Jaime Muñoz Masqué

Maximal Periods of x + c in Fq

The orbits produced by the iterations of the mapping x ? x2 + c, defined over Fq, are studied. Several upper bounds for their periods are obtained, depending on the coefficient c and the number of elements q.

Abbena-Garbiero's class K2 ⊕ K4 of Kähler homogeneous structures | La classe K2 ⊕ K4 d'Abbena et Garbiero des structures homogènes kählériennes

Partiellement supporte par la DGES (Espagne) : P.M.G. et J.M.M. sous Ie Projet PB-95-0124 et A.M.A. sous Ie Projet PB-94-0972

Analysis and algebra on differentiable manifolds

After recalling some definitions and results on the basics of smooth manifolds, this chapter is devoted to solve problems including (but not limited to) the following topics: Smooth mappings, critical points and critical values, immersions, submersions and quotient manifolds, construction of manifolds by inverse image, tangent bundles and vector fields, with integral curves and flows. Functions and other objects are assumed to be of class C∞ (also referred to either as ‘differentiable’ or ‘smooth’), essentially for the sake of simplicity. Similarly, manifolds are assumed to be Hausdorff and second countable, though we have included a section that analyses what happens when one of these properties fails to hold. We thus try to elicit in the reader a better understanding of the meaning and importance of such properties. On purpose, we have sprinkled this first chapter with many examples and figures. As an instructive example, we prove in detail that the manifold of affine straight lines of the plane, the 2-dimensional real projective space RP2 minus a point, and the infinite Mobius strip are diffeomorphic. As important and non-trivial examples of differentiable manifolds, the real projective space RP and the real Grassmannian Gk(R) are studied in detail. Lines of latitude and longitude began crisscrossing our worldview in ancient times, at least three centuries before the birth of Christ. By A. D. 150, the cartographer Ptolemy had plotted them on the twenty-seven maps of his first world atlas. DAVA SOBEL, Longitude, Walker & Company, New York, 2007, pp. 2–3. (With kind permission from the author and from Walker & Company publishers.) A differentiable manifold is generally defined in one of two ways; as a point set with neighborhoods homeomorphic with Euclidean space En, coordinates in overlapping neighborhoods being related by a differentiable transformation (. . . ) or as a subset of En, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions (. . . ). The first fundamental theorem is that the first definition is no more general than the second (. . . ) P.M. Gadea et al., Analysis and Algebra on Differentiable Manifolds, Problem Books in Mathematics, DOI 10.1007/978-94-007-5952-7_1,

Fibred Frobenius theorem

We give a version of Frobenius Theorem for fibred manifolds whose proof is shorter than the “short proofs” of the classical Frobenius Theorem. In fact, what shortens the proof is the fibred form of the statement, since it permits an inductive process which is not possible from the standard statement.

The Isomorphism of Polynomials Problem Applied to Multivariate Quadratic Cryptography

The threat quantum computing poses to traditional cryptosystems (such as RSA, elliptic-curve cryptosystems) has brought about the appearance of new systems resistant to it: among them, multivariate quadratic public-key ones. The security of the latter kind of cryptosystems is related to the isomorphism of polynomials (IP) problem. In this work, we study some aspects of the equivalence relation the IP problem induces over the set of quadratic polynomial maps and the determination of its equivalence classes. We contribute two results. First, we prove that when determining these classes, it suffices to consider the affine transformation on the left of the central vector of polynomials to be linear. Second, for a particular case, we determine an explicit system of invariants from which systems of equations whose solutions are the elements of an equivalence class can be derived.

Some Formulas and Tables

This chapter contains a 56-page-long list of formulae from the calculus on manifolds, and tables concerning different topics: Lie groups, Lie algebras and symmetric spaces, a list of Poincare polynomials of compact simple Lie groups, an overview of real forms of classical complex simple Lie algebras and their corresponding simple Lie groups, a table of irreducible Riemannian symmetric spaces of type I and III, a table of Riemannian symmetric spaces of classical type with noncompact isotropy group, etc. One can find the formulae for Christoffel symbols, the curvature tensor, Bianchi identities, Ricci tensor, the basic differential operators, the expression for conformal changes of Riemannian metrics, Cartan structure equations for pseudo-Riemannian manifolds, and many more. Several of these formulae are used throughout the book; others are not, but they have been included since such a collection might prove useful as an aide-memoire, also to lecturers and researchers.

Tensor Fields and Differential Forms

After providing some definitions and results on tensor fields and differential forms, this chapter deals with some aspects of general vector bundles, including the ‘cocycle approach’; other topics are: Tensors and tensor fields, exterior forms, Lie derivative and the interior product; calculus of differential forms and distributions. Some examples related to manifolds studied in the previous chapter are also present, such as the infinite Mobius strip, considered as a vector bundle, and the tautological bundle over the real Grassmannian. Certain problems intend to make the reader familiar with computations of vector fields, differential forms, Lie derivative, the interior product, the exterior differential, and their relationships. Other group of problems tries to develop practical abilities in computing integral distributions and differential ideals.

Integration on Manifolds

After giving some definitions and results on orientability of smooth manifolds, the problems treated in the present chapter are concerned with orientation of smooth manifolds; especially the orientation of several manifolds introduced in the previous chapter, such as the cylindrical surface, the Mobius strip, and the real projective space ℝP2. Some attention is paid to integration on chains and integration on oriented manifolds, by applying Stokes’ and Green’s Theorems. Some calculations of de Rham cohomology are proposed, such as the cohomology groups of the circle and of an annular region in the plane. This cohomology is also used to prove that the torus T 2 and the sphere S 2 are not homeomorphic. The chapter ends with an application of Stokes’ Theorem to a certain structure on the complex projective space ℂP n .

Non-Hamiltonian actions and Lie-algebra cohomology of vector fields.

Two examples of Diff + S 1 -invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on H 2 (X(S 1 )).

Lie algebra pairing and the Lagrangian and Hamiltonian equations in gauge-invariant problems

Let P → M be a principal G-bundle over a pseudo-Riemannian manifold (M, g). If G is semisimple, the Euler-Lagrange and the Hamilton-Cartan equations of the Yang-Mills Lagrangian defined by g are proved to remain unchanged if the Cartan-Killing metric is replaced by any other non-degenerate, adjoint-invariant bilinear form on the Lie algebra