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MATH 4800 |
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Spring 2010 courseTopic: Symmetry & GroupsInstructor: Sarah Kitchen Catalog Number: MATH 4800-1 Credits: 3 Time:T, Th 2:00-3:20 PM, LCB 323 Email the following items to the course instructor (kitchen@math.utah.edu) to be considered for admission into the course:
Tuition Benefit: The NSF VIGRE program provides a $500 tuition benefit for undergraduates who are US citizens, nationals, and permanent residents. Description: Groups are ubiquitous to mathematics and understanding them and their representations is the cornerstone to understanding both matter and the space around us. For example, consider a square in the plane, centered at the origin. This square can be reflected across the x- or y-axis and still cover the same points of the plane. It can also be rotated by 90-degree increments clockwise or counterclockwise. The symmetry group of the square is the collection of all combinations of these operations. We all know squares are different from triangles, and the fact that a 90-degree rotation does not preserve a triangle, but does preserve a square, gives a mathematical interpretation of "different": Their symmetry groups are not the same, and therefore they cannot be the same. In this example, we come to an understanding about space based on an understanding of the actions, rotation and reflection, of a symmetry group on that space. We were also able to determine the symmetry group based on its actions, so in this sense "understanding" goes both ways and we are led to the general questions:
The focus of this course will be precisely on the study of symmetry groups and their representations, with a view towards applications. We will primarily use the text Linear Representations of Finite Groups, by J.P. Serre, with supplementation from chemistry texts and instructor-provided course notes as needed. Students will be expected to work individually or in small groups to research a special topic related to the course material, and will present their research in the second-half of the semester. Topics covered by the lecturer beyond an introduction to the representation theory of finite groups and in particular the representation theory of symmetry/point groups, will depend on the interests of the class, but could include wavefunctions, crystal symmetry, and other applications. Prerequisites: Basic linear algebra will be used extensively in this course. Students will be expected to have seen at least an introduction to matrices, vectors, and vector spaces provided by a Calculus III (MATH 2210), Linear Algebra (MATH 2270 or MATH 2250), or comparable physics/chemistry courses requiring these as pre/co-requisites. Modern Algebra (MATH 5310) would be ideal, but will not be assumed. A course in complex variables and/or differential equations could also be helpful, but is not strictly necessary. Past Courses: Fall 2006, Math Finance Spring 2007, Fractals Fall 2007, Metric Spaces, The Contraction Mapping Principle, Fractals & Other Applications Spring 2008, Knot Theory Fall 2008, Random Walk: Modeling, Theory, and Applications Spring 2009, Graph Theory Fall 2009, Metamaterials |
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