Workshop
Organizers Visitors
Postdocs
1. ANNUAL PROGRAM ORGANIZERS
2. WORKSHOP ORGANIZERS
Lisa Fauci of Tulane University
(Mathematics) is one of the annual program organizers for the
199899 year in "Mathematics in Biology" She and
Shay Gueron of Technion  I.I.T. (Mathematics) express
the following:
As the organizers of the workshop on Computational Modeling
in Biological Fluid Dynamics that was held during January 2529,
we feel that it is our pleasant duty to thank you and the IMA
staff for this wonderful workshop.
The workshop was a success in all respects: friendly atmosphere,
high quality talks, fruitful scientific interactions, and extraordinary
logistics. Many of the participants addressed us during the
workshop and after it was over, and expressed their satisfaction.
All of them ranked the workshop as one of the best scientific
meetings in their entire academic career. We feel the same.
Although we would happily take the credit for this success,
we feel that it is your perfect organization that is actually
entitled to it.
We were extremely impressed by the friendliness, efficiency,
willingness, and resourcefulness of the entire IMA staff: meticulous
and professional organization contributed immensely to the smooth
operation during the week of the workshop, and the very cordial
staff generated the friendly and casual atmosphere which was
also an essential ingredient.
We would like to take advantage of this opportunity to particularly
acknowledge the important contribution of Dr. Fred Dulles to
the success of the workshop.
As the workshop's organizers and on behalf of the participants
and speakers  thank you again for organizing a wonderful event
which benefited our scientific community.
Philip Maini
of University of Oxford (Mathematics Institute) is one of the
organizers of the following sessions:
 IMA Tutorial: Mathematical and Computational Issues in Pattern
Formation, September 34, 1998
 IMA Workshop: Pattern Formation and Morphogenesis: The Basic
Process, September 812, 1998
 IMA Workshop: Pattern Formation and Morphogenesis: Model Systems
He writes:
I was at the IMA during the period SeptemberDecember, 1998.
During that time I ran, together with Hans Othmer, a 2day tutorial
and two oneweek long workshops on pattern formation. We are
presently nearing the completion of a proceedings volume that
came out of the workshops. I also ran a fourday long workshop
on mathematical modelling in cancer. This was an area I was
not involved in but subsequent to this workshop I have become
involved in it and, together with one of the participants, Dr
Byrne, we are now the UK partner in a European Network that
recently secured funding of approx 1.5 m dollars to work in
this area.
The very good computing and library facilities at Minneapolis
allowed me to make good progress in my own research and grant
writing activities. The workshops enabled me to interact with
coworkers and finish papers or begin new projects, and to meet
many new people in the field. Having a base in the US allowed
me to visit a number of collaborators within the continent.
I also had some interaction with the postdocs but I found that
being a visitor for only a term did not really give enough time
to develop proper collaborative projects.
The IMA, with its excellent support facilities, provided a very
good place to spend a sabbatical.
Lee Segel
of Weizmann Institute of Science (Applied Mathematics and Computer
Science) is one of the organizers of the IMA Period of Concentration
"Forging an Appropriate Immune Response as a Problem in Distributed
Artificial Intelligence" held on October 1923, 1998. He also
delivered a lecture as part of the School of Mathematics Ordway
Lecture on November 10, 1999. He writes:
One feature of the workshop that I organized was participation
of a relatively large number of U Minn faculty: N Papanikolopolous
(Electrical Engineering), M Mescher and Marc Jenkins (Immunology)
and Deborah Richards (Political Science). All have interests
in distributed autonomous systems. My own interaction was particularly
close with Jenkins. We had several conversations, which certainly
helped me with deeper understanding of various relevant aspects
of immunology. Jenkins asserted that he might do some experiments
based on some of my ideas. These are connected with "adjuvants"
that the immunologists use to boost otherwise weak immunological
reactions. Someone called adjuvants "the dirty little secret
of immunology" but in my view adjuvants can be viewed as the
vehicles by which the immune system adjusts its actions in accord
with feedbacks from sensing progress toward a variety of "goals".
If this view is correct, then there may be a long range contribution
here to vaccine design; this is what interested Jenkins. I talked
with several of the post docs, and offered some advice that
was nontrivial I believe, but not in any way decisive: to Tracey,
Patrick and Kathleen. Patrick mediated an interaction with Ron
Siegel, another U Minn faculty member, which resulted in some
genuine collaborative work, and a paper I believe, by Siegel
and another of the post docs.
Aside from general and informative discussions with IMA visitors,
I enjoyed getting up to date on the work of some of my distinguished
friends from U Minn, Dan Joseph (Mechanical Engineering), and
Chemical Engineers Skip Scriven, Gus Aris, and Bob Tranquillo.
I gave a lecture in Mechanical Engineering which seemed well
received. And of course I gave those Ordway lectures in mathematics.
Both in the Ordway lectures and in personal conversations (with
Naresh Jain, Wei Min Ni, Don Aronson, and Hans Weinberg) I pushed
my views on the importance of subjectmatter oriented applied
mathematics. I don't know what influence this had.
Annual Program Organizers
Workshop Organizers
Postdocs
3. VISITORS/SPEAKERS
Graham A. Dunn
of MRC Muscle and Cell Motility Unit, The Randall Institute
King's College London visits the IMA from January 130, 1999.
His report follows:
Analysing
the Marginal Activity of Moving Cells
My stay at the IMA began with the Workshop on "Cell Adhesion
and Motility" (Jan 48) during which I presented a talk on "Using
Microinferometry to Study the Function of the Cytoskeleton in
Cell Motility". This described a project to investigate the
dynamics of protrusion and retraction of the cell margin during
crawling cell motility. The DRIMAPS system of microinferometry
that we have developed enables the margin of living, thinly
spread, vertebrate cells to be located with unprecedented accuracy.
The crawling form of cell locomotion is ubiquitous among vertebrate
cells and plays a central role in embryogenesis and tissue repair.
The main protein involved in crawling locomotion is actin which
interacts with a host of other proteins and provides the basis
of the cell's cytoskeleton. Understanding the dynamics of disassembly,
relocation and assembly of actinbased structures is therefore
the key to understanding how vertebrate cells move. Earlier
work in our laboratory and in that of Wolfgang Alt (one of the
organizers of the Workshop) had demonstrated that temporal and
spatial correlations in the activity of the cell margin could
reveal interesting dynamical properties of the actin cytoskeleton
of the cell. In the case of temporal correlations, there is
often a strong positive correlation between the protrusion activity
of the cell margin and its retraction activity a short time
later after a lag of about one minute.
In collaboration with Michelle Peckham of the University of
Leeds, we have studied how these correlations are affected by
transfecting mouse myoblast cells with the human betaactin
gene. Betaactin and gammaactin are the two forms of actin
commonly present in nonmuscle cells and betaactin is preferentially
located near the cell margin which suggests that it may be more
directly involved in protrusion and retraction of the cell margin.
Transfection with betaactin increases its preponderance within
the cell and probably also leads to a down regulation of gammaactin.
The effects of this transfection on temporal correlations of
marginal activity were analysed during my stay at the IMA. Cross
correlations between protrusion and retraction were obtained
after prewhitening each time series to eliminate autocorrelations
which can give rise to spurious cross correlations. One effect
of the transfection is to reduce dramatically the positive correlation
between the protrusion and the retraction of one minute later.
Another effect is to double the speed of cell locomotion. It
is too early to understand the implications of these observations,
and more experiments will be needed, but it does suggest that
the absolute and/or relative sizes of pools of beta and gammaactins
within the cell may be critical parameters in the regulation
of cell motility. Total cell actin is known to be increased
in most cells when they are activated to perform motile tasks
and these observations indicate that the extent of this increase,
and/or the type of actin increased, may determine the type of
task which the cell is able to perform.
Frithjof Lutscher of Universitat
Tubingen (Lehrstuhl Biomathematik) is a graduate student who
participates in the following sessions:
 Workshop: Local Interaction and Global Phenomena in Vegetation
and Other Systems, April 1923, 1999
 Tutorial: Introduction to Epidemiology and Immunology, May
1314, 1999
 Workshop: From Individual to Aggregation: Modeling Animal
Grouping, June 711, 1999.
He writes:
I am in the middle of my Ph.D. thesis work in biomathematics
on the formation of fish schools through alignment. I found
out about the IMA's special program "Mathematics in Biology"
via the internet. It was clear to me that all of spring quarter's
activities would be very interesting for me, in particular the
last workshop on "Animal Aggregation." All the leading experts
in my field (Schooling and Alignment) were going to be there.
When I first asked about the possibilities to participate I
got an immediate and very encouraging answer. And from then
on the IMA staff was always very helpful during my preparation
of the stay with everything from finding housing to visa information.
I was lucky enough to get a fellowship from the DAAD (German
Academic Exchange Program) and arrived in Minneapolis in the
beginning of April. I was welcomed very warmly and everything
was already set up (office space, computer account,...) so that
I could get to work immediately.
I did participate in all the three workshops and found them
very inspiring. Most of the talks were very good. And at least
equally important was the fact that there was enough time in
between the talks for discussion with other participants. And
not only the time but also the space and an atmosphere to do
so was set up with white boards and tea and coffee. For me this
is much more efficient than on big conferences where twice as
many scheduled talks per day don't permit time to talk individually.
