Zeta functions of groups and rings and Igusa's local zeta function

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The concept of a zeta function is ubiquitous in mathematics, both pure and applied. The idea is to encode an infinite amount of information about a given object into a single function; by studying analytic properties of this zeta function -- such as the position of its zeros or poles -- we hope to learn something about the object we started with. Many classical applications show that arithmetic properties of zeta functions hold the key to understanding important aspects of the object's structure. With the Riemann hypothesis and the Birch- and Swinnerton-Dyer-conjecture, two of the seven Millenium Prize Problems ask about analytical properties of zeta functions. Infinite groups are fundamental objects in Pure Mathematics and are indispensable in describing symmetries in nature. Zeta functions of groups are tools to understand their subgroup growth: Given an infinite group, what can be said about the number of its subgroups of index n as n goes to infinity? Over the past twenty years, the development of the theory of zeta functions of groups has helped to answer asymptotic questions like this, and to address other exciting, more arithmetic aspects of subgroup growth. In a recent paper I pioneered a connection between zeta functions of nilpotent groups and a variant of Igusa's local zeta function. This more traditional type of zeta function comes from classical number theory. It is related to the problem of counting solutions of polynomial equations over finite residue rings. Exploiting this link allowed me to prove a long-standing symmetry conjecture about zeta functions of groups, which was only recently presented as a major open problem to the International Congress of Mathematicians 2006 in Madrid by two of the world's leading researchers in this area. The precise scope of this amazing symmetry-phenomenon, however, is still mysterious. It is known, for instance, that this palindromic symmetry may also collapse in certain cases. Part of my research project aims at understanding exactly when and why this happens.Emboldened by this success, my ambition is now to exploit this new bridge further to try and crack some of the outstanding open problems in the theory of zeta functions of groups. In order to answer the fundamental question of subgroup growth -- How does the number of subgroups of index n grow as n goes to infinity? -- one needs to understand the poles of zeta functions of groups. To pin down the exact location of these poles in the complex plane is a difficult task, and not many general results are known. On the other side of the bridge which I began pioneering, however, a wealth of results on poles of zeta functions is waiting to be exploited. A beautiful and deep conjecture links, for instance, the position of the poles of Igusa's local zeta function with the zeros of so-called Bernstein-Sato polynomials. These polynomials offer a subtle way to measure how singular certain spaces are. In many instances this conjecture is a proven theorem! In my research I envisage to transfer these remarkable results to the realm of zeta functions of groups. Ultimately I work towards the answers to questions like this: Is there a Bernstein-Sato polynomial for zeta functions of groups? What aspects of the group's structure are hidden in the local pole spectrum of its zeta function?