Reference number : GC-2017-Q1-01640, amendment number : , agreement type : G, recipient legal name : McKinnon, David (University of Waterloo), 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 fundamental problems of number theory is to describe the
 
set of rational or integer solutions to Diophantine equations, which
 
are polynomial equations in several variables with integer
 
coefficients. My research program investigates the distribution of
 
rational solutions to systems of Diophantine equations, in several
 
directions.
 
 
Paul Vojta has made some wide-ranging conjectures on what kinds of
 
solutions Diophantine equations should have, based on the geometric
 
properties of the solution sets of these equations. In my future
 
research, I propose to study these conjectures, to improve on my
 
previous proofs of various special cases of them, and to use existing
 
results to gain further insight into the solutions of Diophantine
 
equations.
 
 
In particular, I am interested in the distribution of rational points
 
on K3 surfaces. I have already proven many results in this area,
 
including (with Logan and van Luijk) a proof that the rational points
 
on many diagonal quartic surfaces are dense in the real and Zariski
 
topology, and a proof of the celebrated Batyrev-Manin Conjecture that
 
is conditional on Vojta's Main Conjecture.
 
 
I have, in joint work with Michael Roth, investigated how close two points with rational coordinates can get to one another, in terms of the geometry of the object the points lie on. Even more, we have obtained some results in which one of the points doesn't have rational coordinates, but instead has coordinates that are the roots of polynomials with rational coefficients. This has proven to be deep and interesting work, and I am continuing to work on proving more interesting results in this area., program purpose french : One of the fundamental problems of number theory is to describe the
 
set of rational or integer solutions to Diophantine equations, which
 
are polynomial equations in several variables with integer
 
coefficients. My research program investigates the distribution of
 
rational solutions to systems of Diophantine equations, in several
 
directions.
 
 
Paul Vojta has made some wide-ranging conjectures on what kinds of
 
solutions Diophantine equations should have, based on the geometric
 
properties of the solution sets of these equations. In my future
 
research, I propose to study these conjectures, to improve on my
 
previous proofs of various special cases of them, and to use existing
 
results to gain further insight into the solutions of Diophantine
 
equations.
 
 
In particular, I am interested in the distribution of rational points
 
on K3 surfaces. I have already proven many results in this area,
 
including (with Logan and van Luijk) a proof that the rational points
 
on many diagonal quartic surfaces are dense in the real and Zariski
 
topology, and a proof of the celebrated Batyrev-Manin Conjecture that
 
is conditional on Vojta's Main Conjecture.
 
 
I have, in joint work with Michael Roth, investigated how close two points with rational coordinates can get to one another, in terms of the geometry of the object the points lie on. Even more, we have obtained some results in which one of the points doesn't have rational coordinates, but instead has coordinates that are the roots of polynomials with rational coefficients. This has proven to be deep and interesting work, and I am continuing to work on proving more interesting results in this area., agreement title english : Rational points on algebraic varieties, agreement title french : Rational points on algebraic varieties, agreement number : RGPIN, agreement value : 150000.0, 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