theory was entirely function-theoretic.
Already, though, Osgood can introduce the ideas and theorems that will be
fundamental to the subject for the rest of the century: Weierstrass
preparation, periodic functions and theta functions,
singularities—including Hartogs' phenomenon, the boundary of a domain of
holomorphy, and so on.

#### Table of Contents

# Table of Contents

## The Madison Colloquium

#### Readership

Graduate students and research mathematicians interested in number theory and analysis.