(The only time I found it difficult to follow was when there
were many short talks scheduled for some afternoons.)
In the time between the workshops I was able to get quite some
of my own work done. It was great to have so many experts around,
visiting professors at the IMA as well as faculty of the mathematics
department, and to be able to talk to them. The big open space
at the institute and the coffee breaks there support and facilitate
communication as well as the (joint) seminars. I certainly found
very good technical support (computer system).
In short: it was a very valuable time, inspiring for my own
work. I met many researchers and had fruitful discussions. And
I learned about many new ideas in mathematical modeling.
Annual Program Organizers
Workshop Organizers
Visitors
4. POSTDOCS
Kevin Anderson
reports:
Workshops
and Seminars
I spent the past year as a postdoctoral member of the I.M.A.
participating in the special year of emphasis on Mathematical
Biology. A large portion of my time has been spent participating
in the fifteen workshops which comprised the majority of the
year's programming. In these workshops I gained an outstanding
overview of current research in mathematical biology. This was
both through attending the workshops and tutorials, and through
interacting with workshop participants and long term visitors.
In particular, I enjoyed the workshops in the fall quarter,
which discussed developmental biology and immunity, and the
spring quarter which covered ecosystems and epidemics. Some
of the visitors with whom I had the most useful interactions
have were Carla Wofsy of the University of New Mexico, Phillip
Mani of Oxford University, Lee Segal of the Weizmann Institute
of Science, Yoh Iwasa of Kyushu University, and Mark Lewis of
the University of Utah.
I also participated in the I.M.A. postdoctoral seminar series
on Mathematical Biology. I also organized this seminar during
the winter quarter.
Talks
During the year I also gave three talks. On November 14, I spoke
on the topic ``Estimating Mean Time to Extinction: An Overview
of Some Popular Methods'' at the Western Regional Meeting of
the American Mathematical Society in Tucson Arizona. On April
2, I spoke on the topic "Measure Valued Markov Processes
and Bacteria" at the Probability Seminar of the University
of Minnesota's School of Mathematics, and on May 25 I spoke
on "Spatial Patterns in Gynodioecious Populations"
at the I.M.A. postdoctoral seminar series. Research
I spent a considerable amount of time during the beginning of
the year finishing up projects which I had started while a graduate
student in the Program in Applied Mathematics at the University
of Arizona. The work culminated in the submission of two papers.
The first, "A New Mathematical Approach Predicts Individual
Cell Growth Behavior Using Bacterial Population Information,"
which was written along with Dr Joseph Watkins and Dr. Neil
Mendelson, was submitted to Journal of Theoretical Biology.
This paper described work which I had done as a part of my dissertation.
The second, "A Test of Popular Methods for Estimating Mean
Time to Extinction" was written with Dr Wade Leitner of
the Department of Ecology and Evolutionary Biology at the University
of Arizona. This was submitted to the journal Conservation
Biology.
I also began some new research projects. With Claudia Neuhauser
of the University of Minnesota's School of Mathematics, I began
a theoretical study of bacteriocin producing bacteria. Bacteriocins
are poisonous substances secreted by some bacterial strains.
This is a purely spiteful behavior, as the production is lethal
to the bacteria who produce it. Systems in which bacteriocin
producing strains compete with bacteriocin susceptible strains
make very interesting models for studying competition between
species. This competition has been the focus of our study. Dr
Neuhauser was my faculty mentor, and we met weekly to discuss
our research. In addition, I participated in a graduate ecology
course on Spatial Processes in Ecology which Dr Neuhauser taught.
Another project upon which I worked, was one with Dr Neuhauser
and Dr Yoh Iwasa of Kyushu University's Department of Biology.
Dr Iwasa was a visitor to the I.M.A. during the spring quarter.
This work was initiated during his visit, and continued after
he left. In this project we studied the spatial patterning of
gynodioecious plant populations. Gynodioecious populations are
one which are comprised of both female and hermaphroditic plants.
When observed in nature, the female plants are often clustered
together. We asked what conditions of pollen and seed dispersal
lead to this clustering behavior. To answer this we studied
a system of differential integro equations which models the
population. We have prepared a manuscript on this work entitled
"Spatial Patterns in Gynodioecious Plant Populations"
which we plan to submit to Journal of Mathematical Biology.
A third new project upon which I worked this year was an analysis
of Gott's Formula. Gott's formula gives one confidence intervals
for the time remaining in an epoch, using only the time since
the beginning of the epoch as data. It is a completely nonparametric
method, assuming only that the observer occupies no privileged
place in time. I showed that one makes implicit assumptions
in using the method, and that it is therefore it is not as nonparametric
or as powerful as previously thought. This work resulted in
my submitting the paper "Implicit Assumptions in the Application
of Gott's Formula" to the journal Nature.
Conclusion
I feel that by participating in the special year on mathematical
biology at the I.M.A., I have gained a broad overview of the
field. I am now familiar with many more areas of current research
than I was upon finishing my graduate studies. I have also learned
new techniques for studying such systems, and met many of the
people who are actively researchers in the field of biomathematics.
I feel all of this will provide an invaluable foundation for
my career.
Bruce P. Ayati, formerly
of the University of Chicago, is completing the first of two
years as an IMA Postdoctoral Member.
Workshops
and Seminars
As a Postdoctoral Member of the IMA during the special year
Mathematics in Biology, my main responsibility was participating
in twelve weeks of workshops, seminars and tutorials. These
were meant to give a broad view of the diverse field of mathematical
biology, a goal which I feel was accomplished.
My main area of interest during the special year was population
biology, the theme for the spring quarter. The workshops on
vegetation, epidemiology, and animal aggregation each brought
different problems that had common themes. In particular, the
use of structured population models was prevalent in all three
workshops. Since my current research is in numerical methods
for structured population models, these workshops gave motivation
for future research as well as provided areas of application
for the current numerical methods.
A particular project I am currently working on is the derivation
and numerical solution of size and space structured forest models.
This is in conjunction with Claudia Neuhauser, who also coorganized
the vegetation workshop. As a result, the vegetation workshop
proved invaluable to my current research.
After the completion of the special year at the IMA, I attended
a NATO Advanced Study Institute on Mathematics Arising from
Biology at the Fields Institute in Toronto. This two week workshop
was also coorganized by Claudia Neuhauser. It consisted of
one week of ecology and another week of population genetics.
Presentations
The IMA has a Postdoctoral Seminar Series, which is organized
by the IMA postdocs but which gathers speakers and attendees
from the University of Minnesota as well as longer term visitors
to the IMA. I coorganized the spring seminar series with Ralf
Wittenberg.
I spoke during the winter postdoc seminar on Numerical Analysis
and Population Dynamics. In addition, I spoke at University
of Iowa Applied Mathematical and Computational Sciences Colloquium
on Galerkin Methods for PDE Models of Age and Space Structured
Populations and the University of Iowa Numerical Analysis Seminar
on BuGS for Parabolic Problems (some software I had written).
At the Fields Institute, I presented a poster on The Influence
of Discontinuities in the Population Density and Migration Rate
on Neutral Models of Geographical Variation.
Research
While at the IMA, postdocs are expected to pursue their own
research programs with the aid of University of Minnesota Department
of Mathematics mentors. My mentors are Mitch Luskin, a numerical
analyst, and Claudia Neuhauser, a probabilist who also holds
an appointment in the ecology department.
During my first year at the IMA, I completed and submitted three
papers. The first paper was a generalization of my thesis work
to higher order finite element spaces in age, Galerkin Methods
in Age and Space for a Population Model with Nonlinear Diffusion.
The second paper was the completion of work begun at the University
of Chicago with my thesis advisor, Todd F. Dupont, and Thomas
Nagylaki of the Department of Evolution and Ecology at the University
of Chicago. It was a population genetics paper entitled The
Influence of Spatial Inhomogeneities on Neutral Models of Geographical
Variation IV: Discontinuities in the Population Density and
Migration Rate, which is in press in Theoretical Population
Biology. The third paper was a work in numerical analysis on
an adaptive time stepping method for parabolic PDE's entitled
Convergence of a Stepdoubling Galerkin Method for Parabolic
Problems.
I am currently working on simulations of Proteus mirabilis swarm
colony development, a bacteria that forms interesting spatial
and temporal patters on petri dishes. My main project is the
derivation and numerical solution of size and space structured
forest models. This work is being done with Claudia Neuhauser.
Radu Balan reports:
From September 1998 till August 31 1999, I was an industrial
mathematics postdoctoral member working with the Computer Graphics
group at IBM, T.J.Watson Research Center, Hawthorne in New York.
For the first six months I stayed in residence at T.J.Watson
working with Gabriel Taubin on the problem he assigned to me.
