The investigator will work on two projects connected with arithmetic geometry. The first project is to study the existence of rational points in algebraic families of varieties, and to build a library of diophantine subsets of the field of rational numbers, where a diophantine set in this context means the set of rational parameter values for which the corresponding variety in a family has a rational point. In particular, the investigator will explore families whose fibers are Chatelet surfaces or more complicated conic bundles, for which the Brauer-Manin obstruction produces interesting diophantine sets. A long-term goal of the first project is to construct a model of the integers using diophantine sets, because this would disprove a conjecture of Mazur regarding topology of rational points and simultaneously prove the undecidability of the problem of deciding whether a multivariable polynomial equation has a rational solution. The second project is to study countable unions of subvarieties over a countable algebraically closed field, such as the field of algebraic numbers, and in particular to prove that in naturally occurring situations, there exists a closed point outside the countable union, as required for various constructions. Examples include the union of rational curves in a non-uniruled variety, the moduli space locus of abelian varieties isogenous to a Jacobian, the locus in the base of a family of varieties where the Picard number of the fiber jumps, and unions of subvarieties arising from iteration of endomorphisms of varieties.
Arithmetic geometry lies at the intersection of number theory and algebraic geometry: like algebraic geometry, it studies the solutions to multivariable polynomial equations, but it does so under the number-theoretic restriction that the coordinates of the solutions be integers (whole numbers like -37) or rational numbers (fractions like -3/5) or perhaps elements of some other number system different from the traditional systems of real numbers or complex numbers. Such questions were studied for their intrinsic interest since the time of the ancient Greeks, and in the 20th century they found unforeseen applications to cryptography and error-correcting codes. The investigator's research focuses not on these applications, but on the fundamental questions underlying and surrounding them, such as the question of whether it is possible to write a computer program to decide whether an arbitrary multivariable polynomial equation has a solution in rational numbers. The research covered by this grant will study patterns in families of equations in the hope of deducing a negative answer, while also proving the existence of solutions satisfying infinitely many constraints in a larger number system.
The bulk of the PI's work is in the area of number theory. This area, concerned with properties of whole numbers and their ratios, has been studied for its intrinsic interest since the time of the Greeks, but many of the basic questions (e.g., does a polynomial equation have a rational number solution?) remain unsolved. The development of methods to solve such problems has had the side-effect of leading to applications for security and transmission of data (reading data from DVDs without errors, satellite communications, etc.) The PI's project was not to develop such applications, but to develop the fundamental mathematics surrounding this area. Rational solutions to polynomial equations of degree 2 are fully understood, but the case of degree 3 has remained a mystery for over a century. The PI's work on Selmer groups has led to a new structural understanding of the solutions to such equations, by attempting to explain the statistical behavior of families of such equations instead of studying them one at a time. Also, the PI's work on explicit descent means that now even many equations of degree 4 are within reach. Another aspect of the PI's work concerns a method for finding the minimum of a function when one does not know the derivative. This method, called the Nelder-Mead method, has been used in scientific and engineering applications for over 50 years without knowing whether it works. The PI has given the first demonstration that it works for functions of two variables in a restricted setting.
