UNDERGRADUATE PROGRAM
AN UPDATE ON OUR HONORS PROGRAM
We are very proud and excited
about the advances we have made
in our Honors Program over the
last few years. When Professor
Jay Goldman started the program
about fifteen years ago, his
initial function was to be an
advisor and mentor for students
capable of honors work, oversee
the development of the IT Honors
Calculus sequence, a hard problem
solving oriented course, and
recruit teachers for it.
Every fall Professor Goldman
runs an information meeting
for students planning to apply
to graduate school. However,
over the last few years, he
and the department have adopted
a much more activist role which
includes designing new honors
courses, coordinating programs
of different students so they
could get to know and motivate
each other, and finding study
and research opportunities for
undergraduates.
Our department has funded undergraduate
summer research programs for
several years (see the following
note). Professor Claudia Neuhauser's
Grant currently sponsors undergraduate
research in mathematical biology.
Our students have also participated
in NSF funded programs around
the country. Two students have
spent a semester in the well-known
Budapest Seminar Program in
Mathematics, which included
Paul Erdos among its founders.
The department recently initiated
a Math Club under the direction
of Professor Carme Calderer
and a Junior Colloquium organized
by Professor Richard McGehee
and advanced graduate students
John Hall and James Swenson.
The
new three semester theoretical
honors sequence, which was inaugurated
in the fall of 2002, has just
turned out its first graduates.
The subject of the first two
semesters is primarily linear
algebra and multivariable calculus.
Professor
Steve Sperber, who taught the
course during the academic year
2002-03, characterized it in
the last year's newsletter as
follows. The "...course...is
designed to help develop the
mathematical potential and ability
of promising undergraduates....
The sequence has a much higher
requirement of mathematical
rigor explicitly including proofs
in the treatment of topics than
in the analogous IT sequence."
The
course textbook is the superb
"Vector Calculus, Linear
Algebra, and Differential Forms"
by John H. Hubbard and Barbara
Burke Hubbard, which was developed
for the honors course at Cornell
University. A recent review
in the American Mathematical
Monthly said "It has a
breadth and depth that is rarely
seen in undergraduate texts,
and it teaches real mathematics
from a researcher's point of
view instead of the standard
off-the-shelf recipes that have
little use outside the classroom.
Its definitions and theorems
are carefully formulated so
that the essential content of
the results is clearly manifested."
Professor
Peter Webb is teaching this
course during the current academic
year. Professor Goldman says
that several of the students
in the course told him that
it was their most challenging
course, requiring a big time
commitment, but it was a really
fun course.
The
third semester of the sequence
is a topics course. In the fall
of 2003 the subject was differential
equations and dynamical systems,
with Professor Rick Moeckel
as the teacher. In addition
to students from the first year
of the sequence, the class included
some of our more advanced honors
mathematics majors. He used
the classic text Differential
Equations, Dynamical Systems,
and Introduction to Chaos, by
M. Hirsch and S. Smale, with
R. Devaney co-authoring the
2nd edition. For most of the
students, this was a first course
in differential equations, whereas
the book is really intended
for a more advanced second course.
But with such talented, hard
working and highly motivated
students, it was possible to
get everyone up to speed quickly
and to cover a lot of material,
with emphasis on proofs.
Professor's
Sperber and Webb have made similar
comments about the high quality
of the students.
THE
MATH CLUB
The Math Club has entered its
third year of meetings and activities.
In addition to being a welcoming
center for undergraduate students
majoring in mathematics, it
continues ongoing themes of
mathematics and careers explorations.
The
Math Club sponsors guest speakers
and visitors who meet with students
in an informal setting, over
pizza, to discuss and answer
questions about specific areas
of research and career issues.
During the past year, guests
included faculty members from
the School of Mathematics of
the University of Minnesota
as well as visitors. The 2002-03
list of visitors included Professors
Jay Goldman, Markus Keel, Fadil
Santosa, Peter Olver, Hans Othmer
and Arnd Scheel, as well as
Claudia Neuhauser from the Ecology
Department.
