Thesis advisor: Martin J. Bridgeman / Thesis advisor: Ian Biringer / The first part of this dissertation is on the quasiconformal homogeneity of surfaces. In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces. We then move on to identities on hyperbolic manifolds. We study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

One of the most important tools to study the geometry of irreducible holomorphic symplectic manifolds is the monodromy group. The first part of this dissertation concerns the construction and studyof monodromy operators on irreducible holomorphic symplectic manifolds which are deformation equivalent to the 10-dimensional example constructed by O'Grady. The second part uses the knowledge of the monodromy group to compute the number of connected components of moduli spaces of bothmarked and polarised irreducible holomorphic symplectic manifolds which are deformationequivalent to generalised Kummer varieties.

The accuracy of modern digital human models has led to the development of human simulation engines capable of performing a complex analysis of the biometrics and kinematics / dynamics of a digital model. While the capabilities of these simulations have seen much progress in recent years, they are hindered by a fundamental limitation regarding the diversity of the models compatible with the simulation engine, which in turn results in a reduction in the scope of the applications available to the simulation. This is typically due to the necessary implementation of a musculoskeletal structure within the model, as well as the inherent mass and inertial data that accompany it. As a result a significant amount of time and expertise is required to make a digital human model compatible with the simulation. In this research I present a solution to this limitation by outlining a process to develop a set of mutually compatible human models that spans the range of feasible body shapes and allows for a “free” exploration of body shape within the shape manifold. Additionally, a method is presented to represent the human body shapes with a reduction of dimensionality, via a spectral shape descriptor, that enables a statistical analysis that is both more computationally efficient and anthropometrically accurate than traditional methods. This statistical analysis is then used to develop a set of representative models that succinctly represent the full scope of human body shapes across the population, with applications reaching beyond the research-oriented simulations into commercial human-centered product design and digital modeling.

We present a unifying framework in which "object-independent" modes of variation are learned from continuous-time data such as video sequences. These modes of variation can be used as "generators" to produce a manifold of images of a new object from a single example of that object. We develop the framework in the context of a well-known example: analyzing the modes of spatial deformations of a scene under camera movement. Our method learns a close approximation to the standard affine deformations that are expected from the geometry of the situation, and does so in a completely unsupervised (i.e. ignorant of the geometry of the situation) fashion. We stress that it is learning a "parameterization", not just the parameter values, of the data. We then demonstrate how we have used the same framework to derive a novel data-driven model of joint color change in images due to common lighting variations. The model is superior to previous models of color change in describing non-linear color changes due to lighting.

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For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.

This thesis researches techniques for modelling static facial expressions, as well as the dynamics of continuous facial motion. We demonstrate how static and dynamic properties of facial expressions can be represented within a linear and nonlinear context, respectively. These two representations do not act in isolation, but are mutually reinforcing in conceding a cohesive framework for the analysis, animation, and manipulation of expressive faces. We derive a basis for the linear space of expressions through Principal Components Analysis (PCA). We introduce and formalize the notion of "expression manifolds", manifolds residing in PCA space that model motion dynamics for semantically similar expressions. We then integrate these manifolds into an animation workflow by performing Nonlinear Dimensionality Reduction (NLDR) on the expression manifolds. This operation yields expression maps that encode a wealth of information relating to complex facial dynamics, in a low dimensional space that is intuitive to navigate and efficient to manage.

This thesis researches techniques for modelling static facial expressions, as well as the dynamics of continuous facial motion. We demonstrate how static and dynamic properties of facial expressions can be represented within a linear and nonlinear context, respectively. These two representations do not act in isolation, but are mutually reinforcing in conceding a cohesive framework for the analysis, animation, and manipulation of expressive faces. We derive a basis for the linear space of expressions through Principal Components Analysis (PCA). We introduce and formalize the notion of "expression manifolds", manifolds residing in PCA space that model motion dynamics for semantically similar expressions. We then integrate these manifolds into an animation workflow by performing Nonlinear Dimensionality Reduction (NLDR) on the expression manifolds. This operation yields expression maps that encode a wealth of information relating to complex facial dynamics, in a low dimensional space that is intuitive to navigate and efficient to manage.

The qualitative case study on which this thesis is based was designed to investigate the relationship between migration and identity construction of three young people who immigrated as children and adolescents, two of them as refugees, from the former Yugoslavia to Germany and finally to Canada. The autobiographical narrative interviews of the manifold migration stories were mainly analyzed from the point of view of Vygotsky’s Sociocultural Theory, which considers speech and thought in a close relation, to illustrate how identity must be understood as both individual and social in nature, and as a complex narrative action. The socialization processes in all countries of migration were viewed in order to investigate how the previous acculturation affects the cultural identity of the young people today and how it is unfolded in the story. The study reveals that these subjects with threefold migration position themselves between their lifeworlds which enable them not only to say where they belong to or which is their homeland but to answer the simple question “Who am I?” This is revealed by the way of their narration which contains a lot of contradictions. The individuals deal differently with their dynamic identity construction, while one of them seems to suffer under the instability of his identity, and of being different in all his lifeworlds, other subjects however can see advantages related to it. The way how they deal with this dynamics has a crucial influence on their view of their migrations today and consequently on their identity construction as a narrative action.

In the first part of this thesis, we will explore an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Some attention will be given to finding approximations to solutions which are themselves flows. Moreover, we will address the writing of one component in terms of another in the case of a planar system. In the second part of this thesis, we will explore the Michaelis-Menten mechanism of a single enzyme-substrate reaction. The focus is an analysis of the planar reduction in phase space or, equivalently, solutions of the scalar reduction. In particular, we will prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Also, we will determine the concavity of all solutions in the first quadrant. Moreover, we will establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we will determine the asymptotic behaviour of the slow manifold at infinity. Additionally, we will study the planar reduction. In particular, we will find non-trivial bounds on the length of the pre-steady-state period, determine the asymptotic behaviour of solutions as time tends to infinity, and determine bounds on the solutions valid for all time. In the third part of this thesis, we explore the (nonlinear) Lindemann mechanism of unimolecular decay. The analysis will be similar to that for the Michaelis-Menten mechanism with an emphasis on the differences. In the fourth and final part of this thesis, we will present some open problems.