Combinatorics and Number Theory

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Li 9622938 The applications of Number Theory to other fields have been attracting increasing interest in recent years. This investigation will exploit the rich interplay between Number Theory and Combinatorics. The problems to be investigated either have their origin in Combinatorics or have immediate application to Combinatorics, yet the problems themselves are Number- Theoretic. Roughly speaking, the problems in this investigation can be divided into two parts. The first part concerns the distribution of the character sums occurring as eigenvalues of Ramanujan graphs and their connection with eigenvalues of Hecke operators. The PI will continue her investigation of the Kloosterman conjecture over rational function fields. This conjecture implies that certain Kloosterman sums are eigenvalues of Hecke operators over function fields. The proposed approach includes the study of L-functions attached to curves over function fields and their connection to L-functions attached to representations. The second part is to obtain good results for partial character sums defined using additive and multiplicative characters of p-adic fields. If established, these bounds will have immediate applications in information theory to design better families of periodic sequences with low partial correlation. The proposed approach is a combination of the investigator's technique in estimating full character sums involving characters of p-adic fields and estimating partial character sums involving characters of finite fields. The PI plans to give a graduate course on "Number Theory with Applications" in 1996-97, which will include her past work in this area as well as the results from this proposal. This research falls into the general mathematical fields of Number Theory and Combinatorics. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mat hematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. Combinatorics is also an indispensable tool in modern communications. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.