Participants of the IMA (Institute
for Mathematics and Its Applications)
thematic year also visited the
Math Club. Students got acquainted
with research activities in
Mathematical Optimization and
its applications to problems
in industry, such as design
of airplane wings and transportation
scheduling. Speakers from the
IMA included Professors Collette
Coullard (Northwestern University),
John Dennis (Rice University),
Lisa Miller (University of Minnesota),
William Cooper (University of
Minnesota) and Michael Powell
(University of Cambridge).
During the spring semester of
2003, the Math Club sponsored
two lunch gatherings in the
School Lounge open to faculty,
graduate and undergraduate students
of mathematics. The lunch meetings
provided undergraduate students
with the opportunity to meet
with faculty members and graduate
students. The meetings also
provided feedback to the organizers
of the Math Club on activities
of interest to math majors.
Currently the Math Club sponsors
weekly meetings in the computer
laboratory for students to become
acquainted with MatLab, by working
on special projects. The meetings
take place every Tuesday at
2:30 pm in 314 Vincent; students
are very much encouraged to
join the MatLab group. If you
are interested, please, come
to the Undergraduate Lounge
on Tuesday before 2:30 pm.
Participants in the current
IMA special year on "Probability
and Statistics in Complex Systems:
Genomics, Networks, and Financial
Engineering" are also scheduled
to visit the Math Club.
A day of career and graduate
school exploration in Mathematics
is scheduled for Saturday, April
15.
The Math Club wants to reach
out to all undergraduate math
majors. Please contact the organizers
with suggestions. Also, volunteers
are very much needed and sought
among faculty members, graduate
and undergraduate students of
Mathematics.
Carme Calderer, Professor of
Mathematics
RESEARCH
EXPERIENCES FOR UNDERGRADUATES
The department has offered REU's
for three consecutive summers:
2000, 2001 and 2002. Due to
the recent severe budgetary
constraints there was no departmental
REU program the past summer
2003, but Professors Ionut Ciocane-Fontanine
and Victor Reiner supported
REU's for several students from
their NSF Grants: Reiner supported
four students, one of whom was
a high school student, and Ciocane-Fontanine
supported two. By contrast,
there were 16 undergraduate
participants in 2000, and 18
in each of 2001 and 2002. There
were seven faculty mentors in
2000 (S. Adams, A. Friedman,
P. Garrett, L. Gray, R. Kuske,
V.Reiner, J. Roberts), four
faculty mentors in 2001 (S.
Adams, P. Garrett, R. Kuske,
V. Reiner), and five in 2002
(C. Calderer, P. Garrett, R.
Kuske, V. Reiner, A. Voronov).
Professor Avner Friedman coordinated
the program in the year 2000,
and Professor Paul Garrett did
so the following two summers.
Under their able leadership
it has achieved national recognition
as witnessed by the large roster
of top universities that have
sent participants. We should
not fail to mention the former
department Head, Professor Naresh
Jain, without whose vision and
faith in the validity of such
a large scale endeavor the program
could not have become a reality.
The
Summer REU program is intended
to introduce undergraduates
to genuine research environments
and to put them in close contact
with research mathematicians.
Working in teams or individually,
students carry out their own
projects, write reports on their
findings, and make presentations,
both within their own groups
as well as to all participants.
Some students have participated
for more than one summer and
many go on to pursue graduate
study in mathematics, including
in our department.
Typically,
the students were in residence
here for about 8 weeks and came
from all around the U.S., both
from top research universities
as well as from small colleges.
About a third of the participants
were from Minnesota, both from
the University as well as from
local colleges.
The
students mentored the past summer
by Professor Reiner were U of
M undergraduates Andrew Berget,
Ramon Calderer and Jesse Plautz,
as well a high school student,
Adil Ali. After some preliminary
study of graph theory and algebra,
the group set to work on two
projects in algebraic graph
theory. Calderer and Plautz
studied G-parking functions
and their relation to the spanning
tree number of a graph. They
also gave a simple proof of
a known relation between G-parking
functions and the Tutte polynomial.
In the spirit of some of Reiner's
previous REU's, Berget studied
the critical group of a graph,
which is an isomorphism invariant
of the graph that comes in the
form of a finite abelian group.
