Rational points on varieties in families, and countable unions of varieties over countable fields

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The investigator will work on two projects connected with arithmeticgeometry. The first project is to study the existence of rationalpoints in algebraic families of varieties, and to build a library ofdiophantine subsets of the field of rational numbers, where adiophantine set in this context means the set of rational parametervalues for which the corresponding variety in a family has a rationalpoint. In particular, the investigator will explore families whosefibers are Chatelet surfaces or more complicated conic bundles, forwhich the Brauer-Manin obstruction produces interesting diophantinesets. A long-term goal of the first project is to construct a modelof the integers using diophantine sets, because this would disprove aconjecture of Mazur regarding topology of rational points andsimultaneously prove the undecidability of the problem of decidingwhether a multivariable polynomial equation has a rational solution.The second project is to study countable unions of subvarieties over acountable algebraically closed field, such as the field of algebraicnumbers, and in particular to prove that in naturally occurringsituations, there exists a closed point outside the countable union,as required for various constructions. Examples include the union ofrational curves in a non-uniruled variety, the moduli space locus ofabelian varieties isogenous to a Jacobian, the locus in the base of afamily of varieties where the Picard number of the fiber jumps, andunions of subvarieties arising from iteration of endomorphisms ofvarieties.Arithmetic geometry lies at the intersection of number theory andalgebraic geometry: like algebraic geometry, it studies the solutionsto multivariable polynomial equations, but it does so under thenumber-theoretic restriction that the coordinates of the solutions beintegers (whole numbers like -37) or rational numbers (fractions like-3/5) or perhaps elements of some other number system different fromthe traditional systems of real numbers or complex numbers. Suchquestions were studied for their intrinsic interest since the time ofthe ancient Greeks, and in the 20th century they found unforeseenapplications to cryptography and error-correcting codes. Theinvestigator's research focuses not on these applications, but on thefundamental questions underlying and surrounding them, such as thequestion of whether it is possible to write a computer program todecide whether an arbitrary multivariable polynomial equation has asolution in rational numbers. The research covered by this grant willstudy patterns in families of equations in the hope of deducing anegative answer, while also proving the existence of solutionssatisfying infinitely many constraints in a larger number system.