SCIENTIFIC PROGRAMS AND ACTIVITIES

The Fields Institute is hosting the Fields-Mitacs 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. Non-Canadian 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
Sergio-Iker Martínez-Juárez,
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).

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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.

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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 Fields-Mitacs 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 follow-up 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 signal-to-noise 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.

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Project title: Applications of logic to operator algebras
Supervisors: Ilijas Farah and Bradd Hart
Research Students: (Group report)

Kevin Carlson
Enoch Cheung
Alexander Gerhardt-Bourke
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.

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