In particular, he investigated
the structure of the critical
group of regular line graphs.
This work was motivated by a
result relating the spanning
tree number of a regular graph
and its line graph. His work
is suggestive of a larger result,
currently under investigation.
A summary of the work of Professor
Reiner's group is available
online at http://www.math.umn.edu/~reiner/REU/REU.html.
Professor
Ciocan-Fontanine mentored two
students: Brian Jacobson and
Derek Lee, both from U of M
(in fact Brian had graduated
in June and is now a graduate
student at the U. of Michigan).
They both studied the same problem:
the "quantum Littlewood-Richardson
Rule". This is a hard open
problem with a rather elementary
formulation, and lies at the
intersection of Algebraic Geometry
and Combinatorics, although
the original motivation comes
from string theory in Physics.
It involves finding a positive
combinatorial description of
the number of rational curves
on a Grassmannian. Brian and
Derek explored the connections
between two different ways of
calculating these numbers (which
involve signs and cancellations)
and a recent conjectural rule
involving counting of "puzzles".
Although (as expected) they
didn't solve the problem in
general, one of them came up
with a new proof of a known
case - the quantum Pieri Rule
- and he also wrote a nice (and
fast!) computer program that
generates the puzzles mentioned
above. The program provided
a lot of data, which will be
a basis for future work on this
problem.
We
are grateful to Professors Ciocan-Fontanine
and Reiner for continuing for
another year what has almost
become a tradition in our department,
of providing deserving undergraduates
with an experience that lets
them feel what being a research
mathematician is like, and we
hope that in the near future
our Summer REU program will
be fully restored.
THE ACTUARIAL PROGRAM AT 21
The actuarial program was founded
in 1983, and this is a good
time to look back over the role
it has played in the department.
The total number of participant/graduates
now stands at 440, so we are
looking at a graduation average
of just about 20 per year, or
a participation average of about
40 at any moment of time. The
actual number of graduations
has been a fairly low-variance
random variable, measuring 22,21,19,18,17,27
since 1998 , after starting
slowly and peaking at 36 in
1993 . 56 % (247 of the 440)
have been IT Math majors, and
to put this in perspective,
we have graduated 892 IT Math
majors over the same 21-year
period. This puts the proportion
of actuarial students at 28%
of our majors.
We have placed the "source"
of each of our 440 into five
categories: IT math majors,
CLA math majors, Carlson School
of Management (CSOM) majors,
math graduate students, and
"other". Historically,
at 56% the IT math majors are
the backbone, with CLA coming
in at 19%, and CSOM at 11%.
The source vector for 2003 was
however (12,2,7,3,3), representative
of the clear trends that CLA
is slipping while CSOM is on
the rise. As those who have
followed my occasional reports
in past newsletters will recall,
in 1997 the CSOM introduced
a so-called "Actuarial
Science" major which has
cost us (and IT) a certain number
of students, while we continue
to teach the actuarial courses
taken by all program participants
regardless of college.
We also track our students by
success on Society of Actuaries
(SoA) Examinations and with
respect to employment, and have
observed historical performances
of 73% exam success and 58%
employment. I always consider
the conditional percentage "employed,
given success on an exam"
(79% = 58/73) not only more
favorable, but genuinely more
meaningful than the 58% ratio
cited. There are at most a half
dozen of the 440 who are in
the classification of "working
actuaries who have never passed
an exam". It is safe to
say that the IT students do
better in exam success, but
rather scary is the current
employment count on the class
of 2003: of 9 employed, 6 were
from CSOM. Of course, there
have always been cycles in both
participation and employment,
unfortunately rarely in phase.
At the moment jobs are tight
(note the blip in participation),
and even the conditional employment
rates are down (64% for 2002,
53% for 2003).
Recent stars of the program
include: Matt Gray on the threshold
of full Fellowship in the SoA
after graduating from IT in
2001; Paul Tschida (full Fellow
after graduating IT in 1999);
Brandon Welte (graduated IT
2002 , Associate of the SoA)
; and Caixia Ge (MS 2001, Associate
of the SoA). Paul is a distant
relative of a famous American
League umpire, and Matt is the
son of a certain department
member who bears the same last
name.
