Reference number : GC-2017-Q1-01582, amendment number : , agreement type : G, recipient legal name : Chen, Imin (Simon Fraser University), recipient country : CA, recipient province : nan, recipient city : British Columbia|Colombie-Britannique, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : Number theory, which is motivated by the study of numbers and equations, has existed since the dawn of recorded history. Perhaps because of its fundamental role in mathematics, number theory continues to be an active area of research, with a wide applicability and relevance to applications. 
 
This research program aims to study and answer some fundamental open questions in number theory concerning a natural class of diophantine equations known as modular varieties, with a view towards developing stronger methods for tackling the Fermat-Catalan conjecture, a natural generalization of Fermat's Last Theorem. 
 
The approach is fundamental science and problem motivated but will also involve the development of new methods and theory in the areas of Galois representations, modular symbols, and Frey abelian variety constructions. 
 
In terms of relevance to applications, the proposed research enhances our foundational knowledge concerning elliptic and hyperelliptic curves, number and function fields, as well as explicit aspects of representation theory. 
 
The expertise gained from the fundamental science component of this proposal will also be used to study the application motivated problem of homomorphic encryption and its resistance to quantum algorithms, which is relevant to maintaining privacy in big data applications., program purpose french : Number theory, which is motivated by the study of numbers and equations, has existed since the dawn of recorded history. Perhaps because of its fundamental role in mathematics, number theory continues to be an active area of research, with a wide applicability and relevance to applications. 
 
This research program aims to study and answer some fundamental open questions in number theory concerning a natural class of diophantine equations known as modular varieties, with a view towards developing stronger methods for tackling the Fermat-Catalan conjecture, a natural generalization of Fermat's Last Theorem. 
 
The approach is fundamental science and problem motivated but will also involve the development of new methods and theory in the areas of Galois representations, modular symbols, and Frey abelian variety constructions. 
 
In terms of relevance to applications, the proposed research enhances our foundational knowledge concerning elliptic and hyperelliptic curves, number and function fields, as well as explicit aspects of representation theory. 
 
The expertise gained from the fundamental science component of this proposal will also be used to study the application motivated problem of homomorphic encryption and its resistance to quantum algorithms, which is relevant to maintaining privacy in big data applications., agreement title english : Modular varieties, generalized Fermat equations, and special functions, agreement title french : Modular varieties, generalized Fermat equations, and special functions, agreement number : RGPIN, agreement value : 70000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada