Math 499w-02 - Senior Seminar - Spring
|TTh 2:30-4:00 p.m.
|As has become customary for the mathematics senior seminar, we
our own text, based on an outline provided by the instructor.
The Nature of This Course
The senior seminar is the capstone course for the mathematics major at
Willamette. As such, one of its purposes is to bring together
things learned throughout your mathematical coursework. While
this will inevitably involve specific content or facts you have
learned in prior courses, our primary focus in the seminar is to
increase and polish the student's ability to reason
mathematically, and to enhance the student's ability to reason independently.
Consequently, the specific content of any iteration of the seminar is
seen primarily as a vehicle - something to reason about.
In this edition of the seminar, our topic is Linear Fractional
Transformations or Möbius Transformations, also known
as bilinear maps. Topics include a brief introduction to
complex arithmetic and geometric interpretations thereof, the Riemann
Sphere or one-point compactification of the complex plane and its
geometric relationship with the complex plane, the geometry of linear
fractional transformations on the plane and on the sphere, and the
relationship of linear fractional transformations with the
Poincaré disk or half-plane models of hyperbolic geometry.
Being a seminar, this course will involve virtually no lecturing by the
instructor. Rather, most class sessions will consist of student
presentations of assigned problems or other material. As instructor, I will
provide an outline of the course content that has a great many
carefully-chosen and explicit gaps in its reasoning. Students
will work out how to fill those gaps and present their findings in
class. Early in the semester, the gaps will be very specific and
students will be assigned specific work to prepare in small groups for
presentation in class. Later in the semester the gaps will get
larger, and students will need to develop more autonomy in their
thinking. If things go well, we will come to work on the material
more collectively, with each student bringing to class the progress
they have made on the problems at hand, and the class as a whole
working through to solutions.
For each class session, two students will be assigned as scribes
to record (independently) what the class accomplishes and to write up (collaboratively)
the results using TeX / LaTeX. By the end of the course, we will have
produced a small book or monograph on the subject of Linear Fractional
Transformations, with the entire class as authors.
As the presentation assignments become less rigid, we will have a few
quizzes (four to six total). Some will be announced in advance,
some will not. These will test basic concepts previously covered,
to encourage students to keep
current with the course, and assess how well they are doing so.
Likewise after the first few weeks I will be requiring students to keep
a journal of their work for the class, to be collected periodically, so
that I can get a clear picture of the amount and quality of work you
are putting into the material.
In light of the forgoing, we have the following Course Goals:
At the completion of this course, the successful student will have demonstrated
- The ability to understand both abstract and concrete
mathematical reasoning, and the ability to differentiate between sound
and flawed mathematical reasoning.
- The ability to recognize incomplete or inconclusive
arguments, and the ability to recognize and fill in omitted details in
an otherwise correct but concise exposition.
- The ability to formulate and prove theorems that arise from the
definitions and concepts of the course content, and the ability to
apply those theorems to specific examples.
- The ability to present one’s mathematical proofs and arguments in
clear and compelling manner, both orally and in writing. This
includes the ability to organize one’s reasoning in a clear and
- The ability to carry out all of the above in a semi-autonomous
both in individual study and in collaboration with small groups.
|Write-ups of class results (When assigned as scribes)
|Quizzes and journals
Once we get beyond the assignment of specific problems to specific
students, your presentations will be evaluated both for their quality
and quantity. Strong but infrequent contributions may be
evaluated less generously than frequent contributions of moderate
quality. Regular and steady engagement with the subject is an
absolute requirement. After the first few weeks, when specific
material is no longer assigned to specific students, voluntary class
participation will be the only way to earn your presentation
grade. Therefore, to be successful, students must make every
effort to come to every class
prepared to discuss the material chosen for the day. Students
will receive feedback on a regular basis so that there are no surprises
in this regard.
Departmental Colloquium Attendance Requirement:
As is the case with all 400-level mathematics courses at Willamette,
students in this class are required to attend four of the semester's
departmental colloquia. The purpose of this requirement is to
instill an appreciation of the breadth and variety of mathematical
topics, techniques, people and (academic) programs. If scheduling
conflicts make colloquium attendance impossible, alternative short
written assignments will be made. If a student anticipates
needing this alternative for more than one colloquium session, they
should contact the instructor well in advance so that suitable
assignments can be prepared. Failure to complete this requirement
will result in a modest grade penalty for the course.
Accommodations for students with disabilities: Accomodations
required by students with disabilities will be provided upon reasonable
advance notice and verification of requirement/eligibility from the Office of Disability
Services (Bishop Wellness Center). If you forsee needing an accomodation, it is probably best to inquire at the Office of
Disability Services at the start of the semester.
Policy on in-class distractions and cell phones: It
is important to respect the concentration and attention of your fellow
in the class, especially in a seminar setting. Class time is
limited, precious, and the tuition is
quite expensive per minute. Arriving late for class is
severely frowned upon.
Electronic devices not
required for the course must be turned off during class time. If
your cell phone rings during
regular class time, you will be required to bring cookies or other treats for the entire
class at the next class meeting.
Academic Honesty Expectations
Quizzes are to be taken with books and notes closed
as noted on the quiz paper), completely on your own. Unannounced or
"pop" quizzes will probably allow you to use your notes (at the
instructor's discretion). A calculator may
be allowed, depending on quiz content, but palmtop computers, PDAs,
laptop computers, cell
phones, Blackberries, and especially any equipment that communicates
wirelessly will not be allowed in quizzes
For all other coursework,
you may (and should) discuss the
methods of approach, examples you have found, and even the solution(s),
with anyone. Because this class is intended as an experience of the process
of doing mathematics rather than a content-focussed
course, the use of outside written sources (including internet sources)
is discouraged except as noted in specific assignments. In any
case, any sources you use must be acknowledged explicitly in your work.
Plagiarism is the copying or paraphrasing of any work from another
proper written acknowledgement. All group members are responsible
for knowing all the sources their group's members used in completing an
In keeping with college policy, plagiarism will be reported to the
(see student handbook). Systematic or organized plagiarism will result
in course failure. If you are uncertain about some
aspect of the academic honesty policy, it is your responsibility to get
clarification from the instructor.
Last Modified January 16, 2011.
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