The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory ” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 × E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface

Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents

This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n) . 1.

This paper presents the rst examples of K3 surface automorphisms f : X ! X with Siegel disks (domains on which f acts by an irrational rotation). The set of such examples is countable, and the surface X must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of degree 22 and trace 1, which play the role of the leading eigenvalue for f jH 2 (X). The construction uses the Torelli theorem, the Atiyah-Bott xed-point theorem and results from transcendence theory. Contents 1

As early as the birth of Seiberg Witten invariants [W1], the positive scalar curvature metrics on four dimensional manifolds have played a very important role. It was Witten [W] who first noticed that assuming b + 2> 1 then one could easily derive the vanishing result of Seiberg Witten Invariants

Dedicated to the memory of Gian-Carlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the η-function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing

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Abstract. We give the first examples of infinite sets of primes S such that Hilbert’s Tenth Problem over Z[S −1] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E ′ over Z[S −1] such that the topological closure of E ′ (Z[S −1]) in E ′ (R) has infinitely many connected components. 1.

. Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings. z Heisenberg Fellow p-adic quasicrystals 2 1. Introduction The cut and project method of constructing nonperiodic point sets, as developed by Peter Kramer and others in the early eigthies [10, 11, 9, 3, 12], is one of the basic tools in the mathematical study of quasicrystals and aperiodic order. The intuition behind their use is that quasiperiodic po...

1.1. A fake projective plane is a smooth compact complex surface P which is not the complex projective plane but has the same first and second Betti numbers as the complex projective plane (b1(P) = 0 and b2(P) = 1). It is well-known that such a surface is projective algebraic and is the quotient of the complex two ball in C2 by a cocompact torsion-free discrete subgroup of PU(2, 1). These are surfaces with the smallest Euler-Poincaré characteristic among all smooth surfaces of general type. The first fake projective plane was constructed by Mumford [Mu] using p-adic uniformization, and more recently, two more examples were found by related methods by Ishida-Kato in [IK]. We have just learnt from Keum that he has an example which may be different from the earlier three. A natural problem in complex algebraic geometry is to determine all fake projective planes. It is proved in [Kl] and [Y] that the fundamental group of a fake projective plane, considered as a lattice of PU(2, 1), is arithmetic. In this paper we make use of this arithmeticity result and the volume formula of [P], together with some number theoretic estimates, to make a complete list of all fake projective planes,