The
Fields Institute is hosting the Fields Undergraduate
Summer Research Program being held July and August of
2013. The program supports up to thirty students to
take part in research projects supervised by leading
scientists from Fields thematic programs or partner
universities.
Out
of town students accepted into the program will receive
financial support for travel to Toronto, student residence
housing on the campus of the University of Toronto from
July 1 to August 30, 2014, and a per diem for meals.
NonCanadian students will receive medical coverage
during their stay.
Students will work on research projects in groups of
three or four.
In
addition, supervisors will suggest other topics outside
of these fields. In some cases students may also have
the opportunity to spend a week off site at the home
campus of the project supervisor(s).
Students
participating in the 2014 Program
To be announced after April 15.
LIST
OF PROJECTS to
be announced shortly
Note: projects will be presented by
supervisors on the first day of the program. Students
will ballot their top three choices of project, and
can expect to be in your first or second choice.
Project
1  Spectral
Geometry in Fuzzy Domains
Supervisor: Masoud Khalkhali (Western University)
Spectral geometry is a branch of mathematics which
studies those properties of a space that can be encoded
in terms of eigenvalues of operators like Laplacian.
In a nutshell one wants to know what one can hear
about the shape of a space. The simplest such invariant
is the volume. This was discovered by Herman Weyl
about 100 years ago. Recently there has been some
progress in extending techniques of spectral geometry
beyond its tradiational doamin and to discrete objects
like graphs, to fractals, or even to much more singular
objects. A concrete problem is to develop these techniques
for singular spaces that are defined as limits of
matrix algebras.
Project
2  Modelling of Fetal Neurovascular Coupling
Supervisor:
Huaxiong Huang (York University)
Brain injury acquired antenatally
remains a major cause of postnatal longterm neurodevelopmental
sequelae. There is evidence for a combined role of
fetal infection and inflammation and hypoxicacidemia.
Concomitant hypoxia and acidemia (umbilical cord blood
pH < 7.00) during labour increase the risk for
neonatal adverse outcomes and longerterm sequelae
including cerebral palsy. The main manifestation of
pathologic inflammation in the fetoplacental unit,
chorioamnionitis, affects 20% of term pregnancies
and up to 60% of preterm pregnancies and is often
asymptomatic.
During the first two weeks
of the summer program, students will be introduced
to a recently developed mathematical model that couples
blood circulation with neural responses to investigate
the effect of umbilical cord occlusion on heart rate
variation as well as the development of acidemia in
the fetus. Stuednts will be asked to use the model
and run computer simulations under a variety of occulsion
conditions.
In third week of the program, students will be asked
to participate in a problem solving workshop on neurovascular
coupling and developing brain, and work with other
participants of the workshop to develop mathematical
models, based on their work during the first two weeks
of the program and experimental observations. They
will present a preliminary report at the end of the
oneweek workshop and continue to refine their model
during the reminder of the program, by comparing them
with experiment data when possible, and produce a
final report and present their findings during the
final week of the program.
Project
3  The
model theory of C*algebras
Supervisor: Bradd Hart (McMaster) and Ilijas Farah (York
University)
Model theory is a branch of
mathematical logic which studies classes of structures
or models of theories in the sense of logic. Traditionally
this logic has been classical first order logic and
the techniques of first order model theory have been
used successfully in many areas of algebra, number
theory and geometry. Recently a new logic called continuous
logic has been developed and it is more suited for
applications in analysis. One area of application
is that of C*algebras (algebras of operators acting
on a Hilbert space) and a concrete example of a problem
in this area is understanding the model theory of
strongly selfabsorbing algebras, a class of C*algebras
that have a central place in classification program.
Some familiarity with basic
logic would be helpful and a solid grounding in linear
algebra and analysis would be an asset.
Project
4  Metric Arens irregularity
Supervisor: Matthias Neufang, (Carleton University)
and Juris Steprans (York University)
Banach algebras are fundamental objects in functional
and harmonic analysis. One can think of the product
in a Banach algebra as a generalization of matrix
multiplication. In recent years the study of bidual
Banach algebras has been a very active field. Given
a Banach algebra A, its second dual carries two natural
products extending the multiplication on A, called
the left and right Arens products. If these coincide,
A is called Arens regular. If, on the contrary, A
equals the set of elements in the bidual for which
multiplication with respect to both products is the
same, A is called strongly Arens irregular (SAI).
All operator algebras are Arens regular, while the
group algebra of any locally compact group is SAI.
In this context, Z. Hu, M. Neufang and Z.J. Ruan
have introduced and started to develop
the concept of 'metric Arens irregularity' which measures
the degree of Arens irregularity through a numercial
value g(A), forming an isometric invariant of the
algebra A: indeed, g(A) is the supremum of the norms
of all differences of left and right Arens products
formed by elements in the unit ball of the bidual.
Obviously, g(A) is a
number between 0 and 2, and g(A)=0 precisely when
A is Arens regular. We have shown that g(A)=2 for
many SAI algebras A, but also that there
exist nonSAI algebras A with g(A)=2. Our study gives
rise to fascinating questions to be explored, e.g.,
can g(A) lie strictly between 0 and 2, and which values
are produced by Beurling algebras or algebras of operators,
particularly those that are neither Arens regular
nor SAI?
