Vagn Lundsgaard Hansen

Functional Analysis: Entering Hilbert Space

This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, self-adjoint operators on separable Hilbert spaces. It exhibits a construction of the space of pth power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Holder and Minkowski. The Lp-spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated.

Branched polynomial covering maps

Abstract A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere.

INTEGRABLY PARALLELIZABLE MANIFOLDS

A smooth manifold M is called integrably parallel- izable if there exists an atlas for the smooth structure on Mn such that all differentials in overlap between charts are equal to the identity map of the model for M. We show that the class of con- nected, integrably parallelizable, «-dimensional smooth manifolds consists precisely of the open parallelizable manifolds and manifolds diffeomorphic to the /i-torus. 1. Introduction. In this note Mn is an «-dimensional, paracompact smooth manifold without boundary, and G is an arbitrary subgroup of the general linear group Gl(«, R) on Rn. Definition. M is called G-reducible if the structural group of the tangent bundle for Mn can be reduced from Gl(«, R) to G. Mn is called integrably G-reducible if there exists an atlas {((/ 6()} for the smooth structure on Mn such that the differential in overlap between charts (0? o OJ1)^ belongs to G for all x e B^UiCMJ,)^ Rn and all /, j in the index set for the atlas. It is clear that an integrably G-reducible manifold is G-reducible and therefore the following problem naturally arises. Problem. Classify for a given G those G-reducible manifolds which are integrably G-reducible. Let us illustrate this problem with two examples. Example 1. Let G=0(n) be the orthogonal group. Any manifold Mn is 0(/7)-reducible, since it admits a Riemannian metric. On the other hand it is easy to see that M is integrably 0(«)-reducible if and only if it admits a flat Riemannian metric. (A flat Riemannian manifold is locally isometric to R.) Example 2. Suppose n=2k and let G1(A:, C) be the general linear group on Ck considered as a subgroup of Gl(«, R) under the usual identifi- cation of Ck with Rn. Then M is G1(A, C)-reducible if and only if it admits an almost complex structure and integrably G(k, C)-reducible if and only if it admits a complex structure. The classification of the manifolds admitting a complex structure among those admitting an almost complex structure is far from being complete.

Changes and Trends in Geometry Curricula

It would be convenient to know the dynamics of curricular changes. What brings them about and how can changes occur? Even though it seems hard to say anything of a general nature on these questions, it is a fact that good ideas cannot always be implemented. There will always be boundary conditions determined among other things by what teachers are prepared to teach and what changes governments are prepared to pay for. To make significant changes in a curriculum the level of knowledge and the level of educational and pedagogical consciousness of teachers are decisive factors. It is not enough to have a few enthusiastic and very competent teachers. Eventually, a well qualified teacher is needed in every classroom in every school. In this respect the change of a curriculum has to start at the teacher-level. First one must train a sufficient number of teachers to master the new material. Altogether it takes years of dedicated work to change a curriculum in a subject like mathematics. In this chapter we are primarily concerned with the curriculum in geometry. It goes without saying that introducing more geometry will reduce time for other mathematics, and hence there is a need for an integrated curriculum in mathematics.

The dawn of non‐Euclidean geometry

The emergence of non‐Euclidean geometries in the beginning of the 19th century represents one of the dramatic episodes in the history of mathematics. It may be an often told tale, but although it entails a lot of important and useful geometrical ideas and constructions, it is seldom told with the necessary mathematical details in contemporary teaching of geometry. The purpose of this article is to provide a short, but relatively complete, exposition of the geometry in the Poincare disc model of the hyperbolic plane within the historical perspective of Euclids postulates. We also describe the isometries in the hyperbolic plane and tilings of the hyperbolic plane with congruent regular hyperbolic polygons.

Covering maps with separators

Abstract In the theory of polynomial covering maps it has emerged that the key property for an n -fold covering map π : E → X to be polynomial is that it admits a continuous complex valued function f:E → C which separates points in the fibres of π. Such a separating function leads immediately to the fundamental properties of a polynomial covering map π : E → X , i.e.: the existence of an embedding of π into the trivial complex line bundle over X ; a characteristic homomorphism for π into the braid group on n strings; and the existence of a primitive for the extension of complex function rings C ( E ) of C ( X ) defined by π. In this paper we investigate the existence of general separation spaces for finite covering maps and study possible generalizations of the above properties of polynomial covering maps.

The circle equation over finite fields

Abstract Interesting patterns in the geometry of a plane algebraic curve C can be observed when the defining polynomial equation is solved over the family of finite fields. In this paper, we examine the case of C the classical unit circle defined by the circle equation x2 + y2 = 1. As a main result, we establish a concise formula for the number of solutions to the circle equation over an arbitrary finite field. We also provide criteria for the existence of diagonal solutions to the circle equation. Finally, we give a precise description of how the number of solutions to the circle equation over a prime field grows as a function of the prime.

A note on powers in finite fields

Abstract The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years, the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection, it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation that is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.

Ranking Entities in Networks via Lefschetz Duality

In the theory of communication it is essential that agents are able to exchange information. This fact is closely related to the study of connected spaces in topology. A communication network may be modelled as a topological space such that agents can communicate if and only if they belong to the same path connected component of that space. In order to study combinatorial properties of such a space, notions from algebraic topology are applied. This makes it possible to determine the shape of a network by concrete invariants, e.g. The number of connected components. Elements of a network may then be ranked according to how essential their positions are in the network by considering the effect of their respective absences. Defining a ranking of a network which takes the individual position of each entity into account has the purpose of assigning different roles to the entities, e.g. Agents, in the network. In this paper it is shown that the topology of a given network induces a ranking of the entities in the network. Further, it is demonstrated how to calculate this ranking and thus how to identify weak sub-networks in any given network.

Keeping Mathematical Awareness Alive

Caring for the development of abstract mathematics is not at first sight useful and in modern society the applications of mathematics are more sophisticated and more difficult to grasp than in earlier times. This may easily lead to a decline in the emphasis on mathematical knowledge and thought in society, in particular in the educational system. The increasing pressure from governments for more focus on immediate applications of mathematics in education and in mathematical research at universities makes it a huge task to inform the public about the usefulness and necessity of understanding the abstract mathematical thinking required to develop fundamental new applications of mathematics. How can we handle it?