Master of Science / Department of Mathematics / Diego M. Maldonado / In this work we present the main concepts of Fourier Analysis (such as Fourier series, Fourier transforms, Parseval and Plancherel identities, correlation, and convolution) and illustrate them by means of examples and applications. Most of the concepts presented here can be found in the book "A First Course in Fourier Analysis" by David W.Kammler. Similarly, the examples correspond to over 15 problems posed in the same book which have been completely worked out in this report. As applications, we include Fourier's original approach to the heat flow using Fourier series and an application to filtering one-dimensional signals.

This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any right-angled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the first-documented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

Doctor of Philosophy / Department of Mathematics / David Yetter / We generalize the definition of the framed Kontsevich integral initially presented in [LM1]. We study the behavior of the renormalized framed Kontsevich integral Z[hat]_f under band sum moves and show that it can be further renormalized into some invariant Z[widetilde]_f that is well-behaved under moves for which link components of interest are locally put on top of each other. Originally, Le, Murakami and Ohtsuki ([LM5], [LM6]) showed that another choice of normalization is better suited for moves for which link components involved in the band sum move are put side by side. We show the choice of renormalization leads to essentially the same invariant and that the use of one renormalization or the other is just a matter of preference depending on whether one decides to have a horizontal or a vertical band sum. Much of the work on Z[widetilde]_f relies on using the tangle chord diagrams version of Z[hat]_f ([ChDu]). This leads us to introducing a matrix representation of tangle chord diagrams, where each chord is represented by a matrix, and tangle chord diagrams of degree $m$ are represented by stacks of m matrices, one for each chord making up the diagram. We show matrix congruences for some appropriately chosen matrices implement on the modified Kontsevich integral Z[widetilde]_f the band sum move on links. We show how Z[widetilde]_f in matrix notation behaves under the Reidemeister moves and under orientation changes. We show that for a link L in plat position, Z_f(L) in book notation is enough to recover its expression in terms of chord diagrams. We elucidate the relation between Z[check]_f and Z[widetilde]_f and show the quotienting procedure to produce 3-manifold invariants from those as introduced in [LM5] is blind to the choice of normalization, and thus any choice of normalization leads to a 3-manifold invariant.

Doctor of Philosophy / Department of Mathematics / Charles N. Moore

In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps. In the first part, we construct a lamination of the space of unimodal maps whose critical points have fixed degree d greater than or equal to 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space. This result implies that the statistical description of typical unimodal maps obtained in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in particular, almost every map in such a family is either regular or stochastic. In the second part, we prove the Poincare exponent for the Fibonacci map is less than two, which implies that the Hausdor ff dimension of its Julia set is less than two.

We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent. Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov. Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad. Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.

Several conjectured and proven generalizations of the Brouwer Fixed Point Theorem are examined, the plane fixed point problem in particular. The difficulties in proving this important conjecture are discussed. It is shown that it is true when strong additional assumptions are made. Canonical examples are produced which demonstrate the differences between this result and other generalized fixed point theorems.

This article is an investigation of a recently developed method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as X/M. In the next section, we construct some examples and analyse the topological relationship between X and X/M. In the final section, we apply X/M to get novel results about Lindelof spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.

Several conjectured and proven generalizations of the Brouwer Fixed Point Theorem are examined, the plane fixed point problem in particular. The difficulties in proving this important conjecture are discussed. It is shown that it is true when strong additional assumptions are made. Canonical examples are produced which demonstrate the differences between this result and other generalized fixed point theorems.