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2Analysis of several complex variables
Takeo Ohsawa American mathematical society
Introduction to complex analysis in several variables
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An introduction to complex analysis in several variables
Lars Hormander, North Holland
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Algebraic Topology
Lars Hesselholt
(Evaluation)
Occational exercises reviewed by the teacher.
(textbooks and references)
• M. Hovey: Model Categories, Mathematical Surveys and Monographs, vol. 63, Amer. Math. Soc.
• S. Schwede: An untitled book project about symmetric spectra, available at www.math.uni-bonn.de/people/schwede/SymSpec.pdf
(object of the course)
This course gives an introduction to homotopy theory. Classically, this is the study of the weak homotopy-type of topological spaces, a notion that goes back to H. Poincar´e. A continuous map of topological spaces is called a weak equivalence if it induces an isomorphism of homotopy groups. The weak homotopy-type of a topological space is the isomorphism class of the space in the category obtained by formally introducing an inverse to every weak equivalence. It is the structure of this category, the homotopy category of spaces, that is the main object of study. The main techniques are centered around two classes of maps called the fibrations and the cofibrations which were introduced by J.–P. Serre and J. H. C. Whitehead, respectively. The properties of the category of topological spaces together with the three classes of maps given by the weak equivalences, the fibrations, and the cofibrations were formalized by D. Quillen into the notion of a model category for a homotopy theory. This makes the use of homotopy theoretical methods possible in other areas of mathematics.
(schedule of the course)
We begin with the basic notions of a model category and associated model category. We will consider the model category of symmetric spectra and the associated stable homotopy category in some detail. If time permits, we will treat some additional topics. One possibilty is an introduction to higher algebraic K-theory. The choice of additional topics and the overall pace of the course will depend on the participants.
(key words)
Homotopy, model categories, fibrations, symmetric spectra.
(required knowledge)
An introductory course in topology including the fundamental group and covering spaces.
(attendance)
This course is open for any students at Nagoya University as one of the ”open subjects” of general education.
larsh@math.nagoya-u.ac.jp
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I. Assem, D. Simson, A. Skowro´ nski: Elements of the representation theory of associative
algebras. Vol. 1. (Cambridge),
D. Happel: Triangulated categories in the representation theory of finite-dimensional
algebras. (Cambridge)
Y. Yoshino: Cohen-Macaulay modules over Cohen-Macaulay rings. (Cambridge)
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