The
Fields Institute is hosting the FieldsMitacs Undergraduate
Summer Research Program being held July and August of 2012.
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
25, 2012, 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. Some projects will be related to the Fields Thematic
Program on Inverse Problems and Imaging, the Focus
Program on Geometry, Mechanics and Dynamics, and the Legacy
of Jerry Marsden, and the Thematic
Program on Forcing and its Application. 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).
LIST OF PROJECTS
Project title: Toric varieties
Supervisors: Megumi Harada (McMaster) and Jessie
Yang (U Toronto and McMaster)
Research Students: (group
report)
Ana
Lucía Báez Camargo Aguilar
Alexander Flood
Darren Gooden
SergioIker MartínezJuárez,
Laura
Walton
Project description:
The study of toric varieties is a beautiful part of algebraic
geometry. There are many elegant theorems and connections
with convex geometry (via the theory of polytopes), combinatorics,
commutative algebra, symplectic geometry, and (equivariant)
topology. Toric varieties also have applications in many other
areas of research, such as physics, coding theory, and algebraic
statistics. On the other hand, the concreteness of toric varieties
provides an excellent context for learning some of the powerful
techniques of algebraic geometry for the first time. This
project will first introduce the student(s) to this subject,
and then will focus on obtaining a better understanding of
certain subtle phenomena within the theory of toric varieties
(which can happen when the variety is not, for instance, smooth).
==============
Project Title: Bayesian and Statistical Inverse Problems
Supervisors: Nicholas Hoell and Adrian Nachman
Research Students: Group
Report
Bilal
Abbasi
CarrieBragnalo
Feng Chi
Jiho Han
Iryna Sivak
Project Description:
Inverse problems have had numerous applications in physics
and engineering disciplines as well as in pure mathematics.
One area where they have played a crucial role is medical
imaging. This project will introduce the student(s) to many
key elements of inverse problems in medical imaging, particularly
through the source localization problem in quantitative electroencephalography
(EEG). Participants will gain significant background in modeling,
analysis, numerical analysis, probability theory and applied
statistics. This would be ideal for students with strong mathematical
training looking to learn more about important applications
and current research areas.
==============
Project Title: Understanding Financial Crises  a statistical
perspective
Supervisors: Bei Chen (McMaster), Matheus Grasselli
(McMaster)
Research
Students:
Yi
Lu
Francesc Rul·lan, (report)
Saúl Toscano Palmerin,(report)
Zixuan (Kevin) Wang
Camelia Yazdani
Project description: In their recent book "This time
is different" (2009), Reinhart and Rogoff describe a
rich dataset of financial crises including sovereign and domestic
default, currency devaluation, inflation bouts, bank runs
and stock market crashes over a span of 800 years and 66 countries.
As part of the 2011 FieldsMitacs Summer Research Program,
a group of students reviewed and compiled all publicly available
sources for the dataset described in the book, and independently
reproduce their statistical analysis of the most salient features
of financial crises, including a detailed comparison with
the 2008 crisis and its aftermath.
As a followup to this project, we plan to implement the
signals approach suggest in the book to obtain early warning
indicators for currency, banking and stock markets crises.
Indicators can then be ranked according to a variety of criteria,
such as their signaltonoise ratio, the persistence of the
signal, the probability of a crises conditioned on the occurrence
of a signal versus its unconditional probability, etc. Given
a ranking, one can construct a crisis index composed of a
sum of the indicators weighted by their performance, which
can be used as a macroeconomic input to traditional problems
in financial mathematics, such as optimal portfolio selection.
==============
Project title: Applications of logic to operator algebras
Supervisors: Ilijas Farah and Bradd Hart
Research
Students: (Group
report)
Kevin
Carlson
Enoch Cheung
Alexander GerhardtBourke
Leanne Mezuman
Alexander Sherman
This project will explore the growing interactions between
set theory and model theory, both branches of mathematical
logic, and the study of operator algebras  algebras of linear
operators acting on a Hilbert space. A wide variety of logical
tools can be brought to bear from descriptive set theory,
forcing and continuous model theory  all subjects to be studied
during the project. For example, we will we will study the
structure of C*algebras from the point of view of mathematical
logic and consider questions related to the asymptotic behaviour
of matrix algebras.
Some familiarity with basic logic would be helpful and a solid
grounding in linear algebra and analysis would be an asset.
PROGRAM
Activities start July 3, 2012 at 9:30 a.m. at the Fields Institute,
222 College Street. Map to Fields
If you are coming from the Woodsworth residence, walk south
on St. George to College Street, turn right, Fields is the
second building on your right.
Week
of July 36 
Jul
3


9:30
a.m.

Introductory
Session: Introduction and presentation of the program
(Fields Deputy Director, Matheus Grasselli)
Introduction to supervisors, and overview
of theme areas and projects 
11:00
a.m.

Coffee
break 
11:30
a.m.

Open
time for students to meet informally with supervisors.

12:30
p.m.

Lunch
provided at Fields for students and supervisors 
1:30
p.m.

Orientation
Meeting: Students meet with Fields program staff
Re: computer accounts, offices, expense reimbursements,
and overview of Fields facilities. 

*By
4 p.m. Hand in ranking sheet to Members Liaison, Sharon
McCalla, Room 330*

Jul
46

Students
will meet informally with supervisors and in their groups
to work on research project. 
Week
of July 913 

Students
will meet informally with supervisors and in their groups
to work on research project. 
Week
of July 1620 

Students
will meet informally with supervisors and in their groups
to work on research project.
Introduction to the Fields SMART board and video conferencing
facilities which are useful for remote collaboration.

Week
of July 2327 

Students
will meet informally with supervisors and in their groups
to work on research project.

Week
of July 30Aug. 3 

Students
will meet informally with supervisors and in their groups
to work on research project. 
Week
of August 710 (Note Aug. 6 is a Civic Holiday) 

Students
will meet informally with supervisors and in their groups
to work on research project. 
Week
of August 1317 

Students
will meet informally with supervisors and in their groups
to work on research project. 
Week
of August 20 24 

During
the final week, students are requested to prepare a report
on their projects and their experience in the Program
to be emailed to programs(PUT_AT_SIGN_HERE)fields.utoronto.ca
before August 24. These reports will be used in the Fields
Newsletter and Annual Report. 
Aug
22

MiniConference:
Undergraduate research students will present their work.


An
excursion  sponsored and organized by Fields  is planned
for all students. 
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