More specific the problem was to find efficient compression
algorithms for 3D model geometry. The first method we tried
was based on an eigenfunction decomposition of the discretized
Laplace operator on a fictive uniform mesh. We came up with
an iterative eigenproblem solver adapted to the topological
surgery that is taking place on the mesh. Unfortunately, the
initial intuition was not correct and the algorithm turned out
not to compress the geometry, but rather to expand it. This
aspect, on the other hand, shed more light on some spectral
interrelations between coarser and finer meshes in the level
of detail sequence. Then, at around the time I moved in residence
to Minneapolis, we changed the approach and we looked for linear
filtering techniques on the sequence of meshes. It turned out
as a fairly robust and good compression method with possible
applications to some other problems as well (I would mention
the fairing  or smoothing  problem).
During this period, I regularly visited T.J.Watson to keep in
contact with Gabriel Taubin. I also had fruitful discussion
with several members if IMA or University of Minnesota. Firstly,
I regularly consulted with Fadil Santosa who gave me the idea
of using the simplex algorithm for solving an optimal linear
l1problem. Unrelated to the IBM project, but rather to a paper
that I finished writing this period, I had a very interesting
discussion with Scot Adams who suggested a topologic invariant
(some homology group) as the main obstruction to what is called
the BalianLow nonlocalization theorem. I also had some interactions
with the analysis group on the Monday's RivireFabes Seminars
on Real Analysis. A special mention is for Max Jodeit, the organizer
of these seminars. Lastly, but not least, I had interesting
discussions with Radu Marculescu, who is assistant professor
within the Electrical Engineering department. Our discussions
evolved several stochastic models of digital integrated circuits
(more specific, how the heat dissipation can be modeled on a
Markov chain).
During this year I gave a number of talks and I took part in
several workshops and conferences. First I gave some lectures
on wavelets at IBM T.J.Watson. These ran for about 3 months
during which I had the opportunity to find out of some other
research over there. At IMA I gave a talk at the Vertical Integration
Applied Mathematics Seminar (the Monday one) on the status of
the 3D geometry compression problem. Then I presented to the
analysis group, in a RiviereFabes seminar, a density result
concerning super WeylHeisenberg frames. I presented the same
result at the AMSMAA joint meeting in San Antonio, as an invited
speaker to the special session on harmonic analysis of wavelets
and frames. Toward the end of my stay at IMA, I gave a talk
in the Postdoc Seminar on a signal processing problem, namely
the identification of singular multivariate AR processes with
application to the blind source separation problem. This constituted
my research for Siemens Research Corporation, my next employer.
Finally, I should mention an extraordinary opportunity that
one of the workshops gave it to me. It all started in the first
days I moved to Minneapolis. An workshop on audition was just
about to start and, because of my Siemens project, I was curios
and particularly interested in the subject. However, in one
of the talk breaks I had the chance to meet Zeph Landau, then
a postdoc at UCSF. He happens to be one of the authors of a
particularly important paper on WeylHeisenberg frames, and
also an expert on von Neumann algebras. After a couple of talks
we realized we can do something in this field (WeylHeisenberg
sets) and we decided to write down our ideas. So now we are
about to write a lengthy paper on von Neumann algebra methods
applied to WeylHeisenberg sets (also known as windowed Fourier
transform). Recently we have been invited to a concentration
week at Texas A&M to present some results. There, at College
Station, we had very interesting discussions with the people
in the same field.
Nicholas Coult, IMA Industrial
Postdoc Industrial Partner: Fast Mathematical Algorithms and
Hardware Corp. He writes:
Following is a brief report on my activities for my first year
at the IMA, from September 1, 1998 to August 31, 1999. I have
broken the report into four parts:
 Industrial Research. Overview of research relevant to industrial
problem.
 Local collaborations. Description of collaborations with researchers
at the University of Minnesota.
 Papers. Brief descriptions of papers published, submitted,
or in preparation.
 Presentations. Brief descriptions of presentations given during
the year.
Industrial
Research:
I pursued research on the primary aspect of my industrial problem,
which is compression of threedimensional traveltime data for
seismic applications. The basic difference between this compression
problem and others such as image compression is that one would
like to keep the data in an appropriately compressed form so
that one may use it as a sort of compressed interpolation table.
I wrote prototype code to perform this compression using piecewisepolynomial
wavelets. During the course of developing and testing this code
it became clear that a number of improvements were necessary
both in the mathematics and the implementation. Specifically,
the relationship between the error level and compression ratio
was not as good as one would like. I would like to emphasize
that the biweekly industrial postdoc meetings played an essential
role in the process of my understanding these aspects of the
problem. I spent the remainder of my research time for this
problem on improving these parts of the algorithm, and recently
I have made some significant developments towards achieving
a dramatically improved compressionratio to errorlevel relationship.
Related to this problem is that of traveltime computation for
seismic problems. On this topic I was mostly a novice, so I
spent a good amount of time reading books and literature (and
occasionally discussing it with my industrial colleagues) to
better understand the state of the art. The goal is to eventually
design an algorithm which can compute traveltimes directly
in a compressed format. I did not expect to make much headway
on this problem during this year, but I am planning to pursue
this more vigorously next year.
Local
Collaborations:
As part of the IMA's industrial postdoc program I was assigned
Professor Bernardo Cockburn as my faculty mentor. With some
exceptions, we met regularly and discussed the state of my research
and also his research in the area of discontinuous Galerkin
methods for PDE's. The basis that I used for compression of
traveltime data is the same basis that he uses for DGM. Thus,
we saw potential for some collaboration. We continued meeting
regularly and discussing this matter, and I gave a talk at a
conference that he organized (the talk and associated paper
will be described later). We have committed to writing a paper
together on this topic in the coming months.
Papers:
During the first part of this year at the IMA, I submitted a
paper on computational quantum physics together with my Ph.D.
advisor Gregory Beylkin and Martin Mohlenkamp, a postdoc of
his. We submitted revisions in January, and the paper was published
this spring. I am continuing to collaborate with them remotely
on this research area. Following is the citation:
G. Beylkin, N. Coult, and M. Mohlenkamp. "Fast Spectral Projection
Algorithms for Density Matrix Computations." J. of Comp. Phys.
(152) 1999, no. 1, 3254.
This May I submitted a paper for the refereed proceedings of
the International Symposium on Discontinuous Galerkin Methods,
at which I also gave a presentation. The paper grew out of discussions
with Bernardo Cockburn, and gives a basic description of discontinuous
wavelets:
N. Coult. "Introduction to Discontinuous Wavelets." Accepted
for publication in Proc. Int. Symp. Discontinuous Galerkin Methods.
Finally, I am working on a manuscript which extends the discontinuous
wavelet construction to arbitrary triangles and tetrahedra.
This construction will allow the use of fast wavelet algorithms
for problems on nontrivial domains in two and three dimensions,
something which to my knowledge has not yet been feasible:
N. Coult "Triangular Multiwavelets for the Solution of PDE's
and Integral Equations." In preparation.
Professor Cockburn and I are planning on further collaboration
to tie together his work on discontinuous Galerkin methods with
the automatic adaptivity that wavelets provide. He and I, together
with his student Paul Castillo, will work together in the coming
year on this area.
Presentations:
During the fall, I was invited to give a presentation on wavelets
for the Math Department's Real Analysis seminar. The title of
my talk was "Waveletbased Homogenization of PDE's and Eigenvalue
Problems."
I was also invited to give a talk at Carleton College's Math/CS
colloquium. This took place in January. The title of this talk
was "Data Compression and Fast Algorithms Using Wavelets."
Finally, I was a plenary speaker at the International Symposium
on Discontinuous Galerkin Methods. The title of this talk was
"Waveletbased Discontinuous Galerkin Methods."
Trachette L. Jackson reports:
I was a postdoctoral member of the Institute for Mathematics
and Its Applications during the 19981999 year devoted to highlighting
mathematical challenges emerging from the consideration of issues
in the biological sciences. This general topic has been of great
interest to me for some time and is in line with the subject
of my doctoral dissertation, Mathematical Models in Two Step
Cancer Chemotherapy. After being focused on such a specific
aspect of mathematical biology during my graduate training,
I accepted this postdoctoral position in order to gain a broader
appreciation of the significant applications of mathematics
to medicine.
The IMA brought in world leaders in several areas of mathematical
biology to organize and participant in a selected series of
workshops and tutorials. The first three months of my tenure
were centered around developing new collaborations and studying
theoretical problems in developmental biology and immunology.
During this time I attended tutorials on mathematical and computational
issues in pattern formation, the physiology of the immune system,
and mathematical models of AIDS. These tutorials were an invaluable
introduction to the topics, terminology, and biology of up coming
workshops. Also during this time, I came to know several leaders
in the fields of pattern formation and mathematical immunology.
Applying mathematical models to the field of cancer biology
and treatment has been my main focus. Recognizing the high degree
of interest, the IMA arranged a minisymposium on this subject.
They invited a small group of clinicians, experimentalists,
and theoreticians to present their latest discoveries in the
area in hopes of stimulating crossdisciplinary collaboration.
I was invited to speak during this workshop to represent the
contributions of mathematical models to new chemotherapeutic
approaches. This minisymposium was extremely important to me
and resulted in several collaborations which are discussed in
detail below.
Collaborations
and Research Projects
In the field of mathematical biology, communication and collaboration
with experimental biologists and clinical researchers is crucial
to the development of realistic mathematical models which can
be validated by experimental data. I wanted to use my time at
the IMA to develop collaborations with experimentalist as well
as continue my training with senior mathematicians. To this
end I began work on four distinct projects.