Reporting: Steve Agard, Program
Coordinator
PROFESSOR FRANK'S COURSE WEBSITE
EARNS PRAISE
In the January 2002 issue of
the Newsletter we surveyed the
department's substantial effort
in the area of teaching communication
skills. We suggested at the
time that, compared to the rest
of the university, we would
rank well in that respect. But
we would not have imagined that
our faculty would be complimented
on their work by our colleagues
at the University of Minnesota
Center for Writing.
In fact, a recent letter from
the Center to Professor David
Frank states: "you have
developed (and posted online)
exemplary instructional materials
that we have included in the
Teaching with Writing section
of our Web site.... Your materials
are currently serving as valuable
models to instructors on our
campus.... Our goal is to choose
models that enhance the role
of writing activities in student
learning and, at the same time,
allow instructors and students
to recognize ways in which writing
tasks and expectations differ
from discipline to discipline."
The material referred to is
the course website for Math
2374, IT Multivariable Calculus.
"I take credit for recognizing
that my TA's had talent, for
pointing them in the right direction,
and then for getting out of
their way," says Prof.
Frank, who was course supervisor
of Math 2374 from Fall 2001
through Fall 2003. "But
the real credit goes to my TA's,
especially Jonathan Rogness,
James Swenson, Dan Drake, and
Ryan Gantner. I maintained careful
oversight of their work, making
sure that what they did was
consistent with the goals of
the course. Naturally I was
responsible for producing the
syllabus and exams, and I always
participate with my TA's in
grading the exams, but they
deserve the credit for developing
the materials which are praised
by the Center for Writing. It's
gratifying to know that these
TA's have already make significant
contributions to undergraduate
education and will continue
to do so throughout their professional
careers."
In the summer of 2003, Prof.
Frank became Director of Undergraduate
Studies. Although he was nominally
the course supervisor of Math
2374 last Fall, in fact he turned
over control of the course to
Prof. Duane Nykamp (a new faculty
member) and Jonathan Rogness.
"Now the website is even
better," Frank said. "They
are continually improving the
labs and Nykamp has added WebCT
to the course. It's amazing
how something supposedly under
my control gets better when
I put talented people in charge."
SENIOR PROJECTS
Writing a Senior Project paper
is one of the ways for math
majors to satisfy the writing
requirement. All CLA math majors
are required to do a Senior
Project. A faculty member guides
the student in the writing of
the paper, which must be at
least ten pages long. It is
also required that the project
demonstrate acquisition of new
mathematical material. The goal
is to provide a capstone experience
in which the student combines
mathematics that he or she already
understands with some new mathematics
or applications at a similar
level. Extra one-on-one student-faculty
contact is an added benefit
of the senior projects.
Several faculty members have
recently supervised a number
of these projects. They are
listed below with the project
titles. Their dedication to
guiding students through this
intensive learning experience
is very much appreciated.
Steve
Agard: "Limiting Distributions
of Finite Markov Chains",
"Demystifying Calculus",
"The Interpolation of Reserves
Using Pseudopremiums",
"Renewal Theory",
"Topology: History and
Principles", "Error-correcting
Codes". John Baxter: "Self-Energy
of Charges", "Bayesian
analysis in game theory",
"Mathematical Structure
of Music", "The Central
Limit Theorem". Carme Calderer:"Numerical
simulations of ordering transitions
in liquid crystals", "Differential
equations and population models",
"Multivariable optimization:
sensitivity analysis".
Max Jodeit: "The properties
of the exponential function",
"Mathematics and music".
Karel Prikry: "On the completeness
theorem". Joel Roberts:
"Archimedes and pi (and
some discussion of later results)",
"Mathematics and perspective
drawing", "Comparisons
between Math 5335 and a recent
high school geometry text",
"High school math instruction:
curriculum, expectations, and
outcomes", "Alan Turing:
code breaking and computable
numbers", "Blaise
Pascal: his mathematical work
and later developments".
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