Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Differential geometry is deceptively simple. It is surprisingly easyto get the right answer with unclear and informal symbol manipulation.To address this problem we use computer programs to communicate aprecise understanding of the computations in differential geometry.Expressing the methods of differential geometry in a computer languageforces them to be unambiguous and computationally effective. The taskof formulating a method as a computer-executable program and debuggingthat program is a powerful exercise in the learning process. Also,once formalized procedurally, a mathematical idea becomes a tool thatcan be used directly to compute results.

In this thesis we study the finite quotients of 3-manifold groups, concerning both residual properties of the groups and the properties of the 3-manifolds that can be detected using finite quotients of the fundamental group. A key theme is the analysis of when two 3-manifold groups can have the same families of finite quotients. We make a detailed study of this 'profinite rigidity' problem for Seifert fibre spaces and prove complete classification results for these manifolds. From Seifert fibre spaces we continue on this trajectory and extend our classification results to all graph manifolds. We illustrate this classification with examples and several consequences, including for graph knots and for mapping class groups. The third part of the thesis concerns the behaviour of the finite p-group quotients of 3-manifold groups. In general these quotients may be scarce and poorly behaved. We give results showing that some of these issues may be resolved by passing to finite-sheeted covers of the manifold involved. We also prove theorems concerning the p-conjugacy separability of certain graph manifold groups. The concluding chapter of the thesis collects other results linking low-dimensional topology and finite quotients of groups. In particular we prove that finite quotients of a right-angled Artin group distinguish it from other right-angled Artin groups, and we give an argument detecting the prime decomposition of certain 3-manifold groups from the finite p-group quotients.

Our aim is to study the role of omnioriented quasitoric manifolds in equivariant complex bordism. These are a well-behaved class of even-dimensional smooth closed manifolds with the action of a half-dimensional compact torus and an equivariant stably complex structure. They are beneficial objects to work with as they can be described completely in terms of combinatorial data.We include an overview of equivariant complex bordism, highlighting the relationship between localisation and restriction to fixed point data. By keeping in mind the particularly interesting case when the group in question is the compact torus, we revisit work found in [BPR10], reinterpreting and expanding certain results relating to the universal toric genus.We then consider oriented torus graphs of stably complex torus manifolds and classify these using a boundary operator on exterior polynomials related to geometric equivariant complex bordism classes of the manifolds. We also extend the connected sum construction of quasitoric pairs which allows for a more general notion of the equivariant connected sum of omnioriented quasitoric manifolds.We then consider whether an equivariant version of Buchstaber and Ray’s result in [BR98] holds; that is, does there exist an omnioriented quasitoric manifold in every geometric equivariant complex bordism class in which they naturally exist? We conjecture that this is true showing that we have a combinatorial model for such objects and exhibiting low-dimensional examples.

The study of bifurcations of vectorfields is concerned with changes in qualitative behaviour that can occur when a non-structurally stable vectorfield is perturbed. In a sense, this is the study of how such a vectorfield fails to be structurally stable. Finding a systematic approach to the study of such questions is a difficult problem. One approach to bifurcations of vectorfields, known as "generic bifurcation theory," is the subject of much of the work of Sotomayor (Sotomayor [1973a], Sotomayor [1973b],Sotomayor [1974]). This approach attempts to construct generic families of k-parameter vectorfields (usually for k=1), for which all the bifurcations can be described. In Sotomayor [1973a] it is stated that the vectorfields associated with the "generic" bifurcations of individual critical elements for k-parameter vectorfields form submanifolds of codimension ≤ k of the Banach space ϰʳ (M) of vectorfields on a compact manifold M. The bifurcations associated with one of these submanifolds of codimension-k are called generic codimension-k bifurcations. In Sotomayor [1974] the construction of these submanifolds and the description of the associated bifurcations of codimension-1 for vectorfields on two dimensional manifolds is presented in detail. The bifurcations that occur are due to the parameterised vectorfield crossing one of these manifolds transversely as the parameter changes. Abraham and Robbin used transversality results for evaluation maps to prove the Kupka-Smale theorem in Abraham and Robbin [1967]. In this thesis, we shall show how an extension of these evaluation transversality techniques will allow us to construct the submanifolds of ϰʳ (M) associated with one type of generic bifurcation of critical elements, and we shall consider how this approach might be extended to include the other well known generic bifurcations. For saddle-node type bifurcations of critical points, we will show that the changes in qualitative behaviour are related to geometric properties of the submanifold Σ₀ of ϰʳ (M) x M, where Σ₀ is the pull-back of the set of zero vectors-or zero section-by the evaluation map for vectorfields. We will look at the relationship between the Taylor series of a vector-field X at a critical point ⍴ and the geometry of Σ₀ at the corresponding point (X,⍴) of ϰʳ (M) x M. This will motivate the non-degeneracy conditions for the saddle-node bifurcations, and will provide a more general geometric picture of this approach to studying bifurcations of critical points. Finally, we shall consider how this approach might be generalised to include other bifurcations of critical elements. / Science, Faculty of / Mathematics, Department of / Graduate

