Fall 2006

MTHS 645 Groups and Their Actions

09/11/2006 - 12/16/2006
Wednesday 06:00 PM - 08:30 PM

Science Tower 139

The concept of a group and its actions is fundamental in mathematics, and examples are found throughout the subject. Consider these examples: rearrange the vertices of a cube so that the distances between them are unchanged. Move the points of a plane so that every circle is carried to a circle. Twist the faces of a Rubik's cube. Shuffle a deck of cards. Every one of these is an example of an action of a group.

We will begin by studying specific examples of groups such as those above, and others, and these examples will be used to motivate the abstract development of the subject. The abstract development, which focuses on the ideas that are common to all the examples, can in turn be used to inform us about geometry, and other applications. For example, these ideas lead to a proof that there are only five regular solids, and only seventeen ways to tile the plane with congruent figures. In an application far from geometry, we can see why in Sam Loyd's famous 15 puzzle (the sliding block puzzle of our youth) exactly half of the possible arrangements of the blocks can be attained.

Readings will be taken from various sources. The textbook, Groups: A Path to Geometry, by R. P. Burn will be especially useful, because of its wealth of instructive problems.

Students will be given regular problem sets to complete, and grades will be based on their work on these assignments.

Students will be expected to have a good command of high school algebra, geometry, and pre-calculus, and be ready to follow extended logical reasoning.

Adam Fieldsteel (A.B. Brown University; Ph.D. University of California, Berkeley) is professor of mathematics. His research focuses on ergodic theory and topological dynamics, and his recent publications include: (with A. Blokh), "Sets that force recurrence," Proceedings of the American Mathematical Society (2002); (with K. Dajani), "Equipartition of interval partitions and an application to number theory," Proceedings of the American Mathematical Society (2001); (with R. Hasfura), "Dyadic equivalence to completely positive entropy," Transactions of the American Mathematical Society (1998). Click here for more information about Adam Fieldsteel.

ENROLLMENT INFORMATION

Consent of Instructor Required: No

 Format: Seminar Level: GLSP Credits: 3 Enrollment Limit: 18

Texts to purchase for this course:
R.P. Burn, GROUPS, A PATH TO GEOMETRY (Cambridge University Press), Paperback