this paper, we will find a third conformal field theory of the conifold -- call it the C model. In the C model, there is a Z 2 discrete torsion sitting at the "singularity," which is in fact not a singularity in the conformal field theory sense, but just a region in which stringy effects are essential. The C model has no marginal operators (in particular the analogs of H

Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations)require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman-Bucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families Of external realizations defined on the same probability space and totally ordered with respect to state covariances. 1. Introduction. One

(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

Contents 1. Introduction 2. Vanishing Theorems 3. Singularities of Pairs 4. Bertini Theorems 5. Effective Base Point Freeness 6. Construction of Singular Divisors 7. The L 2 Extension Theorem and Inversion of Adjunction 8. The Log Canonical Threshold 9. The Log Canonical Threshold and the Complex Singular Index 10. The Log Canonical Threshold and the Bernstein-Sato Polynomial 11. Rational and Canonical Singularities 1. Introduction Higher dimensional algebraic geometry has been one of the most rapidly developing research areas in the past twenty years. The first decade of its development centered around the formulation of the minimal model program and finding techniques to carry this program through. The proof of the existence of flips, given in [Mori88], completed the program in dimension three. These results, especially the progress leading up to [Mori88], are reviewed in several surveys. A very general overview is given in [Koll'ar8

Floer theory assigns, in favourable circumstances, an abelian group HF(L0, L1) to a pair (L0, L1) of Lagrangian submanifolds of a symplectic manifold (M, ω). This group is a qualitative invariant, which remains unchanged under suitable deformations of L0 or L1. Following Floer [7] one can equip HF(L0, L1) with a canonical

Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we use calculate the invariant for several ten-crossing knots. 1.

Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information in turn can be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p, q) which arise as integral surgeries on knots in S 3 with |p | ≤ 1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge’s knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts. 1.

Abstract. In this note we observe that while all overtwisted contact structures on compact 3–manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to the Weinstein conjecture [1] and invariants of contact structures. 1.

We study the topology of the fibres of polynomials f with "singularities at infinity ". We first define W-singularities at infinity, where W refers to a certain Whitney stratification on the space X which is the union of compactified fibres of f . We show that the general fibre of a polynomial with isolated W-singularities at infinity has the homotopy type of a bouquet of spheres and that the number of these spheres is ¯ f + f , where ¯ f is the Milnor number and f is the sum of some polar numbers at infinity. As a byproduct, we improve the connectivity estimation of the general fibre of any polynomial. In the last part we define t-regularity at infinity of a polynomial and show that it implies topological triviality at infinity. We relate isolated W-singularities at infinity to t-nonregularity via polar curves at infinity. 1 Introduction It has been remarked by Ren'e Thom [T] that a polynomial f : C n ! C induces a locally trivial fibration f : C n n f \Gamma1 () ! C n above t...

Contents 1. Introduction 2. Moduli of smooth cubic surfaces 3. Moduli of stable cubic surfaces 4. Proofs of lemmas 5. Topology of nodal degenerations 6. Fractional dierentials and extension of the period map 7. The monodromy group and hyperplane conguration 8. Cuspidal degenerations 9. Proof of the main theorem 10. The universal cubic surface 11. Automorphisms of cubic surfaces 12. Index of notation 1: Introduction A classical theorem of great beauty describes the connection between cubic curves and hyperbolic geometry: the moduli space of the former is a quotient of the complex hyperbolic line (or real hyperbolic plane). The purpose of this paper is to exhibit a similar connection for cubic surfaces: their space of moduli is a quotient of complex hyperbolic four-space. We will make a precise statement below. The theorem for cubic curves can be established using periods of integrals, or, in modern language, Hodge structures. Indeed