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- Group theory
- Series: Graduate Texts in Mathematics
A kaleidoscope is an example of an optical device for visualizing certain reflection groups in the plane. The theory of reflection groups links linear algebra, abstract algebra particularly group theory , Lie algebras, and representation theory in a beautiful way. The main topics covered will be orthogonal transformations, reflections in real Euclidean space, Coxeter groups, crystallographic groups, root systems, and the classification of finite Coxeter groups. Along the way, the course will cover the basics of finite group theory as well as certain advanced topics in linear algebra.
This will be a heavily proof-based course with homework requiring a significant investment of time and thought. This course is only appropriate to students who have taken a first course in linear algebra.
A previous course in abstract algebra is not necessary. While the course is primarily targeted at students interested in studying higher mathematics, the subject matter would be of interest and possible use in subjects such as chemistry, computer science, materials science, and theoretical physics.
Expected background: The official prerequisite is linear algebra, either Math or The unofficial prerequisites are a mature mathematical mind, some experience with writing proofs, and the desire to work hard. Homework: Weekly homework will be due at the start of class on Friday.
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Each assignment will be posted on the syllabus page the week before it's due. Late or improperly submitted homework will not be accepted. If you know in advance that you will be unable to submit your homework at the correct time and place, you must make special arrangements ahead of time.
Under extraordinary circumstances, late homework may be accepted with a dean's excuse. It seems that you're in Germany. We have a dedicated site for Germany.
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Authors: Grove , L. Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites.
Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. There is a discussion offundamental regions in Chapter 3. The actual classification and construction of finite reflection groups takes place in Chapter 5.
Witt and B. Prospects in topology Princeton, NJ, , p. Press, Princeton, NJ Dedicata , p. London Math.
Zbl Topology 40, no. Scuola Norm. Pisa Cl. Special issue on braid groups and related topics Jerusalem, Topology Appl. Fourier Grenoble 53, no. Russian Uspekhi Mat.
Series: Graduate Texts in Mathematics
Nauk 57 , no. Translation in Russian Math. Surveys 57, no. Translation in Functional Anal.
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