The PI proposes to extend recent work with Belabas and Bhargava on counting the number of algebraic number fields within the complex numbers of a given degree over the field of rationals, and with the absolute value of the discriminant below a given bound. This work also has an algorithmic side where one counts the exact number and not just a main term with an error estimate. In addition, the PI proposes to study the chaotic behavior of the multiplicative order function. Here one studies the arithmetic function which assigns to an odd natural number the multiplicative order of 2 to this modulus (with obvious generalizations). This function appears intrinsically in many problems, and it has important applications in cryptography. In these projects and others in the proposal, the PI will involve graduate students, undergraduates, and junior faculty, as he has done with success in the past.
The rational numbers (fractions) are so basic that they are taught in elementary school. For the past few centuries, number theorists have found natural ways of expanding them to larger domains that involve throwing in roots of particular polynomials. The program to classify these new fields was begun by Gauss over two centuries ago, where he was able to solve the problem for the case where the polynomial has degree two. In the past century we learned how to do this for degree three, and in the past decade for degrees four and five. Building on this work, the PI proposes a deeper study of these fields from a statistical point of view, and also an algorithmic point of view. We have very few hard data in connection with higher degree fields, and the new perspectives for counting them seem ripe for development as computational tools. In addition, the PI proposes to study the lengths of the periods of certain cyclic processes that are important in cryptography. These lengths have chaotic behavior (for nearby parameters, the lengths can be wildly different). There have been certain conjectures proposed for both the normal lengths of these cycles, and the lengths on average. The PI hopes to settle some of these conjectures, perhaps in the negative. In these projects and others in the proposal, the PI will involve undergraduates, graduate students, and junior faculty. The problems proposed are fundamental and some have been studied for centuries. Some of the more recent problems are entwined with cryptography, a topic of great importance in the world's economy.
This grant supported a number of interesting projects in number theory. Here are a few specifics: (1) With Belabas and Bhargava, we published a paper detailing the first power-saving error terms for the distribution of cubic and quartic fields. Let me explain. A polynomial can be used to define an extensiion of the familiar field of fractions that we all studied in school. For example x^2-2 enlarges that field by allowing the squareroot of 2 to be included. Gauss gave us a way of classifying all such fields of degree 2, like in this example, but higher degree fields have proved to be much thornier. Davenport and Heilbronn in the mid 20th century found the broad outlines for cubic fields, and Bhargava did the same for quartic fields about a decade ago. In this paper we showed that these older estimates are extremely accurate. The work has already spurred further research. (2) With Kurlberg and Lagarias we almost completely solved an interesting problem about product-free sets. We are given a large subset of the positive integers such that no product of two members is again in the subset. E.g., one can take the numbers that leave a remainder of 2 when divided by 3. In this example we have 1/3 of the integers. In our paper we very accurately pinpoint how "dense" such a set may be: in fact, it can have more than 99.99% of all numbers. There were a number of other interesting projects completed under this award, for details a reader might check out my homepage. The broader impacts of my work were mostly connected with training students. During this award I had 3 students finish their PhD's, and 2 more are nearly finished now. One of these students is from an underrepresented group (he is Mexican/American). I also worked with a number of talented undergraduates in completing an honors thesis. In fact, during the term of this award I published or submitted 3 papers written with undergraduates and each on a totally different topic. For example, with Noah Lebowitz-Lockhard, we considered the problem of generalizing to quadratic fields an algorithm of Kalai that allows one to choose a random number of a given magnitude together with its prime factorization, and do so in polynomial time. In a quadratic field, one must more generally be dealing with "ideals", but this is a technicality. In addition to one-on-one work with students, during the term of this award I gave a number of talks around the country aimed at students. For example, I gave the Rademacher Lectures at the University of Penssylvania and I gave the keynote lecture at the Hudson River Undergraduate Mathematics Conference held recently at Skidmore College.
