Fall 2021 Geometry

Another useful source is Geometry by Its History by Alexander Ostermann and Gerhard Wanner. A free electronic version can be obtained for free via the Rutgers Springer Mathematics E-books package---Rutgers affiliated users can also order each print copy for $24.95 (Shipping and handling are included). The specific Springer link for the book is here. This book will not be used systematically in this course, but it provides a lot of interesting geometric arguments which complement the more algebraic approach of our textbook, and one can browse through and work on specific topics without too much difficulty.

Lecture: Online in Canvas Conferences MW 5:00-6:20 AM

Office Hours: Online via zoom Monday 4:00-5:00 (see Canvas)

Course Syllabus

Grading Scheme:

Exam I: Wednesday 10/13/21 online during classtime
Exam II: Monday 11/29/21 online during classtime
Final Paper : Due Monday 12/20/21 7:00PM

Homework: There will be weekly written homework assignments, turned in via Canvas. Late homework will not be accepted. You may discuss your homework with other classmates, but the work you hand in must be your own.

General Comments on the Course

This course uses the classical Euclidean plane geometry as an anchor point to study some of its natural outgrowth: affine, projective, spherical, and non-Euclidean geometries. The study of these geometries will, in turn, deepen our appreciation for the classical Euclidean geometry. One unifying theme in this course's approach to these geometries is Klein's transformation groups. Technically we will use a lot of analytic methods (setting up and analyzing equations in appropriate coordinates, matrix manipulations), so it is essential that students review the material in Math 250 (Linear Algebra)--- most of the material will be needed in an essential way for the material starting from chapter 3. You can find a list of review problems here. On the other hand, our approach will not be purely computational. We will emphasize the geometric flavor of the subject, and whenever possible and beneficial, will provide direct geometric argument. In particular we will blend in fair amount of deductive proofs (also called axiomatic or synthetic proofs).

Out hope is that, after the course, you will have an appreciation for the liveliness, diversity and connectedness of mathematics, and the excitement and pleasure of discovering mathematics, and that you would be comfortable to attack geometric problems using a combination of methods learned in this course.

Emphasis will be placed on geometric understanding and logical reasoning. As such, mere memorization of facts would be of little help. Nor can most regular assignments be completed by simply looking up a magic formula on a page from the texts. Instead, you should be prepared to fully participate in the discussions(in-class and out-class), do extra readings and research, develop and communicate your ideas. You are also encouraged to try to use a combination of geometric exploration, model making, and thought experiments to help you in the learning process. Group discussions and brainstorms will be strongly encouraged. An important aspect of the course is to help you sort out your ideas and present them in a logical way. So it is expected that you present your work in a coherent way, using complete English sentences. More guidelines are given below.

Course Material

This course uses a fair amount of linear algebra, especially from chapter 3 on. The attached file here contains a list of review problems for the relevant material. Please review these topics on your own in the first two weeks. You can focus your review to 2x2 or 3x3 matrices.

Structure of Assignments

Homework and Quizzes: You will have weekly regular assignments. The regular assignments are to help you work through the ideas discussed in class and gain a fuller understanding of the technical aspect of the ideas. Discussion and cooperation with each other is strongly encouraged at every stage of the course work, except at the writing-up. In your submitted work, ideas that come from other people should be given proper attribution. If your work has emerged from work with other people, write down whom you have worked with. If you have referred to some sources, cite them. Short quizzes may occasionally be given to test basic understanding on concepts.

Assignment Grading Guidelines

The grading of the regular assignments will be based on correctness of concepts and content, soundness of logical structure in your arguments, and exposition. You should present your work just as you would do for a writing assignment in any other subject, giving the necessary background information, definition of terms you are about to explain, logical arguments, and your conclusions. More specifically, if you are presenting the solution to a problem, explain first what the problem is; if you are going to use some terms or concepts, try to give as clear a definition as possible(because mathematical concepts in a typical student's mind are often vague and may change from context to context, but in scientific discussions precision of concepts is needed); if you are giving an argument, try to explain the main points before you launch into the details. This may seem hard in the beginning. But you can improve quickly if you keep a journal to record what you have been thinking and doing in your work, and then try to organize the ideas in a coherent way as if you wanted to explain your ideas to a friend or to convince him/her of your arguments. It is important that you learn to explore geometry on your own, instead of limiting yourself to answering the questions raised by the instructor. It is good practice to raise further questions of your own at the end of each assignment (as simple as questions like 'what if we are in a different situation like...').

Attendance and Make-up Policy: Class attendance is expected. Poor attendance will be used to decide borderline grade situations. Any changes to the syllabus, homework assignment and any announcement for the midterms and final exam will be made in the lectures. No late work will be accepted. There will be no make-ups for quizzes. A make-up midterm will be given only if you have a valid reason such as serious illness (not a slight cold) or a family emergency, and provide an acceptable, written excuse (not an email message), or you will receive a grade of zero. If possible (particularly if you want to be sure that your excuse is an acceptable one), contact me before missing an exam.

Math 435:01 Geometry , Fall 2021

General Comments on the Course

Course Material

Structure of Assignments

Assignment Grading Guidelines