This dissertation discusses the time and place that Brownian motion on a Riemannian manifold first exit a normal ball of small radius. A general procedure is given for computing asymptotic expansions of joint moments of the first exit time and place random variables as the radius of the geodesic ball decreases to zero. The asymptotic expansion of the joint Laplace transform of exit time and spherical harmonics of exit position is derived for a ball of small radius. A generalized Pizetti’s formula is used to expand the solution of the related partial differential equations. These expansions are represented in terms of curvature in the manifold. Asymptotic Independence Conditions (AIC) and Asymptotic Uncorrelated Conditions (AUC) are defined for the joint distributions of exit time and place. Computations using the methods developed in this work demonstrate that AIC and AUC produce the same curvature conditions up to a certain level of asymptotics. It is conjectured that AUC implies AIC. Further, a generalized method is given for computing the Laplace transform, and therefore the moments of the exit time. This work is related to and also extends the work of M. Liao and H. R. Hughes in stochastic geometric analysis.

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order Fourier-Taylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of a-posteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2021. / FAU Electronic Theses and Dissertations Collection

In this thesis, we study three problems in complex geometry. In the first part, we study the behavior of the Kahler-Ricci flow on complete non-compact manifolds with negative holomorphic curvature. We show that Kahler-Ricci flow converges to a Kahler-Einstein metric when the initial manifold admits a suitable exhaustion function, thus improving upon a result of D. Wu and S.T. Yau. These results are partly obtained in joint work with S. Huang, M.-C. Lee and L.-F. Tam. In the second part of this thesis, we introduce a new Kodaira-Bochner type formula for closed (1, 1)-form in non-Kahler geometry. Based on this new formula, We propose a new curvature positivity condition in non-Kahler manifolds and proved a strong rigidity type theorem for manifolds satisfying this curvature positivity condition. We also find interesting examples non-Kahler manifolds satisfying the curvature positivity condition in a class of manifolds called Vaisman manifolds. In the third part of this thesis, we study the degenerations of asymptotically conical Calabi-Yau manifolds as the Kahler class degenerates to a non-Kahler class. Under suitable hypothesis, we prove the convergence of asymptotically conical Calabi-Yau metrics to a singular asymptotically conical Calabi-Yau current with compactly supported singularities. Using this, we construct singular asymptotically conical Calabi-Yau metrics on non-compact singular varieties and identify the topology of these singular metrics with the singular variety. We also give some interpretations of these asymptotically conical Calabi-Yau metrics from the point of view of physics. These results are obtained in joint work with T. Collins and B. Guo.

We study hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. We consider individual hyperbolic structures as well as special regions in the space of all such hyperbolic structures. We use some properties of the boundary hyperbolic structures on C to establish an interesting property of cusp shapes of tunnel number one manifolds. This extends a result of Nimershiem in [26] to the class of tunnel number one manifolds. We also establish convergence results on the geometry of compression bodies. This extends the work of Ito in [13] from the punctured-torus case to the compression body case.