The typical experience of a potential mathematics major who survives the calculus sequence with enough curiosity remaining to continue their study of mathematics, is to explore discretely the broad array of topics which constitute modern mathematics: linear and abstract algebra, real, complex, and functional analysis, algebraic and differential topology, geometry, probability, statistics, combinatorics, logic, set theory, number theory, as well as a myriad of focused applied subjects.

Unfortunately, to the undergraduate this often appears to be exactly what was intimated above: discretely chosen, if disconnected, subjects. However this is far from the truth. As one takes more advanced undergraduate and certainly graduate courses, one begins to see the interconnections between many of these subjects which makes mathematics so vibrant a subject.

It is the goal of this course to attempt to present some of the vista of modern mathematics, and some of the interconnections between some of these subjects at an early stage in your careers, which hopefully will invigorate your view of mathematics and its potential.

The broad theme of this offering of Math 17 is algebraic structures in mathematics and perhaps even in the world at large. We will develop a number of basic and pervasive algebraic and number theoretic tools with an eye to intertwining applications in algebra, geometry, cryptography, and number theory. A focal point will be to understand how certain curves in the plane (elliptic curves) have both algebraic and geometric connections to problems as diverse as Fermat's Last Theorem to major efforts in modern cryptography.

There is no official textbook for this course; class notes will be pulled from a number of sources many of which will be on reserve.

All students will write a research term paper, and present the contents of that paper to the class late in the term; the term paper is due on the last day of class. The paper should be approximately 10-15 pages in length, and cover a topic closely related to the course. A list of potential topics will be provided later in the term. Each student will have discussion of potential topics with the instructor who will offer guidance about what topics and scope are reasonable.

• Homework will be assigned weekly and will be due at the beginning of class on Wednesdays.


• All homework assignments will be posted on the course assignment's web page.


• Late homework is rarely accepted, and never without prior arrangement with the instructor. Starting assignments early will assure you have at least some work to submit for grading.


• Homework is to be written neatly using one side of 8 1/2 x 11 inch paper. Do not use paper from a spiral notebook unless you can tear off the ragged edge. All papers are to be stapled.


• Use English. If you can't read your solutions aloud as fluently as if you were reading your textbook, try using nouns and verbs in your write ups! Give references for theorems or propositions you use from the text and class.


• Consult the honor principle (below) as it applies to this course.

The course grade will be based upon the weekly homework, class participation, a term paper and its oral presentation. As there is no textbook for this course and part of your grade depends upon class participation, attendance is strongly encouraged. Grades will be determined as follows:

On Homework: Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. The honor principle applies to homework in the following way. What a student turns in as a homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. Students are discouraged from using solutions to problems that may be posted on the web, and as just stated, must reference them if they use them. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code.

Moreover, if in working with someone they have provided you with an important idea or approach, they should be explicitly given credit in your writeup. Hints I give in office hours need not be cited. Note: It is not sufficient to annotate your paper with a phrase like ``I worked with Joe on all the problems.'' Individual ideas are to be credited at each instance; they represent intellectual property.

On Term Papers: Term papers should be written in a manner consistent with the procedures and policies listed in Sources and Citations at Dartmouth College.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me, and I will be glad to help clarify things. It is always easier to ask beforehand.

I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me before the end of the second week of the term to discuss appropriate accommodations.

T. R. Shemanske
Last updated June 27, 2012 12:25:54 EDT