Math 401          Modern Algebra and Applied Geometry

                    Course Syllabus for Fall semester 2000 – prepared by Marie Vanisko

Instructor:        Marie Vanisko, Professor of Mathematics

                    Office: Room 118 Science Building,  447-4451,  Email:  mvanisko@carroll.edu

                    Class Hours:  MA 121 – MWF (2:00-2:50)  (Room 110)

                                    MA 233 – MWF (10:00-10:50), TTh (9:30-10:45) (Room 123)

      MA 401 – MWF (1:00-1:50)  (Room 120)

                    Office Hours:  M, W & F: 8:30-10:00, 11:00-11:30, 3:00-4:30 (M, W)

                                     T & Th:        8:30-9:30, 11:00-11:30, 2:00-4:30

      Other times by appointment.

Pre-Requisite:                Math 301 (Foundations of Mathematics)

Required Texts:        Contemporary Abstract Algebra by Joseph Gallian (Heath, 4th ed, 1998)

                          Geometry and Its Applications by Walter Meyer (Academic Press, 1999)

Course Description and Objectives:

This course covers the traditional topics from abstract algebra, including groups, rings, integral domains, fields, and homomorphic and isomorphic relationships, as well as standard topics from geometry, including axiomatic systems in both Euclidean and Non-Euclidean geometries and transformational geometry with vectors and matrices, The focus for the class is the contemporary applications of the concepts presented, together with the weaving together of geometric and algebraic themes. Applications include, robotics, boolean algebra, symmetry groups, coding, adjacency matrices, the global positioning system (GPS),etc. By the end of this course, students should be able to write proofs, do computations, and carry out applications of the topics covered. They should see the beauty in the mathematical structures and their use “…for understanding the 21st century world that is unfolding around us.” (Meyer)

Course Format and Policies:

   A portion of most classes will be interactive, with individual and group presentations. Grades for these presentations will be based on preparation, participation, and creativity.

   Writing about what is being learned will be an essential part of every assignment.

        Papers with no comments will not be accepted.

Exams will be given approximately every three weeks. 

Some Fridays will be spent in the computer lab using Mathematica.

Grading scale used:     A (90-100),  B (80-89),  C (70-79),  D (60-69),  F (below 60)

Relative weights:          Exams (60%), 

Homework & Interaction (30%),

                                  Final Project (10%)*

*      For the final project, students may select any topic that is introduced in either the geometry or the algebra book, do some outside research on that topic, and present it to the class during the last week of class. This project may be done individually, in pairs, or in groups of three. Individuals or groups must meet with the instructor just after midterm to select a topic. It is suggested that topics selected relate to your interests and/or cognate concentrations.

Course Outline:

Weeks 1, 2, 3         Chapters 1, 2, & 3 of Meyer’s book (Axiomatic Geometry)

                        Students should learn to appreciate just how “undefined” undefined terms are and be able to develop models for axiomatic systems. The power and the limitations of visualizations in proving theorems will be anlayzed. Hyperbolic, Spherical and Finite geometries will be explored.

Exam

Weeks 4, 5, 6                Part 2 of Gallian’s book (Groups) (Chapters 7, 8, & 9 will be omitted)

        Students should learn what a group is, how to determine if a set of elements under a given operation satisfies the conditions for a group. They will consider permutation and cyclic groups and groups of symmetries and transformations. Applications will be presented.

Exam

Weeks 7, 8. 9         Part 3 & Chapter 19 of Gallian’s book (Rings,Integral Domains, Fields)

                                (Chapters 14, 17, & 18 will be omitted)

        Students should understand the meaning of and be able to appreciate the significance of rings, integral domains, and fields and vector spaces. Applications will be presented.

Exam

Weeks 10, 11, 12            Section 5-3, Chapter 6, Section 8-1 of Meyer’s book (Isometries with Vectors and Matrices)

        Students will see vectors and matrices used in applications involving robotics, medicine,

        computer graphics, cartography and navigation (the global positioning system), computer

        graphics, and engineering.

Exam

Weeks 13 & 14            Selected Applications from Gallian’s and/or Meyer’s book

        Students will present robotic projects and final projects.

Week 15                     Final Exam (Tuesay, December 19, 10 am.)

*      This will be the final project referred to in the grading scheme.