Investigating
the Timescales Involved in the Angiogenic Switch:
With Carla Wofsy (Longterm Visitor) and Sundaram Ramakrishnan
(Dept of Pharmacology, UM)
Carla Wofsy of the department of mathematics at the University
of New Mexico was a long term visitor in residence at the IMA
for three months. We both became interested in tumorinduced
angiogenesis after attending a UM Medical School sponsored lecture
given by Judah Folkman, a world leader in the field. After a
bit of research into the subject, we sought out local researchers
who were studying vessel formation and antiangiogenic treatment
strategies. Sundaram Ramakrishnan of the University of Minnesota's
department of pharmacology, is associated with the University
of Minnesota Cancer Center and is designing novel therapeutics
to specifically kill endothelial cells in the tumorassociated
neovasculature. Carla and I began attending the weekly lab meetings
of Professor Ramakrishnan and his group. They showed us experimental
data which suggest that there are \textit{in vivo} differences
in the time it takes for cell lines which grow at the same rate
in culture to make the angiogenic switch. This led us to the
research project described below.
There are two distinct stages of growth during solid tumorigenesis
 the avascular stage and the vascular stage. During the avascular
stage, the tumor cells obtain nutrients from surrounding tissues
and can grow to a few millimeters in diameter, containing about
one million cells. As nutrient levels are depleted, the tumor
forms a necrotic core consisting of oxygen starved cells and
dead cellular debris; viable, proliferating cells are usually
only found in a thin layer near the periphery of the tumor.
There is some experimental evidence that it is these hypoxic
cells in the necrotic core that begin to secrete certain chemicals
which promote the formation of new blood vessels from the existing
vasculature in the surrounding tissue. When the generation of
new blood vessels is not regulated, this increased blood supply
can sustain the progression of many neoplastic diseases. We
are developing mathematical models that describe the mechanisms
by which an avascular tumor can make the transition to a vascularized
state. Particular attention paid to the timescales involved.
From the models and the analysis we hope provide an explanation
for the in vivo differences observed in the time it takes for
cell lines with similar growth dynamics in culture to make the
angiogenic switch. We also wish determine critical parameters
that lead to ways of controlling the angiogenic switch, thereby
forcing tumors to remain in the avascular state.
Mathematical
Modeling of Benign Tumor Encapsulation: With Sharon Lubkin (North
Carolina State University)
As I was finishing up my doctoral dissertation at the University
of Washington, I began looking for new problem in tumor biology
which I could investigate mathematically. I began reading papers
on benign tumor encapsulation and found that there is very little
experimental research on the underlying mechanisms which cause
the capsule to form. I later brought this modeling idea to Sharon
Lubkin of North Carolina State University's department of mathematics
during one of her many visits to the IMA; our work is discussed
briefly below.
The phenomenon of tumor encapsulation is well documented. In
fact, the presence of a capsule surrounding a tumor mass is
a significant morphological determinant of the clinical outcome.
Encapsulated tumors are generally benign and have a favorable
prognosis. Tumors which do not form this containing capsule
are generally malignant and the cells can invade the surrounding
normal tissue and even enter the blood stream to relocate to
other sites in the body. We are working to develop a mathematical
model which describes the process of tumor encapsulation. From
the model we hope to determine ways of eliciting the encapsulation
of otherwise invasive cancer cells.
The
Response of Vascular Tumors to Chemotherapeutic Treatment: With
Helen Byrne (University of Nottingham)
Helen Byrne of the Mathematics Department at the University
of Nottingham, UK, was one of the invited speakers for the minisymposium
on cancer. After hearing each other speak we set up a meeting
to discuss a possible collaboration. Our mutual interests include
tumor growth and treatment; a description of our first joint
effort follows.
There are several features inherent in the physiology of vascular
tumors which make them difficult treat successfully via blood
borne strategies. These barriers to treatment include the spatial
heterogeneity of the vascular network which nourishes the tumor.
This leads to high density regions of blood vessels to which
the chemotherapeutic agent is accessible, and low density regions
of blood vessels where the agent can reach in only small quantities.
Another barrier to treatment with a single agent is the frequent
mutation of cancer cells to phenotypes which are resistant to
the anticancer agent being administered. We are developing
mathematical models to study how vascular tumors respond to
specific chemotherapeutic treatment strategies. Particular attention
is paid to the dynamics that result from the incorporation of
a semiresistant tumor cell type. The models consist of a system
of partial differential equations governing intratumoral drug
concentrations and cell densities. In the model the tumor is
treated as a continuum of cells which differ in their proliferation
rates and their responses the the chemotherapeutic agent. The
balance between cell proliferation and death with the tumor
generates a velocity field which drives expansion or regression
of the spheriod. Insight into the tumor's temporal response
to therapy is gained by applying a combination of analytical
and numerical techniques to the model equations.
Spatiotemporal
Studies of the Mitotic Clock in Avascular Tumor Growth and Treatment:
With Helen Byrne (University of Nottingham)
After completing our paper on the response of vascular tumors
to chemotherapy, Helen Byrne and I decided to continue our collaboration
and we are now studying the effects of nutrient mediated cellcycle
progression on the growth and treatment of avascular tumors.
A brief summary is provided below.
It has been suggested that a single biochemical mechanism involving
the activation and inactivation of the maturation promoting
factor (MPF) by the protein cyclin underlies the progression
of all eukaryotic cells through the cell cycle. We are developing
and analyzing a mathematical model that incorporates the details
of the intracellular biochemistry of MPF and cyclin into a
model for avascular tumor growth. We hope to determine the spatial
variation in the cell cycle as we move toward the center of
the tumor where the nutrient concentration is lower. We then
wish to include the effects of a cellcycle specific drug and
a drug resistant cell type into the model.
Scientific
Contributions

Jackson, T.L., Lubkin, S.R, and Murray, J.D. Theoretical Analysis
of Conjugate Localization in TwoStep Cancer Chemotherapy.
(Accepted: To appear in J. of Math. Bio., 1999)
Revised
and accepted for publication while at the IMA.
 Jackson,
T.L., Senter, P.D., and Murray, J.D. Development and Validation
of a Mathematical Model to Describe Anticancer Prodrug Activation
by AntibodyEnzyme Conjugates. (Accepted: To appear in J.
of Theo. Med., 1999)
Revised
and accepted for publication while at the IMA.
 Jackson,
T.L., and Byrne H. A Mathematical Model to Study the Effects
of Drug resistance and Vasculature on the Response of Solid
Tumors to Chemotherapy. (submitted)
Started,
completed, and submitted for publication while at the
IMA.
 Jackson,
T.L., and Byrne H. A Spatiotemporal Model of the Mototic
Clock in Avascular Tumor Growth and Treatment. (In preparation)
Started
while at the IMA.

Lubkin, S.R., and Jackson, T.L. Mechanics of Capsule Formulation
in Tumors. (submitted)
Started
and submitted for publication while at the IMA.
Invited
Presentations
 Association
for Women in Mathematics, Minisymposium on Mathematical Biology.
In conjunction with the SIAM annual meeting, Atlanta, GA.
(May)
 Duke
University Mathematical Biology Fest, Center for Mathematics
and Computation in the Life Sciences and Medicine, Department
of Mathematics, Duke University, Durham, NC. (May)

Minority Access to Research Careers, Departments of Mathematics
and Zoology, Arizona State University, Tempe, Az. (April)

Cancer Minisymposium, Institute for Mathematics and Its Applications,
University of Minnesota, Minneapolis, MN. (November)
Accepting a postdoctoral appointment at the IMA for the special
year in Mathematical Biology was the best way I could have chosen
to begin my postgraduate career in the Mathematical Sciences.
During my one year stay, I developed several fruitful collaborations
with researchers here and abroad. As a result, I have coauthored
two papers which were submitted for publication during my appointment
and I have two more in preparation. Due to the diversity of
long term visitors and invited speakers, I have also come to
know the most prominent and influential scientists in the field
of Mathematical Biology and have broadened my own mathematical
and biomedical knowledge.
Bingtuan Li is one of
the IMA Postdocs. His report follows:
The year I spent at the IMA was really exciting, rewarding,
and productive. I spent part of my time attending workshops
and seminars, interacting with people, and part of my time doing
research. I learned a great deal about several areas, such as
pattern information, dynamics of the immune response, hormone
secretion and control, animal grouping, cell adhesion and mobility,
etc. I benefited very much from interacting with people from
different fields. The research I conducted at the IMA included
chemostat modeling, drug deliver modeling, and the spread of
competing species in the habitat, involving ordinary differential
equations, integral differential equations, difference equations,
and partial differential equations.
In the fall of 1998, I prepared a paper about a chemostat model
with two perfectly complementary resources and n species. This
was a continuation of my Ph.D thesis. The difference is that
my Ph.D thesis deals with models with one limiting resource.
I established predicted biological conditions for the survive
of one species and coexistence of two species for the model.
The results generalize those of Hsu, Cheng, and Hubbell (SIAM
J. Appl. Math. 41 (1981), 422444) and Butler and Wolkowicz
(Math. Biosci. 83 (1987), 148) where only two species are considered.
