The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate leve l.

Complex Dimensions of Ordinary Fractal Strings. Generalized Fractal Strings Viewed as Measures.

Explicit Formulas for Generalized Fractal Strings. The Geometry and the Spectrum of Fractal Strings. Periodic Orbits of Self-Similar Flows. Fractal Tube Formulas. Riemann Hypothesis and Inverse Spectral Problems.

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Generalized Cantor Strings and their Oscillations. Critical Zero of Zeta Functions. Recent Results and Perspectives.

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Zeta Functions in Number Theory. Zeta Functions of Laplacians and Spectral Asymptotics. An Application of Nevanlinna Theory. The algebraic techniques developed by Kakde will almost certainly lead eventually to major progress in the study of congruences between automorphic forms and the main conjectures of non-commutative Iwasawa theory for many motives. Non-commutative Iwasawa theory has emerged dramatically over the last decade, culminating in the recent proof of the non-commutative main conjecture for the Tate motive over a totally real p-adic Lie extension of a number field, independently by Ritter and Weiss on the one hand, and Kakde on the other.

## Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

The initial ideas for giving a precise formulation of the non-commutative main conjecture were discovered by Venjakob, and were then systematically developed in the subsequent papers by Coates-Fukaya-Kato-Sujatha-Venjakob and Fukaya-Kato. There was also parallel related work in this direction by Burns and Flach on the equivariant Tamagawa number conjecture. Kakde's proof is a beautiful development of these ideas of Kato, combined with an idea of Burns, and essentially reduces the study of the non-abelian main conjectures to abelian ones.

The approach of Ritter and Weiss is more classical, and partly inspired by techniques of Frohlich and Taylor. Since many of the ideas in this book should eventually be applicable to other motives, one of its major aims is to provide a self-contained exposition of some of the main general themes underlying these developments. The present volume will be a valuable resource for researchers working in both Iwasawa theory and the theory of automorphic forms.

## Bibliography

Kakde on the non-commutative main conjectures for totally real fields. Sujatha: Reductions of the main conjecture. This result is generalized to apply to many other zeta-functions. This highly original, self-contained monograph will appeal to geometers, fractalists, mathematical physicists, and number theorists, as well as to graduate students in these fields. Complex Dimensions of Ordinary Fractal Strings 1.

Generalized Fractal Strings Viewed as Measures 3. Explicit Formulas for Generalized Fractal Strings 4. The Geometry and the Spectrum of Fractal Strings 5. Tubular Neighborhoods and Minkowski Measurability 6.

## NSF Award Search: Award# - Fractal Geometry and Dynamical Systems, with Applications

Generalized Cantor Strings and their Oscillations 8. The Critical Zeros of Zeta Functions 9. Concluding Comments

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