Project Details
Projekt
GRK 1150: Homotopy and Cohomology
Subject Area
Mathematics
Term
from 2005 to 2014
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 796242
Topology stands out amongst other branches of mathematics for the way it bridges the gap between the realm of continuous phenomena (geometry and analysis) and the discrete world (algebra and combinatorics). Topology uses discrete techniques to study continuous objects; it has assimilated methods from many areas of mathematics, and methods developed by topologists have in turn contributed significantly to advances in other areas.
Topology is concerned with the study of geometric objects (such as manifolds) and of the continuous maps between them. Typically one asks whether there are any geometrics objects having certain specified properties; and if so, how many different objects share these properties. The technical term is that one seeks a classification of such objects, or of the maps between them. One example would be classifying the covering spaces of a given manifold. Many of the properties one is interested in are retained when the object under investigation is subjected to a deformation. One example is the degree of a mapping: the degree (an integer) is unchanged when the mapping is continuously deformed. One says that the degree is invariant up to homotopy. The power of Homotopy Theory lies in this invariance up to homotopy: often it allows one to replace complicated objects by simple models of them.
In short, the strategy of Homotopy Theory is to establish this invariance for as many properties as possible, and then exploit this invariance to obtain a classification.
Classifying objects usually involves calculating homotopy groups. As the direct approach is typically prohibitively difficult, one uses cohomology to calculate these homotopy groups by indirect means. That is, one makes use of both ordinary (i.e. singular) cohomology and extraordinary cohomology theories, together with the associated ring structure and cohomology operations.
Recent years have seen several new and promising developments leading to even greater interconnections between Homotopy Theory on the one hand and Algebra, Algebraic Geometry and Theoretical Physics on the other. The areas of Homotopy Theory in question are Stable Homotopy Theory and Elliptic Cohomology. These new theories and methods will provide the thesis topics in the Research Training Group.
Topology is concerned with the study of geometric objects (such as manifolds) and of the continuous maps between them. Typically one asks whether there are any geometrics objects having certain specified properties; and if so, how many different objects share these properties. The technical term is that one seeks a classification of such objects, or of the maps between them. One example would be classifying the covering spaces of a given manifold. Many of the properties one is interested in are retained when the object under investigation is subjected to a deformation. One example is the degree of a mapping: the degree (an integer) is unchanged when the mapping is continuously deformed. One says that the degree is invariant up to homotopy. The power of Homotopy Theory lies in this invariance up to homotopy: often it allows one to replace complicated objects by simple models of them.
In short, the strategy of Homotopy Theory is to establish this invariance for as many properties as possible, and then exploit this invariance to obtain a classification.
Classifying objects usually involves calculating homotopy groups. As the direct approach is typically prohibitively difficult, one uses cohomology to calculate these homotopy groups by indirect means. That is, one makes use of both ordinary (i.e. singular) cohomology and extraordinary cohomology theories, together with the associated ring structure and cohomology operations.
Recent years have seen several new and promising developments leading to even greater interconnections between Homotopy Theory on the one hand and Algebra, Algebraic Geometry and Theoretical Physics on the other. The areas of Homotopy Theory in question are Stable Homotopy Theory and Elliptic Cohomology. These new theories and methods will provide the thesis topics in the Research Training Group.
DFG Programme
Research Training Groups
Applicant Institution
Rheinische Friedrich-Wilhelms-Universität Bonn
Co-Applicant Institution
Heinrich-Heine-Universität Düsseldorf; Ruhr-Universität Bochum
Participating Researchers
Professor Dr. Gerd Laures; Professor Dr. Holger Reich; Professor Dr. Stefan Schwede; Professor Dr. Wilhelm Singhof; Professorin Dr. Catharina Stroppel; Professor Dr. Peter Teichner
Spokesperson
Professor Dr. Carl-Friedrich Bödigheimer, since 10/2009