The difficulty was that the reduced system is no longer a two
dimensional system and therefore I had to examine a high dimensional
system. The key idea was to divide the relevant region into
disjoint sets. The global stability of steady states was then
captured on each set. While preparing this paper, I began collaborating
with Ronald Siegel (department of Pharmaceutics and Biomedical
Engineering, this university). The problem is about a model
for a drug delivery system. It involves negative feedback action,
with hysteresis, of an enzyme on a membrane through which substrate
diffuses to reach the enzyme. The model was proposed by Zou
and Siegel (J. Chem. Phys. 110 (1999), 22672279) and some primary
analysis and extensive simulations were given there. In the
joint work with Ronald Siegel, we gave rigorous mathematical
proofs regarding the global stability of steady states as well
as the existence and global stability of the limit cycle solution.
The globally asymptotic behavior of this model is fully understood.
The final version of a paper on the drug delivery model is in
preparation.
In the spring of 1999, I started studying a chemostat model
with two perfectly complementary resources, two species and
delay. The model incorporates distributed time delay in the
form of integral differential equations in order to describe
the time delay involved in converting nutrient to biomass. This
study was motivated by Ellermeyer (SIAM J. Appl. Math. 54 (1994),
456465) and Wolkowicz, Xia, and Ruan (SIAM J. Appl. Math. 57
(1997), 10191043) where only one limiting resource is involved.
I worked with Gail Wolkowicz (McMaster University) and Yang
Kuang (Arizona State University), and we prepared a paper. We
gave sufficient conditions to predict competitive exclusion
for certain parameters ranges and coexistence for others. We
demonstrated that when delays are large, ignoring them may result
in incorrect predictions. The main technique we used was the
linear chain trick which allowed us to transform the original
system into a huge system of nonlinear ordinary differential
equations.
Mark Lewis (University of Utah) visited the IMA in the spring
of 1999, and we had many discussions on the spread of two competing
species in the habitat. This led us to begin working with Hans
Weinberger (School of Mathematics, this university). We had
regular discussions every week. I learned a lot from our discussions
and from Hans Weinberger's early publications. Currently, we
are working on the spreading speed of twospecies diffusioncompetition
(continuous and discrete) models. In many applications, the
asymptotic speed of propagation of a nonlinear system is the
same as that of its linearization. This principle has been referred
to as the linear conjecture. Hosono (Bull. Math. Bio. 60 (1998),
435448) showed that for the twospecies diffusioncompetition
model, the linear conjecture is not true for certain parameters
by using extensive numerical simulations. We first gave a general
result for the discrete model, and applied that to the continuous
model (a nice feature of this analysis is that the continuous
model can be put in the form of discrete one). We obtained sufficient
conditions for various cases which guarantee the linear conjecture
is true. We then obtained the speed of propagation for each
case. We are now preparing a paper about these results.
I had several discussions with Carlos CastilloChavez (Cornell
University) about some models in epidemiology, and with John
Mittler (Los Alamos National Laboratory) about predatoryprey
models and chemostat models. I also had many interesting discussions
with George Sell and WeiMing Ni (School of Mathematics, this
university). All the discussions are very invaluable, and will
certainly influence my future studies.
I visited Arizona State University in April of 1999, and gave
a talk on the chemastat modeling in the mathematical biology
seminar there. I gave a talk about my Ph.D thesis in the IMA
postdoc seminar. I was invited to give a lecture in two international
conferences ( in Canada and Bulgaria). But I was not able to
make to the conferences.
During my stay at the IMA, I participated in other activities.
I became a reviewer for Mathematical Reviews and for Zentralblatt
fur Mathematick, and reviewed a number of papers. I refereed
papers for several journals, including SIAM Journal on Mathematical
Analysis, Proceedings of the American Mathematical Society,
Applied Mathematics Letters, and Journal of Theoretical Biology.
Xianfeng (Dave) Meng submits
the following report:
As a postdoctoral member for the year 19981999 on Mathematics
in Biology, I spent a large part of my time attending workshops,
seminars, and tutorials. Coming to IMA, my major goal was to
learn more about mathematical biology, to see how other people
use mathematics to explain biological phenomena. I feel I mostly
achieved that goal. Through attending the tutorials, workshops
and through interacting with the other workshop participants,
I became acquainted with various biological area that mathematics
plays an important role in. My knowledge was definitely broadened.
I saw many applications of mathematics in biology. For example,
reactiondiffusion systems are used in pattern formation to
describe how patterns are generated reliably in the face of
biological variation, systems of nonlinear ordinary differential
equations, partial differential equations, cellular automata
and stochastic processes are involved in the modeling of immune
systems and cell signaling, traditionally well studied NavierStokes
equations are used to model the swimming of some living organisms
and deterministic and stochastic mathematical models are used
to describe spectral patterns of plant communities, etc. The
opportunities of meeting leading researchers in their respective
field and of seeing how and what they are doing in their research
fields are priceless. It was an exciting year for me. I particularly
like the idea of tutorials. It gives us a chance to learn some
basic concepts which are not familiar to us and prepares us
for the workshop talks.
I started to collaborate with my mentor Dr. John Lowengrub in
the fall of 1998 on an interesting computational fluid dynamics
problem. We are considering a case of fluid flowing through
a channel with the fluid outside the channel flowing in the
opposite direction as opposed to the fluid inside the channel.
It is a two dimensional problem. The channel walls are assumed
immersed in the fluid. At first, We don't consider any tensions
that hold the the channel walls together against the moving
fluid. With this assumption, the channel walls will eventually
roll up. Later, we will add tensions to the walls to make them
more rigid to move. We are interested in comparing how different
numerical computational methods handle the rollup phenomenon
and how accurately the methods can simulate it. On my part,
I will use Immersed Boundary method to simulate the rollup
phenomenon. The Immersed Boundary method was proposed by Dr.
Charles Peskin of Courant Institute to model blood flow in the
heart. I have done the case without the tension in the channel
walls. Currently, I am still working on the case with tensions
added to the channel walls. The main difficulty I encountered
is the accuracy problem of my program. No paper has come out
from this work yet. We plan to continue our collaboration after
my job at IMA. I enjoyed working with Dr. Lowengrub a lot. As
my mentor, he has helped me both in my research and in my making
career choices. I like IMA's mentor system very much. As a newly
graduate just starting my career, I feel this system helped
me tremendously. My mentor's help definitely has a very positive
impact on my career development.
I also spent part of my time writing papers based on my dissertation
research. One is about proposing a computational model of porous
media at the pore level. It is going to be submitted to the
Conference Proceedings of the IMA Workshop on Computational
Modeling in Biological Fluid Dynamics. Another one is about
the comparison between the computational method I used and the
theoretical analysis. It is still under writing.
I participated the postdoc seminars and gave a talk in February.
I am always interested in mathematical applications in industry.
I attended all the talks of Seminar on Industrial Problems.
As an applied mathematician, I think it is always important
and also helpful to understand how mathematical researches are
done and what kind of mathematics is needed in industrial environment.
Personally, I think pursuing a career in industry is also a
very good option. It is as challenging as that in academia.
At the end of June, I had a job interview with Louisiana Tech
University. It is a regular tenure track position and I got
the job.
Looking back, I feel this is a pretty fruitful year for me.
I learned a lot in mathematical biology. I also established
some contacts with people which I hope will last. My oneyear
stay at IMA is definitely an important point in my career development.
It certainly will help to shape my future academic career.
Patrick Nelson writes:
The past year I have participated, as a postdoctoral fellow,
in the Institute for Mathematics and Its Applications (IMA)
year long program on Mathematical Biology. My year can
be sectioned into three areas.
Area
I
I participated in numerous week long workshops dealing with
a wide range of topics in the biological sciences. My participation
included attending lectures, panel discussions, poster sessions
and social events.
Area
II
I spent time developing a new collaboration with some of the
long term visitors. In the fall quarter, I worked with Dr. Byron
Goldstein, of the Los Alamos National Laboratory. This collaboration
is continuing and we are currently developing models which explain
the cellular response to allergens. I spent a week in March
visiting Byron at the Los Alamos National Laboratory to continue
this research. During the winter quarter, I started working
with Dr. Harold Layton, of Duke University. He introduced me
to models of renal flow in the kidneys. The modeling included
partial differential equation with delays, an area I was currently
working on with another project. This collaboration is continuing
as I am starting a position at Duke University as a postdoctoral
fellow in August and one of my projects will be the continued
work with Dr. Layton. Finally, in the spring quarter, I started
working with Dr. Jorge VelascoHernandez, of the Universidad
Autonoma Metropolitan. During a discussion we determined that
we were each working on models which could be combined to examine
Chagas Infection. This work will be continued after I arrive
at Duke University.
Area
III
The third area and most important was the continued work on
my research. Most of this time I continued my research in the
area of HIV modeling with Dr. Alan Perelson of the Los Alamos
National Laboratory. But there were a few other projects which
were completed.
Project
I
I completed and submitted the following work in June to The
Bulletin of Mathematical Biology on Models of macrophage activation
in response to pathogens.
