Colloquium Publications 1914; 238 pp; softcover Volume: 4 ISBN10: 0821845985 ISBN13: 9780821845981 List Price: US$55 Member Price: US$44 Order Code: COLL/4
 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 numbertheoretic consequences. He is able to find a fundamental set of invariants for both settings. For binary forms, Dickson introduces semiinvariants 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 functiontheoretic. 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, singularitiesincluding Hartogs' phenomenon, the boundary of a domain of holomorphy, and so on. Readership Graduate students and research mathematicians interested in number theory and analysis. Table of Contents  L. E. Dickson  On invariants and the theory of numbers
 W. F. Osgood  Topics in the theory of functions of several complex variables
