Showing papers in "Bulletin of the American Mathematical Society in 2003"

Spectra of hyperbolic surfaces

Abstract: These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey.

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

Abstract: This paper is a handyman’s manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups. Three objectives: Pleasant writing, easy reading, good understanding. One topic: How to prove resolution of singularities in characteristic zero. Statement to be proven (No-Tech): The solutions of a system of polynomial equations can be parametrized by the points of a manifold. Statement to be proven (Low-Tech): The zero-setX of finitely many real or complex polynomials in n variables admits a resolution of its singularities (we understand by singularities the points where X fails to be smooth). The resolution is a surjective differentiable map ε from a manifold X̃ to X which is almost everywhere a diffeomorphism, and which has in addition some nice properties (e.g., it is a composition of especially simple maps which can be explicitly constructed). Said differently, ε parametrizes the zero-set X (see Figure 1). Figure 1. Singular surface Ding-dong: The zero-set of the equation x + y = (1 − z)z in R can be parametrized by R via (s, t)→ (s(1− s) · cos t, s(1− s) · sin t, 1− s). The picture shows the intersection of the Ding-dong with a ball of radius 3. Received by the editors June 25, 2002, and, in revised form, December 3, 2002. 2000 Mathematics Subject Classification. Primary 14B05, 14E15, 32S05, 32S10, 32S45. Supported in part by FWF-Project P-15551 of the Austrian Ministry of Science. c ©2003 American Mathematical Society

Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

Abstract: Typical large unitary matrices show remarkable patterns in their eigenvalue distribution. These same patterns appear in telephone encryption, the zeros of Riemann’s zeta function, a variety of physics problems, and in the study of Toeplitz operators. This paper surveys these applications and what is currently known about the patterns.

On a theorem of Jordan

Abstract: The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology. 1. Statements 1.1. Number theory. Let f = ∑n m=0 amx m be a polynomial of degree n, with coefficients in Z. If p is prime, let Np(f) be the number of zeros of f in Fp = Z/pZ. Theorem 1. Assume (i) n ≥ 2, (ii) f is irreducible in Q[x]. Then (a) There are infinitely many p’s with Np(f) = 0. (b) The set P0(f) of p’s with Np(f) = 0 has a density c0 = c0(f) which is > 0. [Recall that a subset P of the set of primes has density c if lim X→∞ number of p ∈ P with p ≤ X π(X) = c, where π(X) is as usual the number of primes ≤ X .] Moreover, Theorem 2. With the notation of Theorem 1, one has c0(f) ≥ 1 n , with strict inequality if n is not a power of a prime. Example. Let f = x + 1. One has p ∈ P0(f) if and only if p ≡ −1 (mod 4); this set is well-known to have density 1/2. We shall see more interesting examples in §5. 1.2. Topology. Let S1 be a circle. Let f : T → S be a finite covering of a topological space S. Assume: (i) f has degree n (i.e. every fiber of f has n elements), with n ≥ 2, (ii) T is arcwise connected and not empty. Theorem 3. There exists a continuous map φ : S1 → S which cannot be lifted to the covering T (i.e. there does not exist any continuous map ψ : S1 → T such that φ = f ◦ ψ). Received by the editors March 1, 2003. 2000 Mathematics Subject Classification. Primary 06-XX, 11-XX, 11F11. This text first appeared in Math Medley 29 (2002), 3–18. The writing was done with the help of Heng Huat Chan. c ©2002 Singapore Mathematical Society. Reprinted with permission.

