SCIENTIFIC PROGRAMS AND ACTIVITIES

Students participating in the 2013 Program

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)

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 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.

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