The immune response to infection can be classified into two
compartments; innate and cellmediated. Macrophages, part of
the innate system, recognize and digest foreign particles. This
leads to a cascade of events, one of which is the signalling
of the cellmediate system. In the past decade, mathematical
models have become an integral part in the study of infection
and the immune response. Models have been developed which examine
the interactions of the innate and cellmediated system to infection
and have provided much insight into the disease dynamics. Unfortunatly,
there have only been a few mathematical works which focus on
the innate system's response. We study the changes in the dynamics
of macrophages in response to a pathogen and extend the previous
works by including two compartments for macrophages, resident
and activated. The model is then applied to experimental data
and estimates for certain, previously unknown, kinetic parameters
are obtained.
Project
II
I completed and submitted the following work in June to The
Journal of Emerging Infectious Diseases on Predicting the effect
of judicious antibiotic use on drugresistant Streptococcus
pneumoniae colonization among children in daycare. This work
is in collaboration with Tao Sheng KwanGett, M.D. and Dr. James
P. Hughes, both at The University of Washington. The objective
was to predict the effects of reducing antibiotic use on the
prevalence of drugresistant S. pneumoniae colonization in a
child daycare population.
Work
on HIV modeling included the following:
Publication
Myself and Dr. Alan Perelson had a paper published in March,
1999 in Siam Review title Mathematical models of HIV dynamics
in vivo.
Abstract:
Mathematical models have proven valuable in understanding the
dynamics of HIV1 infection in vivo. By comparing these models
to data obtained from patients undergoing antiretroviral drug
therapy, it has been possible to determine many quantitative
features of the interaction between HIV1, the virus that causes
AIDS, and the cells that are infected by the virus. The most
dramatic finding has been that even though AIDS is a disease
that occurs on a time scale of about 10 years, there are very
rapid dynamical processes that occur on time scales of hours
to days, as well as slower processes that occur on time scales
of weeks to months. We show how dynamical modeling and parameter
estimation techniques have uncovered these important features
of HIV pathogenesis and impacted the way in which AIDS patients
are treated with potent antiretroviral drugs.
Project
III A
Extensions of this work included examining a model which included
intracellular delays and recently we submitted, to Mathematical
Biosciences, "A model of intracellular delay used to study HIV
pathogenesis."
Abstract:
Mathematical modeling combined with experimental measurements
have provided profound results in the study of HIV1 pathogenesis.
Experiments in which HIVinfected patients are given potent
antiretroviral drugs that perturb the infection process have
provided data necessary for mathematical models to predict kinetic
parameters such as the productively infected T cell loss and
viral decay rates. Many of the models used to analyze data have
assumed drug treatments to be completely efficacious and that
upon infection a cell instantly begins producing virus. We consider
a model which allows for less then perfect drug effects and
which includes a delay process. We present detailed analysis
of this delay differential equation model and compare results
between a model with instantaneous behavior to a model with
a constant delay between infection and viral production. Our
analysis shows that when drug efficacy is not 100%, as may be
the case in vivo, the predicted rate of decline in plasma virus
concentration depends on three factors: the death rate of virus
producing cells, the efficacy of therapy, and the length of
the delay. Thus, previous estimates of infected cell loss rates
can be improved upon by considering more realistic models of
viral infection.
Project
III B
Concurrently with the above work we submitted to The Journal
of Virology a paper titled "Effect of the eclipse phase of the
viral life cycle on estimation of HIV viral dynamic parameters."
This work focuses on using potent antiretroviral therapy to
perturb the steady state viral load in HIV1 infected patients
has yielded estimates of the lifespan of virally infected cells.
Here we show that including a delay that accounts for the eclipse
phase of the viral lifecycle in HIV dynamics models decreases
the estimate of the productively infected cell lifespan. Thus,
productively infected cells may have a halflife that is shorter
than the estimate of 1.6 days published by Perelson et al.
Project
III C
I
am currently working on extensions of both models above and
am planning to submit, when completed, a detailed paper on the
analysis of the delay models to Siam journal of Applied Mathematics.
This work will be completed while I am at Duke University.
Professional
Activities
Besides research, the past year I coorganized a tutorial on
HIV modeling which was given in November of 1998. I was invited
to give a seminar at Arizona State University on my delay models
in HIV. I visited Dr's Goldstein and Perelson, at Los Alamos
National Laboratory in March. I was also invited to participate
in a week long workshop on Mathematical Biology given at Duke
University in May where I mentored a group of students on a
modeling project and gave a lecture on my current research.
I also spent time this year reviewing research articles for
The Journal of Theoretical Biology and Mathematical Biosciences
and I am currently reviewing a book, titled "Investigating Biological
Systems Using Modeling for the Society of Mathematical Biology.
To conclude, I found this year to be very productive and stimulating
and the most important aspect of this year and what I am most
proud of was the birth of our baby boy, Joshua in January.
Kathleen A. (Rogers) Hoffman
reports:
Workshops
and Seminars
As a Postdoctoral Member of the IMA during the theme year in
Mathematical Biology, my main responsibilities included attending
workshops, seminars and tutorials. This provided a unique opportunity
for me to gain a broad perspective of the latest developments
in the area of mathematical biology. This broad perspective
allowed me to sample the very different research areas that
exist in this broad field. Identifying interesting research
areas in mathematical biology is of particular interest to me
since I have previously worked in the field and I hope to continue
to expand my research program.
My biggest professional accomplishment this year culminated
in accepting a tenure track position at UMBC. After months of
submitting applications and traveling thousands of miles to
interviews, I was thrilled to accept the offer from UMBC. Not
only was it a tenure track position in an applied math department,
it was also in a location in which my husband found a job as
well. So although the seemingly endless hours dedicated to the
job search didn't produce any quantifiable mathematics (theorems
or papers), I believe that it was worth it since it produced
a (nonunique) solution to one of the biggest problems in mathematicsthe
two body problem!
Talks
In addition to participating in the workshops and seminars associated
with the theme year, postdoctoral members are expected to organize,
attend and speak at the weekly `Postdoc Seminar'. As with all
the postdocs, I gave a talk in the postdoc seminar. In addition,
I gave talks at the following series of talks:
 SIAM
Annual Meeting, Atlanta GA, May 1999

Mathematics Colloquium, Drexel University, Philadelphia, PA,
February 1999

NonLinear Science Seminar, Naval Research Lab, Washington
DC, February 1999

Mathematics Colloquium, George Mason University, Fairfax VA,
February 1999

Research Colloquium, Southern Methodist University, Dallas
TX, February 1999
 Mathematics
Colloquium, University of Florida, Gainesville FL, February
1999

Dynamics Seminar, Boston University, Boston MA, February,
1999

Mathematics Colloquium, UMBC, Baltimore MD, January, 1999

Mathematics Colloquium, Case Western Reserve University, Cleveland,
OH, January 1999

Postdoc Seminar, Institute for Mathematics and its Applications,
University of Minnesota, Minneapolis MN, January, 1999
I was also invited to give the keynote address at the annual
Sonia Kovalevsky Day at the University of Minnesota in October
1998. I found this particularly challenging since the audience
consisted of high school students, some of whom hadn't even
had algebra!
Research
My research accomplishments for this year encompassed three
separate projects. The first project involved research on the
stability of twisted elastic rods as a model for supercoiling
in DNA minicircles. The second project was an industrial problem
presented to the IMA by General Motors involving welding and
clamping of beams. The third project investigated a system of
four ordinary differential equations that serve as an idealized
model of two reciprocally inhibitory neurons.
DNA: A twisted elastic rod is widely accepted to be a qualitative
model of supercoiled DNA. Mathematically, a twisted elastic
rod is represented by an isoperimetrically constrained calculus
of variations problem. That is, the equilibria of the rod exactly
correspond to critical points of a certain functional subject
to integral constraints. Similarly, critical points which correspond
to constrained minima are said to be stable equilibria. My thesis
comprises a series of practical tests which determine which
critical points correspond to constrained minima, or equivalently,
which equilibria are stable. My research goals in this particular
area were to complete papers that were based on my thesis research.
During this year, one of the papers
 R. S. Manning, K. A. Rogers, & J. H. Maddocks, Isoperimetric
Conjugate Points with Application to the Stability of DNA
Minicircles
appeared in the Proceedings of the Royal Society of London:
Mathematical, Physical and Engineering Sciences Vol 454, No.
1980, p. 30473074, Dec. 1998. Additionally, a paper with Leon
Greenberg and John Maddocks
 L. Greenberg, J.H. Maddocks, & K.A. Rogers, The Bordered
Operator and the Index of a Constrained Critical Point.
was accepted to Mathematische Nachrichten.
During a trip to visit my advisor, the results for the final
paper from my dissertation were strengthened and generalized
to include stability exchange results at nonsimple folds as
well as simple folds. These generalizations required a significant
rewrite of the paper
 K.A. Rogers & J.H. Maddocks, Distinguished Bifurcation
Diagrams for Isoperimetric Calculus of Variations Problems and
the Stability of a Twisted Elastic Loop.
This last paper is still in preparation, but should be submitted
shortly.
