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Scattering Operator, Eisenstein Series, Inner Product Formula and ``Maass-Selberg'' Relations for Kleinian Groups
In this memoir we have introduced and studied the scattering operator and the Eisenstein series and we have formulated and proved the inner product formula and the "Maass-Selberg" relations for Kleinian groups.
Contributions in Analytic and Algebraic Number Theory
The text that comprises this volume is a collection of surveys and original works from experts in the fields of algebraic number theory, analytic number theory, harmonic analysis, and hyperbolic geometry. A portion of the collected contributions have been developed from lectures given at the "International Conference on the Occasion of the 60th Birthday of S. J. Patterson", held at the University Göttingen, July 27-29 2009. Many of the included chapters have been contributed by invited participants. This volume presents and investigates the most recent developments in various key topics in analytic number theory and several related areas of mathematics. The volume is intended for graduate students and researchers of number theory as well as applied mathematicians interested in this broad field.
Heat Kernels and Spectral Theory
Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators.
Kernel Functions, Analytic Torsion, and Moduli Spaces
This memoir is a study of Ray-Singer analytic torsion for hermitian vector bundles on a compact Riemann surface [italic]C. The torsion is expressed through the trace of a modified resolvent. Thus, one can develop perturbation-curvature formulae for the Green-Szegö kernel and also for the torsion in terms of the Ahlfors-Bers complex structure of the Teichmuller space and Mumford complex structure of the moduli space of stable bundles of degree zero on [italic]C.
Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras
We present a new proof of the identities needed to exhibit an explicit [bold]Z-basis for the universal enveloping algebra associated to an affine Lie algebra. We then use the explicit [bold]Z-bases to extend Borcherds' description, via vertex operator representations, of a [bold]Z-form of the enveloping algebras for the simply-laced affine Lie algebras to the enveloping algebras associated to the unequal root length affine Lie algebras.
Sum of Even Powers of Real Linear Forms
This work initiates a systematic analysis of the representation of real forms of even degree as sums of powers of linear forms and the resulting implications in real algebraic geometry, number theory, combinatorics, functional analysis, and numerical analysis. The proofs utilize elementary techniques from linear algebra, convexity, number theory, and real algebraic geometry and many explicit examples and relevant historical remarks are presented.
Selberg Trace Formulae and Equidistribution Theorems for Closed Geodesics and Laplace
This work is concerned with a pair of dual asymptotics problems on a finite-area hyperbolic surface. The first problem is to determine the distribution of closed geodesics in the unit tangent bundle. The second problem is to determine the distribution of eigenfunctions (in microlocal sense) in the unit tangent bundle.
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