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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.
Non-Canadian 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
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Diego Caudillo Amador,
Jonathan Berger,
Mark Chaim Freeman,
Eric Hu,
Jamal Kawach
Se-jin Kim,
Xiaozhu Li,
Georg Maierhofer,
Adam Mauskopf,
Shuyan Mei,
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Leanne Mezuman,
Roberto Hernández Palomares,
Pranav Rao,
Kimberly Stanke,
Kateryna Tatarko,
Minliu Wu,
ZihaoYan,
Ruoqi Yu,
Yushen Zhang
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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)
Research
Group:
Xiaozhu
Li, Leanne Mezuman, Kateryna Tatarko, Minliu Wu, Ruoqi Yu
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.
Resources for Research Group.
1. Our primary background text will be this.
It has a good selection of material on aspects of
spectral geometry for noncommutative spaces. I shall
cover all background and preparatory stuff in my lectures
and provide many examples. A good textbook on classical
spectral geometry is this.
During our meetings we shall also discuss recent papers
directly related to our project.
2. More advanced texts on noncommutative geometry
for your own self study include: this,
and this.
Project
2 - Modelling of Fetal Neurovascular Coupling
Supervisor:
Huaxiong Huang, Qiming Wang (York University)
Research
Group:
Mark Freeman, Adam Mauskopft, Shuyan Mei, Kimberly Stanke, Zihao
Yan
Brain injury acquired antenatally
remains a major cause of postnatal long-term neurodevelopmental
sequelae. There is evidence for a combined role of
fetal infection and inflammation and hypoxic-acidemia.
Concomitant hypoxia and acidemia (umbilical cord blood
pH < 7.00) during labour increase the risk for
neonatal adverse outcomes and longer-term sequelae
including cerebral palsy. The main manifestation of
pathologic inflammation in the feto-placental 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
one-week 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)
Research
Group:
Jonathan
Berger, Diego Caudillo Amador, Jamal Kawach, Se-jin
Kim, Yushen Zhang
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 self-absorbing 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.
Reading list:
Introduction to continuous model theory
1. Ben Ya'acov, Berenstein, Henson and
Usvyasatov,
http://math.univ-lyon1.fr/~begnac/articles/mtfms.pdf
2. Bradd Hart lecture notes: http://ms.mcmaster.ca/~bradd/courses/math712/index.html
Introduction to operator algebras with a
logic perspective
1. Ilijas' notes: http://www.math.yorku.ca/~ifarah/Ftp/singapore-final-r.pdf
More advanced reading:
1. Farah, Hart and Sherman: http://arxiv.org/pdf/1004.0741v5.pdf
Project
4 - Metric Arens irregularity
Supervisor: Matthias Neufang, (Carleton University) and Juris
Steprans (York University)
Research
Group:
Eric Hu, Georg Maierhofer, Roberto Hernández Palomares,
Pranav Rao
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 non-SAI
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?
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