This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S- curvature. We show that the class of all such subsets can be naturally divided into two subclasses: The first subclass consists of those sets S for which the following holds: If two Riemannian manifolds have positive S- curvature then their connected sum also admits a Riemannian metric of positive S- curvature. The other subclass consists of those sets for which the normalized Ricci flow on a closed Riemannian manifold with positive S-curvature converges to a metric of constant positive sectional curvature. In the second part of the thesis, we study the behavior of Ricci flow for a manifold having positive S - curvature, where S is in the first subclass. More specifically, we study the Ricci flow for a special class of metrics on Sp+1 x S1 , p ≥ 4, which have positive isotropic curvature.

In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 1982 / Bibliography: leaves 78-79. / by John Marshall Lee. / Ph. D. / Ph. D. Massachusetts Institute of Technology, Department of Mathematics

Στη μεταπτυχιακή αυτή διπλωματική εργασία, αρχικά εισάγουμε τις έννοιες των μετρικών πολλαπλοτήτων σχεδόν επαφής και των μετρικών πολλαπλοτήτων επαφής, δίνοντας και μερικά παραδείγματα από κάθε κατηγορία. Στη συνέχεια, αναφέρουμε και αποδεικνύουμε λεπτομερώς μερικές βασικές γεωμετρικές ιδιότητες που χαρακτηρίζουν τις μετρικές πολλαπλότητες επαφής και, οι οποίες, εμπλέκουν τον τανυστή καμπυλό- τητας. Τέλος, δίνεται έμφαση σε ειδικές κατηγορίες μετρικών πολλαπλοτήτων επαφής που παρουσιάζουν ιδιαίτερο γεωμετρικό ενδιαφέρον και, κυρίως, είναι: πολλαπλότητες K- επαφής, πολλαπλότητες του Sasaki, (κ, μ) – πολλαπλότητες επαφής και μετρικές πολλαπλότητες Η – επαφής. / In this Master Thesis, we initially introduce the notions of almost contact metric manifolds and contact metric manifolds, giving some examples from each category. In sequel, we mention and prove explicitly some basic geometric properties of contact metric manifolds, which involve the curvature tensor. Summarizing, we focus on special classes of contact metric manifolds which have particular geometric interest and, mainly, are: K – contact manifolds, Sasakian manifolds, (κ, μ) – contact manifolds and H – contact metric manifolds.

A simplicial cell complex K is the face poset of a regular CW complex W such that the boundary complex of each cell is isomorphic to the boundary complex of a simplex of same dimension. If a topological space X is homeomorphic to W then we say that K is a pseudotriangulation of X. For d 1, a (d + 1)-colored graph is a graph = (V; E) with a proper edge coloring : E ! f0; : : : ; dg. Such a graph is called contracted if (V; E n 1(i)) is connected for each color A contracted graph = (V; E) with an edge coloring : E ! f0; : : : ; dg determines a d-dimensional simplicial cell complex K( ) whose vertices have one to one correspondence with the colors 0; : : : ; d and the facets (d-cells) have one to one correspondence with the vertices in V . If K( ) is a pseudotriangulation of a manifold M then ( ; ) is called a crystallization of M. In [71], Pezzana proved that every connected closed PL manifold admits a crystallization. This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings. We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of vertices of any crystallization of a connected closed 3-manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3-manifolds RP3, L(3; 1), L(5; 2), S1 S1 S1, S2 S1, S2 S1 and S3=Q8, where Q8 is the quaternion group. Moreover, there is a unique such vertex minimal crystallization in each of these seven cases. We have also constructed crystallizations of L(kq 1; q) with 4(q + k 1) vertices for q 3, k 2 and L(kq +1; q) with 4(q + k) vertices for q 4, k 1. In [22], Casali and Cristofori found similar crystallizations of lens spaces. By a recent result of Swartz [76], our crystallizations of L(kq + 1; q) are vertex minimal when kq + 1 are even. In [47], Gagliardi found presentations of the fundamental group of a manifold M in terms of a crystallization of M. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M, we have constructed a crystallization of M. These results are in Chapter 3. We have de ned the weight of the pair (hS j Ri; R) for a given presentation hS j R of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of (hS j Ri; R) is n then our algorithm constructs all the n-vertex crystallizations which yield (hS j Ri; R). As an application, we have constructed some new crystallization of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For m 3 and m n k 2, our generalized algorithm gives a 2(2m + 2n + 2k 6 + n2 + k2)-vertex crystallization of the closed connected orientable 3-manifold Mhm; n; ki having fundamental group hx1; x2; x3 j xm1 = xn2 = xk3 = x1x2x3i. These crystallizations are minimal and unique with respect to the given presentations. If `n = 2' or `k 3 and m 4' then our crystallization of Mhm; n; ki is vertex-minimal for all the known cases. These results are in Chapter 4. We have constructed a minimal crystallization of the standard PL K3 surface. The corresponding simplicial cell complex has face vector (5; 10; 230; 335; 134). In combination with known results, this yields minimal crystallizations of all simply connected PL 4-manifolds of \standard" type, i.e., all connected sums of CP2, CP2, S2 S2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair 4-manifolds which are homeomorphic but not PL-homeomorphic. We have also presented an elementary proof of the uniqueness of the 8-vertex crystallization of CP2. These results are in Chapter 5. For any crystallization ( ; ) the number f1(K( )) of 1-simplices in K( ) is at least d+1 . It is easy to see that f1(K( )) = d+1 if and only if (V; 1(A)) is connected for each d 2 2 1)-set A called simple. All the crystallization in Chapter 5 (. Such a crystallization is are simple. Let ( ; ) be a crystallization of M, where = (V; E) and : E ! f0; : : : ; dg. We say that ( ; ) is semi-simple if (V; 1(A)) has m + 1 connected components for each (d 1)-set A, where m is the rank of the fundamental group of M. Let ( ; ) be a connected (d +1)-regular (d +1)-colored graph, where = (V; E) and : E ! f0; : : : ; dg. An embedding i : ,! S of into a closed surface S is called regular if there exists a cyclic permutation ("0; "1; : : : ; "d) (of the color set) such that the boundary of each face of i( ) is a bi-color cycle with colors "j; "j+1 for some j (addition is modulo d+1). Then the regular genus of ( ; ) is the least genus (resp., half of genus) of the orientable (resp., non-orientable) surface into which embeds regularly. The regular genus of a closed connected PL 4-manifold M is the minimum regular genus of its crystallizations. For a closed connected PL 4-manifold M, we have provided the following: (i) a lower bound for the regular genus of M and (ii) a lower bound of the number of vertices of any crystallization of M. We have proved that all PL 4-manifolds admitting semi-simple crystallizations, attain our bounds. We have also characterized the class of PL 4-manifolds which admit semi-simple crystallizations. These results are in Chapter 6.

Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)

The hippocampus is a complex brain structure that has been studied extensively and is subject to abnormal structural change in various neuropsychiatric disorders. The highest definition in vivo method of visualizing the anatomy of this structure is structural Magnetic Resonance Imaging (MRI). Gross structure can be assessed by the naked eye inspection of MRI scans but measurement is required to compare scans from individuals within normal ranges, and to assess change over time in individuals. The gold standard of such measurement is manual tracing of the boundaries of the hippocampus on scans. This is known as a Region Of Interest (ROI) approach. ROI is laborious and there are difficulties with test-retest and inter-rater reliability. These difficulties are primarily due to uncertainty in designation of the hippocampus boundary. An improved, less labour intensive and more reliable method is clearly desirable. This thesis describes a fully automated hybrid methodology that is able to first locate and then extract hippocampal volumes from 3D 1.5T MRI T1 brain scans automatically. The hybrid algorithm uses brain atlas mappings and fuzzy inference to locate hippocampal areas and create initial hippocampal boundaries. This initial location is used to seed a deformable manifold algorithm. Rule based deformations are then applied to refine the estimate of the hippocampus locations. Finally, the hippocampus boundaries are corrected through an inference process that assures adherence to an expected hippocampus volume. The ICC values of this methodology when compared to the manual segmentation of the same hippocampi result in a 0.73 for the left and 0.81 for the right hippocampi. These values both fall within the range of reliability testing according to the manual ‘gold standard’ technique. Thus, this thesis describes the development and validation of a genuinely automated approach to hippocampal volume extraction of potential utility in studies of a range of neuropsychiatric disorders and could eventually find clinical applications.

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

This thesis is concerned with Calabi-Yau threefolds and vector bundles upon them, which are the basic mathematical objects at the centre of smooth supersymmetric compactifications of heterotic string theory. We begin by explaining how these objects arise in physics, and give a brief review of the techniques of algebraic geometry which are used to construct and study them. We then turn to studying multiply-connected Calabi-Yau threefolds, which are of particular importance for realistic string compactifications. We construct a large number of new examples via free group actions on complete intersection Calabi-Yau manifolds (CICY's). For special values of the parameters, these group actions develop fixed points, and we show that, on the quotient spaces, this leads to a particular class of singularities, which are quotients of the conifold. We demonstrate that, in many cases at least, such a singularity can be resolved to yield another smooth Calabi-Yau threefold, with different Hodge numbers and fundamental group. This is a new example of the interconnectedness of the moduli spaces of distinct Calabi-Yau threefolds. In the second part of the thesis we turn to a study of two new `three-generation' manifolds, constructed as quotients of a particular CICY, which can also be represented as a hypersurface in dP6 x dP6, where dP6 is the del Pezzo surface of degree six. After describing the geometry of this manifold, and especially its non-Abelian quotient, in detail, we show how to construct on the quotient manifolds vector bundles which lead to four-dimensional heterotic models with the standard model gauge group and three generations of particles. The example described in detail has the spectrum of the minimal supersymmetric standard model plus a single vector-like pair of colour triplets.