Topic

Hilbert's Tenth Problem

At the turn of the last century, in 1900, mathematician David Hilbert presented to the International Congress of Mathematicians a list of 23 open problems, the now-famous "Hilbert's problems." The tenth problem on the list was this: Find a method for determining whether a polynomial in several variables with integer coefficients has integer roots. This problem remained open until 1970, when it was solved by Yuri Matiyasevich, building on work of Julia Robinson and of Martin Davis. Actually, in Davis's words, the problem was "unsolved": Matiyasevich produced a proof that the problem is unsolvable. There is no method (that is, no algorithm) that will work.

In Math 98 we will work our way through an expository paper in which Martin Davis presents a complete proof of the unsolvability of Hilbert's tenth problem. [1] This paper assumes very little in the way of prerequisites, being intended, says Davis, for mathematicians who "when a long outstanding problem is finally solved [...] would like to share in the pleasure of discovery by following [...] what has been done [but too often are] stymied by the abstruseness of so much of contemporary mathematics." Depending on the interests of the class, we will explore both mathematical and historical/philosophical aspects of the problem, as well as finding out what related work has been done and what related probems still remain open.

As usual, students will prepare and present short talks, and each student will make a formal oral presentation and prepare a written report on a topic chosen by the student together with the instructor. (Students doing an honors project may submit their project in lieu of the final written report.)

[1] Davis, Martin, "Hilbert's tenth problem is unsolvable," American Mathematical Monthly 80, 1973, 233-629.

Scheduled Lectures

Groszek

MWF 8:45 - 9:50 (x-hour) Thu 9:00 - 9:50

Room TBA

Instructor

Professor Marcia Groszek

Office: 104 Choate House

Office Hours: TBA.

Homework Policy

• Homework will consist of preparations for in-class presentations, an occasional short written assignment, and one final report.

Grades

The course grade will be based upon class participation and the final report.

The Honor Principle

Your written report must acknowledge any sources you consult, as well as any help or ideas you get from any source. If you are unsure about what consultation is acceptable, please talk to the instructor.

Consulting the departmental writing editor on written reports is encouraged. If you do so, please submit the draft commented by the writing specialist along with your revised draft.

You can get any help you like on preparing class presentations. Collaboration is encouraged. If you get help in understanding the material from any person or source, you must acknowledge the source. You will not be penalized for consulting other sources.

Disabilities

Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see the instructor as soon as possible. Also, they should stop by the Academic Skills Center in Collis Center to register for support services.

Marcia J. Groszek
Last updated May 31, 2008 12:24:49 EDT