Dr. Lagarias proposes to investigate various analytic problems related to Eisenstein series and L-functions in number theory. There are three topics. The first topic concerns the Lerch zeta function and its relation to Eisenstein series. The second topic concerns properties of certain de Branges spaces and associated operators attached to automorphic L-functions. A third topic is to investigate Fourier coefficients of various nonholomorphic Eisenstein series.
This proposal is in the area of number theory. Number theory has in recent years has provided many useful applications in communications, coding theory and cryptography. Recent developments have observed parallels between problems in number theory, including the distribution of prime numbers, and various topics in mathematical physics. This proposal investigates possible connections of this kind. The problems considered may stimulate interactions between researchers in these fields. The proposer is training graduate students in related areas.
The research area of this grant is number theory (the study of properties of integers) and its relation to analysis and geometry. A connection to geometry comes through connection to packings (``Geometry of Numbers"), which are important incommunications and cryptography. The research in this grant covered diverse topics. One topic concerned study of properties of Eisenstein series and zeta functions, and how these objects encode number theory data. Another topic investigated was to quantify the assertion that it is not possible for three relatively prime numbers in arithmetic progression to have all three numbers completely factor into very small prime divisors compared to their size. This work was done with K. Soundararajan (Stanford). It quantifies a certain basic incompatibility of the addition and multiplication operation on integers. The research in this grant resulted in two edited books, one of these concerning a number theory problem, thesecond one on sphere packing,at least 13 papers, and four PhD. theses. I trained six graduate students,and below would like to feature the work of two of them. Elizabeth Chen (PhD 2010, U. of Michigan) worked on the problem of the densest packing of space by congruent regular tetrahedra. This is a very old problem. Aristotle (``On Heavenly Bodies", Book III, Part 8) stated that regular tetrahedra fill space. Aristotle's assertion was commented on by Averroes in 1190 AD and argued to be incorrect in a 1543 AD manuscript by Franciscus Maurolyctus.In 1900 David Hilbert included the problem of finding the densest packing of regular tetrahedra in his famous problem list (part of problem 18). In 2008 Elizabeth found the then densest known packing (of density about 77 percent), disproving a speculation of J. Conway and S. Torquato that regular tetrahedra might be harder to pack than spheres (whose maximal packing density is about 74 percent). Her discovery stimulated great activity by physicists and materials scientists. (Another motivation for the interest of scientists and engineers arises from the possibility of making nanomaterials using small tetrahedra, and investigating what properties such materials might have.) This led to several groups finding improved tetrahedral packings with different structures. After a succession of improvements by Salvatore Torquato's Chemistry group at Princeton, Sharon Glotzer's Chemical Engineering group at the Univ. of Michigan and Veit Elser's Physics group at Cornell, in January 2010 Elizabeth, together with Michael Engel and Sharon Glotzer (Chemical Engineering-U. of Michigan), found the current record density packing of regular tetrahedra,having density 4000/4671 (about 85.6 percent). This work was featured in the New York Times(Science Times, Jan. 5, 2010). Jonathan Bober (PhD 2009, U. of Michigan) studied a number theory problem involving the question of when (parametrized) products of ratios of factorials can be integers; normally such a ratio will be a fraction, not an integer. This problem is related to several different areas of mathematics, including the Riemann hypothesis, a central question in pure mathematics. It is also related to the structure of singularities in algebraic geometry, part of the minimal model program, which aims to classify algebraic manifolds up to birational equivalence. Thirdly it has a connection to hypergeometric functions, and indirectly with the phenomenon of mirror symmetry in string theory. His work established the truth of two conjectures made by V. Vasyunin and A. Borisov on the structure of such products, advancing the field. Concerning broader impact, this grant involved the training of six graduatestudents, four of whom have now completed thePhD. degree. In addition I mentored three undergraduate students in mathematics research, at least one of which has resulted in published work. The Ph.D. work of Elizabeth Chen has an ongoing influence of the work of some physicists and materials scientists, motivated in part by the current ability to construct exotic materials which have mathematically nice microscopic structures.
