MATH 3330: Elements of Topology
Description: Topology is the study of properties of shape that persists under "nice" continuous perturbations---stretching, shrinking and twisting, the description given by the author of the textbook. A disk and a triangle, for instance, are regarded as the same shape in topology. This presents a stark contrast to another field in the study of shape, namely, geometry. Geometry, in general, studies the properties of shape that are more "visually" rigid and distinguishable, such as length, angle, area etc. These quantities are somewhat more familiar to most of us, whereas the criteria used in topology demand a certain degree of trained awareness and effort. This course begins with a brief introduction to the tools for the study. The first application is to see how many distinct closed surfaces exist in this world under the eyes of topologists. We will introduce a topological invariant called "Euler characteristic", which assigns an integer to an individual surface. Surprisingly, the Euler characteristic alone completely determines the topological shape of surfaces in spite of all those possible "geometric-shape-deforming" continuous perturbations. This process, which is called the "classification of closed compact surfaces", epitomizes one nature of topology, namely, simplicity. By the way, do you now see how many distinct compact closed surfaces exist from the topological point of view? When time allows, we will introduce more "topological invariants" such as fundamental groups, homology etc for the study of graphs, simplicial complex, knots etc. These concepts have major applications to computer science, biology and other fields. As the final remark, we will try to visit as many websites, some of which are suggested in the textbook, to enjoy the visual aspects of topology during the course.
Prerequisites: A grade of C or better in MATH 2142 or 2710.
Offered: Spring (even years)
Sections: Spring 2019 on Storrs Campus