Can one hear the shape of a drum

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Definition. A (topological) n-manifold M is a Hausdorff topological space with a countable basis of open sets, such that each point of M lies in an open set homeomorphic to R or R+ = {(x1, . . . , xn) ∈ R n : xn ≥ 0}. The boundary ∂M of M is the set of points not having neighbourhoods homeomorphic to R. The set M − ∂M is the interior of M , denoted int(M). If M is compact and ∂M = ∅, then M is closed.

Three-dimensional manifolds

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Elementary differential geometry

We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric spaces.

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Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

In this article, we relate word and normal subgroup growth to certain functions that arise in the quantification of residual finiteness. One consequence of this endeavor is a pair of results that equate the virtual nilpotency of a finitely generated group with the asymptotic behavior of these functions. The second half of this article investigates the asymptotic behavior of two of these functions. Our main result in this area resolves a question of Bogopolski from the Kourovka notebook concerning lower bounds of one of these functions for non-abelian free groups.

Asymptotic growth and least common multiples in groups

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).

Absolute profinite rigidity and hyperbolic geometry

For X = R, C, or H, it is well known that cusp cross-sections of finite volume X-hyperbolic (n + 1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n +1)-dimensional Heisenberg group n 2 n + 1 orthe (4n + 3)-dimensional quaternionic Heisenberg group n 4 n + 3 (H) We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n + 1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.

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Peripheral separability and cusps of arithmetic hyperbolic orbifolds.

Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equivalent and primitive length equivalent Riemannian manifolds. For example we show that every finite volume hyperboli c n‐manifold has pairs of eigenvalue equivalent finite covers of arbitrar ily large volume ratio. We also show the analogous result for primitive length equivalence.

Length and Eigenvalue Equivalence

In this short article, we study the extremal behavior ${{\mathrm{F}}}_\Gamma (n)$ of divisibility functions ${{\mathrm{D}}}_\Gamma $ introduced by the first author for finitely generated groups $\Gamma $ These functions aim at quantifying residual finiteness and bounds give a measurement of the complexity in verifying a word is non-trivial We show that finitely generated subgroups of ${{\mathrm{GL}}}(m,K)$ for an infinite field $K$ have at most polynomial growth for the function ${{\mathrm{F}}}_\Gamma (n)$ Consequently, we obtain a dichotomy for the growth rate of $\log {{\mathrm{F}}}_\Gamma (n)$ for finitely generated subgroups of ${{\mathrm{GL}}}(n,\mathbf {C})$ We also show that if ${{\mathrm{F}}}_\Gamma (n) \preceq \log \log n$, then $\Gamma $ is finite In contrast, when $\Gamma $ contains an element of infinite order, $\log n \preceq {{\mathrm{F}}}_\Gamma (n)$ We end with a brief discussion of some geometric motivation for this work

D. B. McReynolds

Papers

Asymptotic growth and least common multiples in groups

Absolute profinite rigidity and hyperbolic geometry

Peripheral separability and cusps of arithmetic hyperbolic orbifolds.

Length and Eigenvalue Equivalence

Extremal behavior of divisibility functions