SCIENTIFIC PROGRAMS AND ACTIVITIES

August 22, 2014

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

FIELDS UNDERGRADUATE SUMMER RESEARCH PROGRAM
July 2 to August 23, 2013

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 24, 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 Focus Program on Noncommutative Distributions in Free Probability Theory, the
Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry and Physics and the Focus Program on Commodities, Energy and Environmental Finance.

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 2013 Program

Reza Asad, Toronto
John Campbell, York
William Cook, Cambridge
Malors Emilio Espinosa-Lara, Guanajuato-CIMAT
Mark Freeman, Harvard
Jonathan Herman
, Western Ontario
Junrui (Thomas) Hu, Western Ontario
Boris Kadets, Kharkiv National
Vladyslav Kalashnyk, Kharkiv National
Xinyu Li, Minnesota, Twin Cities
James McVittie, Toronto
Meghan Miholics
, McMaster
Alexander Molnar, Queen's
Jiewon Park, Seoul National
Ashwath Rabindranath, Princeton
Nigel Sequeira, McMaster
Paul Sacawa, Toronto
Iryna Sivak, Taras Shevchenko National University of Kyiv
Yihui Tian, Toronto
Kimsy Tor, Manhattan College
Junho (Peter) Whang, Queen's

LIST OF PROJECTS
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 - Modular forms around string theory
Supervisor: Noriko Yui (Queen's University)

Description: This project will include topics ranging from modularities of Galois representations, potential modularity of families of Calabi-Yau varieties, arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, geometric modularity of families of Calabi-Yau varieties, Mathieu/K3 moonshine, automorphic black hole entropy, physicial implications of special values of L-functions of Calabi-Yau varieties.

Modular forms and more generally automorphic forms appear in various places in string theory landscape. Some are more traditional than others. For instance, the modularirty or automorphy of Galois representations of Calabi-Yau varieties defined over number fields is one of the most prominent open problem in modern number theory, known as the Langlands Philosphy. For geometric modularity question, we aim to describe moduli spaces of K3 surfaces and Calabi-Yau threefolds in terms of some modular invariants.

The other modularity problems stem from string theory. Often, generating functions of counting some physical quantities turn out to have modular properties. These include Gromov-Witten invariants, Donaldon-Thomas invariants, wall-crossing numbers, and most recently, black hole microstate counting, etc, where various modular forms (elliptic modular forms, Siegel modular forms, and automorphic forms) enter the scene. To enhance our understanding why modular forms play prominent roles in string theory will be one of our main goals.

*Course information for Project Participants*


Project 2 - The spatio-temporal spread of social media
Supervisor: Jianhong Wu (York University)

Project Presentation slides

Description: This project aims to examine simple differential equations to characterize the spatiotemporal patterns of diffusion in online social networks using established mathematical methodologies for biological invasion in ecology, and to adapt the modern theory of mathematical epidemiology to quantify the influence and transmission dynamics of news along social networks.

Modular forms and more generally automorphic forms appear in various places in string theory
landscape. Some are more traditional than others. For instance, the modularirty or automorphy of Galois representations of Calabi-Yau varieties defined over number fields is one of the most prominent open problem in modern number theory, known as the Langlands Philosophy. For geometric modularity question, we aim to describe moduli spaces of K3 surfaces and Calabi-Yau threefolds in terms of some modular invariants.

The other modularity problems stem from string theory. Often, generating functions of counting some physical quantities turn out to have modular properties. These include Gromov-Witten invariants, Donaldon-Thomas invariants, wall-crossing numbers, and most recently, black hole microstate counting, etc, where various modualr forms (elliptic modular forms, Siegel modular forms, and automorphic forms) enter the scene. To enhance our understanding why modular forms play prominent roles in string theory will be one of our main goals.

Project 3 - Logic and operator algebras
Supervisors: Ilijas Farah (York University) and Bradd Hart (McMaster University)

Project Presentation Slides

Reading list and resources

Description: 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.

Project 4 - Dynamical Systems Models in Macroeconomics
Supervisor: Matheus Grasselli (McMaster University)

Project Presentation Slides

Description: The role of the financial sector is poorly understood in DSGE (Dynamic Stochastic General Equilibrium) models, the dominant paradigm in modern macroeconomics. This project will attempt at a systematic review of several alternative modelling approaches, primarily based on dynamical systems. These include the well-knonw Goodwin model, a predator-prey system describing the wage share and employment rate, and several of its extensions incorporating banks, households, governments, foreign sectors, etc. An overarching theme throughout the project is the endogenous fragility created by cycles of leverage and deleverage and the corresponding periods of bubbles and crashes. Students are expected to summarize and compare the mathematical properties of different models (equilibria, local stability, bifurcation, etc), address possible inconsistencies, identify open problems, and hopefully obtain new results.

No previous knowledge of finance or economics beyond basic concepts is required, but a strong foundation in differential equations, linear algebra and probability is essential.


PROGRAM
Activities start July 2, 2013 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 2-5
Jul 2
 
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
2:30 pm Orientation Meeting: Students meet with Fields program staff
Re: computer accounts, offices, expense reimbursements, and overview of Fields facilities.
3:00 p.m. Fields Tea break
Open time
4 p.m.

*By Hand in ranking sheet*

Jul 3-5

Students will meet informally with supervisors and in their groups to work on research project.
Week of July 8-12
  Students will meet informally with supervisors and in their groups to work on research project.
Week of July 15-19
  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 22-26
  Students will meet informally with supervisors and in their groups to work on research project.
Week of July 29-August 2
  Students will meet informally with supervisors and in their groups to work on research project.
Week of August 6-9 (Note Aug.5 is a Civic Holiday)
  Students will meet informally with supervisors and in their groups to work on research project.
Week of August 12-16
  Students will meet informally with supervisors and in their groups to work on research project.
Week of August 19- 23
  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 23. These reports will be used in the Fields Newsletter and Annual Report.
Aug 21
Mini-Conference: Undergraduate research students will present their work.
Aug 22
An excursion - sponsored and organized by Fields - is planned for all students.

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