(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

Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t,..., t n) projected onto the surface of the unit sphere, divided by π. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric. Contents 1.

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

We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described.

We study families of quantum eld theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g 1, and for the case g = 0 | by smooth closed Jordan curves on the complex plane. In both cases we dene a notion of -function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the -function is an analytic torsion [3], and for the case g = 0 | the regularized energy of a certain natural pseudo-measure on the interior domain of a closed curve. For these cases we rigorously prove the Ward identities for the current correlation functions and determine them explicitly. For the case g > 0 these functions coincide with those obtained in [21, 36] using bosonization. For the case g = 0 the -function we have dened coincides with the -function introduced in [29, 44, 24] as a dispersionless limit of the Sato's -function for the two-dimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44, 24] on relations between conformal maps of exterior domains and -functions. For this case we also dene a Hermitian metric on the space of all contours of given area. As another corollary of the Ward identities we prove that the introduced metric is Kahler and the logarithm of the -function is its Kahler potential. Contents

. Let X ! Y be a double covering of a non{singular complex algebraic manifold branched along a non{singular (reduced) divisor D. In this paper we shall prove that there is a natural isomorphism H i( j X ) = H i( j Y ) H i( j Y (log D)( 1 2 D)): We shall also give some methods to compute the second summand of the righthand side of the above formula. 1991 Mathematics Subject Classication. Primary: 14B05, 14J30; Secondary 32B10. Key words and phrases. Hodge number { logarithmic dierential forms { double covering { rsolution of singularities. Partially supported by KBN grant no 2P03A 022 17. 2 S LAWOMIR CYNK 1. Introduction Let X ! Y be a double covering of a non{singular complex algebraic variety Y branched along a non{singular (reduced) divisor D. Then X is determined by the branch locus D and a line bundle L on Y that can be described in the following way: the natural involution on X (exchanging sheets of the covering) gives the splitting OX = O ...

In this paper, we prove that on any non-singular algebraic variety, the characteristic classes of Cheeger-Simons and Beilinson agree whenever they can be interpreted as elements of the same group (e.g. for at bundles). In the universal case, where the base is BGL(C)^δ, we show that the universal Cheeger-Simons class is half the Borel regulator element. We were unable to prove that the universal Beilinson class and the universal Cheeger-Simons classes agree in this universal case, but conjecture they do agree.

Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. According to Akivis, [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan, [21], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric

this paper, the functions # i will be assumed to be normalized as above, i.e. so that all second-order derivatives of # 0 vanish at 0. 3.4. Conclusion

In some fundamental papers Davenport, Lewis and Schinzel [DLS], Schinzel [Sch1, Sch3] and Fried [Fr1, Fr2, Fr3] have shown how irreducibility criteria for polynomials f(X) g(Y ) in combination with results of Runge or Siegel can be used to prove the niteness of the solutions of the corresponding diophantine equation f(x) = g(y) in integers x; y. In the present paper we are particularly interested in the case f(X) = X(X+d1 ) (X+(m 1)d1 ), g(Y ) = Y (Y +d2) (Y +(n 1)d2 ), i.e. the diophantine equation x(x + d1 ) (x + (m 1)d1) = y(y + d2 ) (y + (n 1)d2 ): (1) We rst give some history on this equation and indicate how results for this equation can be derived from general irreducibility theory in the literature. Then we give direct proofs of the results using only basic facts on algebraic curves. 1 When do nite arithmetic sequences have equal products of terms? The question, under the restriction that the arithmetic progressions have equal lengths, was posed in Poland by Gabovich in 1966 [Ga]. He mentioned the example 2 6 10 = 4 5 6 and gave an innite class of examples of length 4 including 7 20 33 46 = 20 21 22 23 and 18 37 56 75 = 24 37 50 63. Some innite classes of solutions of length 5 were given by Szymiczek [Sz] and Choudhry [Ch]. In 1968 Makowski [Ma] observed that for every positive integer m 2 6 10 (4m 2) = (m + 1)(m + 2) (2m): (1) An opposite result was obtained by Saradha, Shorey and Tijdeman [SaST1]. 1 2 Theorem A. For xed integers d 1 > d 2 > 0 there are only nitely many positive integers m > 2, x; y gcd(x; y; d 1 ; d 2 ) = 1 and x(x + d 1 ) (x + (m 1)d 1 ) = y(y + d 2 ) (y + (m 1)d 2 ) (2) except for the solutions (1). The other solutions are eectively comput...