Reference number : GC-2017-Q1-03364, amendment number : , agreement type : G, recipient legal name : Shankar, Arul (University of Toronto), recipient country : CA, recipient province : nan, recipient city : Ontario, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : One of the most fundamental questions in Number Theory is: given an equation (in possibly many variables) with integer coefficients, how many integer solutions does it have? A general answer to this question will have a tremendous impact across the fields of Mathematics, Cryptography, and Computer Science. However, this question turns out to be extremely difficult. For example, we do not know how to answer it even when the equation is a polynomial equation of degree-3 in two variables. 
 
The question becomes somewhat more manageable if instead of considering only one equation, we consider an entire family of equations, and ask about the average number of solutions. The study of solutions across a family of equations is called Arithmetic Statistics. 
 
Equations may also be studied from a different approach. Namely, the solutions to some equations are intimately related to the analytical properties of complex valued functions, called zeta functions or L-functions. The study of such functions, and their relation to these equations is part of what is called Analytic Number Theory. 
 
Having worked in both the fields of Arithmetic Statistics and Analytic Number Theory, my proposal is to combine the tools of these two subjects to tackle fundamental problems in both fields. This will have a broad impact across Number Theory., program purpose french : One of the most fundamental questions in Number Theory is: given an equation (in possibly many variables) with integer coefficients, how many integer solutions does it have? A general answer to this question will have a tremendous impact across the fields of Mathematics, Cryptography, and Computer Science. However, this question turns out to be extremely difficult. For example, we do not know how to answer it even when the equation is a polynomial equation of degree-3 in two variables. 
 
The question becomes somewhat more manageable if instead of considering only one equation, we consider an entire family of equations, and ask about the average number of solutions. The study of solutions across a family of equations is called Arithmetic Statistics. 
 
Equations may also be studied from a different approach. Namely, the solutions to some equations are intimately related to the analytical properties of complex valued functions, called zeta functions or L-functions. The study of such functions, and their relation to these equations is part of what is called Analytic Number Theory. 
 
Having worked in both the fields of Arithmetic Statistics and Analytic Number Theory, my proposal is to combine the tools of these two subjects to tackle fundamental problems in both fields. This will have a broad impact across Number Theory., agreement title english : Arithmetic Statistics and Analytic Number Theory, agreement title french : Arithmetic Statistics and Analytic Number Theory, agreement number : RGPIN, agreement value : 125000.0, 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