 Welding and Clamping of Beams: Experiments on shells
have demonstrated that the sequence in which two shells are
clamped and welded affects the final shape of the shells. Such
a situation arises in assembling automobiles. In that setting,
the consequences of different final shapes can be costly if,
for instance, the final shape of the two shells (or automobile
parts) causes the larger structure not to meet required specifications.
In order to understand why this sequence dependence arises Dr.
Danny Baker and Dr. Samuel Marin of General Motors Research
and Development Center, Fadil Santosa, Associate Director for
Industrial Programs at the IMA, and I proposed models of clamping
and welding of beams which demonstrate this sequence dependence.
In the model that we propose, a series of rigid links connected
by torsional springs represents a simplified model of the beam.
Using the simplified model, we are able to show that the horizontal
sliding that occurs during the clamping process gives rise to
the sequence dependence. Although the model that does not allow
horizontal sliding, and hence does not produce sequence dependence,
can be solved analytically the model that does allow horizontal
sliding is solved numerically using a constrained optimization
routine from Matlab. Additionally, we are able to perform variational
simulations that statistically demonstrate that clamping and
welding the beams from the inside out produces the least amount
of variation in the final assembly.
Currently, we are in the process of writing an article
 F. Santosa & K.A. Rogers, A Simple Model of Sheet
Metal Assembly
that we intend to submit to the Education segment of SIAM Review.
In summary, we are proposing that this problem could serve as
one topics in an applied math topics class. The problem provides
a great mix of modeling, computation and simulation while solving
a real world problem.
 Reciprocal Inhibitory Neurons: Many of the behaviors
observed in the solutions of the Hodgkin and Huxley equations
can also be seen in simpler, yet still biologically reasonable,
models. In particular, simple models of the action potential
of neurons connected by reciprocally inhibited synapses have
been studied to further understand such biological phenomena
as heartbeat, swimming, and feeding. Two identical oscillatory
neurons connected by reciprocally inhibitory synapses will oscillate
exactly out of phase of each other, that is, while one neuron
is active the other is quiescent. John Guckenheimer, Warren
Weckesser and I studied an idealized model of a pair of reciprocally
inhibited neurons in the gastric mill circuit of a lobster.
Our goal is to understand solutions of a set of four differential
equations which model two asymmetric oscillators in terms of
geometric singular perturbation theory, an effective tool for
understanding equations with multiple time scales. Essentially,
singular perturbation theory pieces together solutions from
the fast system and solutions from the slow system to get a
solution of the singularly perturbed system. Behavior of a singularly
perturbed system consists of motion on the slow manifold (the
set of equilibria of the fast system) and fast jumps between
different parts of the slow manifold. These fast transitions
occur at folds in the slow manifold. A periodic solution to
the system of equations which describes a pair of identical
reciprocally inhibitory neurons can be described in terms of
singular perturbation theory as consisting of two fast transitions.
These fast transitions correspond to one neuron jumping from
an active to a quiescent state and the other jumping from a
quiescent state to an active state.
As we investigated the solution space of the asymmetric problem,
we found many solutions that were qualitatively similar to the
solutions of the symmetric system. We also found other very
different types of behavior. For instance, in a small parameter
range, there exists (at least) two stable periodic orbits of
the full system. Both of these periodic solutions correspond
to more complicated behavior than the typical reciprocally inhibitory
behavior described above. Instead of the orbit consisting of
two fast transitions, the periodic orbits consist of nine and
eleven fast transitions, respectively, and the behavior of the
two neurons can no longer be classified simply as active or
quiescent. The possible implications of the existence of two
stable periodic solutions as well as the structure of these
solutions are a source of continued research.
In addition to bistability in the system, we also found canard
solutions, that is, solutions in which part of the orbit occurs
on an unstable portion of the slow manifold. We identified two
different types of canard solutions. One type of canard solution
consists of a fast transition to an unstable part of the slow
manifold. In the other type of canard solution, the orbit continues
past a fold in the slow manifold onto the unstable part of the
manifold. For a small parameter range, the family of canard
solutions is stable. Continuation of the two stable periodic
solutions reveals that the canard solutions persist for a much
larger parameter regime but are unstable.
We have observed that existing integration algorithms have difficulty
computing accurate numerical representations of canard solutions.
The portion of the canard solution on the unstable part of the
slow manifold is simply too sensitive to small changes to be
computed using the most sophisticated initial value problem
sovlers. Instead, we have had limited success in calculating
canard solutions using AUTO, a package for solving boundary
value problems that uses continuation methods to track solutions.
However, even this approach is not without its computational
difficulties. For instance, the Floquet multiplier routine in
AUTO seems to have trouble with this problem, thus making stability
information difficult to obtain.
Marina Osipchuk, Industrial
Postdoctoral Member shares the following:
My second year as an Industrial postdoc affiliated with the
Honeywell Technology Center (HTC) I worked on disturbance rejection
control in decentralized systems. It was a joint project with
Dr. Michael Elgersma and Dr. Blaise Morton, HTC. We developed
an algorithm and implemented it in a software package that finds
a decentralized control achieving the maximum attenuation of
a disturbance signal.
I presented the results of our project on a research seminar
at the Honeywell Technology Center.
In addition I continued my collaboration with Edriss Titi and
Yannis Kevrekidis on finitedimensional control of reactiondiffusion
systems. I also gave an invited presentation on the results
at the Dynamical Systems and Control Seminar at the Aerospace
Engineering and Mechanics department, UMN. The further results
on this project were presented at the SIAM Conference on Applications
of Dynamical Systems.
Research
Disturbance Rejection Control in Systems with Decentralized
Control Architectures
Many industrial control problems are associated with the control
of complex interconnected systems such as those for electric
power distribution, chemical process control, and expandingly
constructed systems. Such systems are often characterized by
practical restrictions on information flow, rendering conventional
centralized control impractical. This system architecture calls
for decentralized control, wherein a given controller observes
only local subsystem outputs and controls only local inputs
and all controllers function in concert to regulate the composite
system. Compared to centralized control, this architecture provides
for implementation simplicity and tolerance of many types of
failures.
In practice, the decentralized control should demonstrate adequate
performance in the presence of disturbances as well as internal
stability. We designed a decentralized control that achieves
the global minimum of the system response to a disturbance output.
The developed efficient algorithm reduces the optimal control
problem to a sequence of polynomial systems and subsequently
formulate them as matrix equations. The solutions of the matrix
equations corresponding to all local minima were found using
eigensystem solvers. Simulations performed using this algorithm
for a variety of systems and disturbance types indicate that
it has significant promise for practical application to decentralized
control.
Overall, the two years spent at the IMA were productive and
stimulating for me. The environment of the IMA had a broadening
influence on my research. I have established several contacts
that I believe will have a strong impact on my career.
Anthony Varghese writes:
I was an industrial postdoctoral fellow working on projects
from Medtronic Inc. from Oct. 1, 1998 to Sept. 30, 1999. The
year was very fruitful for me as I have learned a great deal
about industrial collaborations. The industrial project I was
working on was the general area of atrial fibrillation. Atrial
fibrillation is a condition that afflicts a large number of
individuals and although not immediately lifethreatening as
ventricular fibrillation is, it can cause chronic fatigue and
can greatly increase the risk of stroke. Atrial fibrillation
takes place in the atria, the upper chambers of the heart and
is characterized by travelling waves of electrical activity
that appear unorganized compared to the regular pattern of activation
in the normal heart. Although there are drugs as well as implantable
devices made by companies like Medtronic to control atrial fibrillation,
both approaches have serious sideeffects. My task was to set
up models of cellular excitability to understand how atrial
fibrillation is initiated with a view towards coming up with
ways of terminating fibrillation. Given that there were two
recent cellular models of human atrial cell electrical activity
published last year, my project was to code these models and
study its properties. This approach has been taken since the
1960s when some prominent cardiologists together with Werner
Rheinboldt set up a coupled cellular automata lattice to study
spiral waves. The difference in my case is that much has been
learned in the intervening years about the properties of heart
cells and so instead of a discretestate cellular automaton
model for each cell, I used systems of coupled nonlinear ordinary
differential equations for each cell. Propagation of electrical
activity can be modeled by combining the above mentioned cell
models with a parabolic equation. A chronic condition like chronic
atrial fibrillation can be modeled by changing a number of parameters
based on experiments on cells from human hearts. The interesting
feature of atrial fibrillation is that it starts as short periods
of fibrillation that occur spontaneously but the more that these
short events occur, the more likely it is for the fibrillation
to persist. It is not clear why this is so.
In addition to modeling atrial fibrillation, I was also involved
in modeling the kinetics of ion channels with Linda Boland who
was in the Dept. of Physiology at first and who switched to
the newly formed Dept. of Neuroscience in June. I set up a Markov
state model based using a published model and used a robust
numerical scheme to handle the resulting stiff equations. Dr.
Boland acquires data from experiments and we were able to compare
the data with the results of the model. During the summer I
helped supervise an undergraduate who set up a scheme to search
in parameter space for model parameters that gave the best fit
in a leastsquares sense. Some of these results will be presented
in abstract form at a meeting on the Biology of Potassium Channels
in Sept. 99 at Colorado and a paper is in preparation. I also
worked with a pain researcher named George Wilcox on setting
up partial differential equation models of conduction of electrical
activity in primary afferent neurons responsible for conducting
pain signals. In addition the action of estrogen on certain
potassium channels were investigated along with a cardiologist,
Scott Sakaguchi, in the Medical School and these results were
presented as an abstract. I was able to collaborate very fruitfully
with a researcher in Bristol, UK, on the effect of genetic mutations
on a particular potassium channel that is responsible for deadly
arrhythmias of the heart. These results were published last
December. I also worked with experimenters in Oxford, UK, on
modeling certain arrhythmias related to excess nervous input
and this work has been submitted for publication.
Other
activities:
I was asked to recruit speakers for the Mathematical Physiology
seminars and managed to get speakers from Chemical Engineering
as well as Biochemistry and Physiology.
Conferences:
I was invited to a conference on Modeling and Defibrillation
organized by a Swiss cardiologist and Medtronic in Lausanne,
Switzerland in December 1998. I was able to present results
that indicated that simple square domains were insufficient
as far as being able to reconstruct atrial fibrillation. This
was an extremely useful and interesting meeting since it brought
together a small group of mathematical modelers with experimenters
and clinicians to discuss cardiac arrhythmias and the contribution
of modeling in particular. In June I was invited to a workshop
organized by Craig Henriquez at the Dept. of Biomedical Engineering
at Duke University on numerical methods for models of heart
electrical activity. This meeting brought together mathematicians,
computer scientists and engineers to examine numerical schemes.
I was surprised to find that a number of schemes relied at least
in part on the simple forwarddifference scheme. In July I visited
Linda Petzold to discuss the use of her numerical codes for
the problems I am investigating. Since I had been using her
codes since January, I had a number of very specific problems
to discuss and the meeting was very fruitful.
Publications:
(Work during the year)
1. Varghese, A., In Press for 2000,`Membrane Models', In: Biomedical
Engineering Handbook}, Second Edition, CRC Press, Boca Raton.
2. Hancox, J.C., H.J. Wichtel, and A. Varghese, 1998, Alteration
of HERG current profile during cardiac ventricular action potential,
following a pore mutation" Biochemical and Biophysical Research
Communications, vol. 253, pp. 719724.
Submitted:
1. Nash, M.P., J.M. Thornton, C.E. Sears, A. Varghese, M. O'Neill,
and D.J. Paterson, "Epicardial Activation Sequence During a
NorepinephrineInduced Ventricular Arrhythmia and its Computational
Reconstruction."
Abstracts:
1. Nash, M.P., J.M. Thornton, A. Varghese, and D.J. Paterson,
1999, "Electromechanical Characterization and Computer Simulation
of a Noradrenaline Induced Ventricular Arrhythmia." FASEB J.
13 (5) A 1075, Mar. 1999
2. Varghese, A., G.L. Wilcox, S. Sakaguchi, 1999, "Modulation
of I_Ks by Estradiol." European Working Group on Cellular Cardiac
Electrophysiology, September, 1999.
3. Hancox, J.C., H.J. Witchel, J.S. Mitcheson, and A. Varghese,
1999, "Insights into the Rapid Delayed Rectifier K Current,
IKr, from Action Potential Clamp Experiments." European Working
Group on Cellular Cardiac Electrophysiology, September, 1999.
4. Boland, L.M. and A. Varghese, 1999, "Immobilization of Shaker
Potassium Channel Gating Currents by the Beta Subunits Kvbeta1.1
and Kvbeta1.3" Biology of Potassium Channels: From Molecules
to Disease, September, 1999.
Warren Weckesser is another
IMA postdoc. He writes:
My activities this year focused on attending the many exciting
workshops in mathematical biology, continuing my collaboration
with Kathleen Rogers and John Guckenheimer on a study of a model
of two coupled neurons, studying some interesting properties
of an inverted pendulum with a rapidly vibrating suspension
point, and continuing my research on the stability of whirling
modes in rotating mechanical systems.
This year's theme of Mathematical Biology provided a fascinating
variety of mathematical applications in many fields of biology.
The workshops on pattern formation, immunology (including AIDS
and cancer), cell motility, renal physiology, ecosystems, epidemics,
and more have been an invaluable educational experience. My
next academic position is at the University of Michigan, and
while there I will maintain an active research program in mathematical
biology.
My primary research effort this year has been a collaboration
with John Guckenheimer and Kathleen Rogers on an intensive study
of the rich dynamical behavior found in a system of two coupled
relaxation oscillators. More specifically, we are considering
two nonidentical Van der Pollike oscillators. The coupling
is based on reciprocal inhibition, as occurs in membrane models
of neurons. Our model of two coupled neurons results in a singularly
perturbed system of differential equations, with two fast variables
and two slow variables. Our observations so far include several
families of complicated periodic orbits, a range of parameters
for which there are two stable periodic orbits, families of
orbits that exhibit a variety of canards (solutions that track
an invariant unstable slow manifold for long times), a possible
homoclinic explosion associated with a homoclinic bifurcation
from a periodic orbit, and several mechanism for the formation
of canards. One goal of this research is to classify the types
of bifurcations that occur in singularly perturbed systems with
more than two dimensions. We have gained great insight into
the importance of canards in the bifurcation of periodic orbits
in singularly perturbed systems. Especially important for this
work are the methods of geometric singular perturbation theory.
Another important component of the work so far has been the
numerical continuation of periodic orbits with the software
package AUTO. We are currently preparing a paper that focuses
on the numerical aspects of this problem; subsequent papers
will discuss bifurcation theory for singularly perturbed systems.
Kathleen and I presented a poster on this research at a conference
in mathematical biology at the University of Pittsburgh.
This year I began working on a new project with Mark Levi (my
thesis advisor and a visitor last year). We are considering
a generalization of the much studied pendulum with a rapidly
vibrating suspension point. In the classical problem, the suspension
point vibrates in a straight line. We are extending the analysis
to the case where the vibration is periodic, but the path of
the suspension point is an arbitrary closed curve. We find that
the averaged equations contain terms that have interesting geometric
interpretations. One is purely geometric, depending on the area
of the closed curve followed by the suspension point. Another
term can be interpreted as the effective force exerted by a
nonholonomic constraint, even though the full system is holonomic.
I have also continued my research on mechanical systems composed
of symmetric rigid bodies coupled with constant velocity joints.
This is a study that I began last year, during a visit by Mark
Levi. Unlike a universal joint, a constant velocity joint creates
a kinematic constraint that directly couples the angular velocities
of the rigid bodies. I am investigating the bifurcation and
stability of whirling configurations of chains of coupled rigid
bodies. This work may shed new light on certain gyroscopic phenomena
in spinning beams and related mechanical systems.
Aleksandar Zatezalo is
one of the industrial postdoctoral associates. He reports:
My postdoctoral appointment started on June 15th of 1998. During
199899 academic year I was working closely on problems and
development of passive surveillance system (PSS) as part of
the research and development group at Lockheed Martin Tactical
Defense Systems, Eagan. We derived and stated mathematical models
for the bistatic Doppler radar system in order to localize in
three dimensional space positions of flying objects in Twin
Cities area using several transmitters and determining direction
of arrival (DOA) of electromagnetic waves which scatter from
them by using standard beamforming techniques. We developed
algorithms and simulations for several scenarios of localizations
and tracking on the real trajectories, the so called geotracker
which for example localizes and tracks flying objects using
the data from six transmitters, four transmitters, or the direction
of arrival in conjunction with two transmitters. We also developed
algorithms for only tracking using simultaneously three transmitters
or one transmitter together with the direction of arrival. Straight
forward analysis of these algorithms is performed. We proceeded
with analysis of the real data collected in December of 1998
and January of 1999 since when we have been working on mathematical
models of the signals and determining statistical parameters
important for their extraction from the noisy environment. We
developed and implemented Bayesian update tracker using these
mathematical models and calculating statistical prediction by
applying the Alternating Direction Implicit methods where we
were dealing with problematic boundary condition by using Markov
process approximations. Since we needed the track initializer
and because of the computational complexity of the straight
forward Bayesian approach which partially comes from the uniform
noise in the phase which appears in the mathematical model we
developed simplified line tracker for tracking Dopper lines
which were appearing on the timefrequency grams. We are still
developing the simplified line tracker for the best possible
performance in order to compare our measurements with the data
which were collected from the radars located in Twin Cities
area and to associate signals which are coming from the different
transmitters but from the same scatterer.
The goal of our research is to demonstrate localization and
tracking of flying objects in Twin Cities area using several
transmitters and directions of arrivals by the end of the current
year.
By the end of the last year (1998) Professor Nicolai Vladimirovic
Krylov and I submitted paper under title A direct approach
to deriving filtering equations for diffusion processes
to Applied Mathematics & Optimization as natural continuation
of the research from my Ph.D. thesis. Several publications connected
with my work on industrial problems are in preparation.
I benefited from discussions with Professor Walter Littman,
Professor Fernando Reitich, Professor John Baxter, Dr. Marina
Osipchuk, Dr. Nicholas Coult, and Dr. Marco Fontelos.
Annual Program Organizers
Workshop Organizers
Visitors Postdocs