Measures and dimensions in conformal dynamics

Abstract: This survey collects basic results concerning fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere. Frequently these results are compared with their counterparts in the theory of Kleinian groups, and this enlarges the famous Sullivan dictionary. The topics concerning Hausdorff and packing measures and dimensions are given most attention. Then, conformal measures are constructed and their relations with Hausdorff and packing measures are discussed throughout the entire article. Also invariant measures absolutely continuous with respect to conformal measures are touched on. While the survey begins with facts concerning all rational functions, much time is devoted toward presenting the well-developed theory of hyperbolic and parabolic maps, and in Section 3 the class NCP is dealt with. This class consists of such rational functions f that all critical points of f which are contained in the Julia set of f are non-recurrent. The NCP class comprises in particular hyperbolic, parabolic and subhyperbolic maps. Our last section collects some recent results about other subclasses of rational functions, e.g. Collet-Eckmann maps and Fibonacci maps. At the end of this article two appendices are included which are only loosely related to Sections 1-4. They contain a short description of tame mappings and the theory of equilibrium states and Perron-Frobenius operators associated with Hölder continuous potentials. 1. Dimensions of Julia sets A first issue we will be dealing with in this article is to describe various fractals of Julia sets as captured by the Hausdorff, packing and box dimensions. Afterwards, we address the question of when the corresponding Hausdorff and packing measures are positive and finite. Given a subset A of a metric space (X, d) a countable family {B(xi, ri)}i=1 of open balls centered at points of A is said to be a packing of A if and only if for any pair i 6= j d(xi, xj) ≥ ri + rj . The supremum sup{ri : i ≥ 1} is called the radius of the packing {B(xi, ri)}i=1. Given in addition a positive radius r > 0, denote by N(A, r) the minimal number of open balls with radius r needed to cover A and by P (A, r) the maximal number of open balls with radius r forming a packing of A. In order to get an idea of what the box dimension is, imagine a two-dimensional smooth surface A in the Euclidean space R. It is reasonable to expect the minimal number N(A, r) and the maximal number P (A, r) to be some multiples of r−2. The Received by the editors December 22, 1999, and, in revised form, January 8, 2003. 2000 Mathematics Subject Classification. Primary 35F35, 37D35; Secondary 37F15, 37D20, 37D25, 37D45, 37A40, 37A05. Research partially supported by NSF Grant DMS 9801583. c ©2003 American Mathematical Society 281 282 MARIUSZ URBAŃSKI coefficient is not important to understand dimensionality of A, but the exponent is crucial, and it is captured by the following limits: lim r→0 logN(A, r) − log r = lim r→0 logP (A, r) − log r . In general these limits fail to exist, and the following two quantities are defined: BD(A) = lim inf r→0 logN(A, r) − log r = lim inf r→0 logP (A, r) − log r

Riemann's zeta function and beyond

Abstract: In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of $L$-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of L-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of L-functions.

The interplay between analysis and topology in some nonlinear PDE problems

Abstract: The “balayage” method introduced by H. Poincaré in his proof of Theorem 1 relies heavily on tools of Potential theory: maximum principle, Harnack’s inequality, explicit representation formulas for the Dirichlet problem in a ball (Poisson integral), etc. In 1900, D. Hilbert [39], in a celebrated address, followed by a (slightly) more detailed paper in 1904, announced that he had solved the Dirichlet problem (0.1) (0.2) via the Dirichlet principle which had been discovered by G. Green in 1833, with later contributions by C. F. Gauss (1837), W. Thomson (=Lord Kelvin) (1847) and G. Riemann (1853). Dirichlet’s principle asserts that any solution u of (0.1) (0.2) is a minimizer of the Dirichlet integral

Catalan’s Conjecture: Another old Diophantine problem solved

Abstract: Catalan’s Conjecture predicts that 8 and 9 are the only consecutive perfect powers among positive integers. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu. A deep theorem about cyclotomic fields plays a crucial role in his proof. Like Fermat’s problem, this problem has a rich history with some surprising turns. The present article surveys the main lines of this history and outlines Mihăilescu’s brilliant proof.

Rotation numbers and instability sets

Abstract: Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows. Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G.D. Birkhoff was led to the concept of “regions of instability” for twist maps. We discuss the closely related notion of instability sets for a generic area preserving surface diffeomorphism and develop their properties.

Making light of mathematics

Abstract: Summary of AMS Gibbs Lecture, delivered at San Diego, CA, 6 January 2002.