Thesis advisor: Robert Meyerhoff / Thurston showed that for all but a finite number of Dehn Surgeries on a cusped hyperbolic 3-manifold, the resulting manifold admits a hyperbolic structure. Global bounds on this number have been set, and gradually improved upon, by a number of Mathematicians until Lackenby and Meyerhoff proved the sharp bound of 10, which is realized by the figure-eight knot exterior. We improve this result by proving a stronger version of Gordon’s conjecture: that excluding the figure-eight knot exterior, cusped hyperbolic 3-manifolds have at most 8 non-hyperbolic Dehn Surgeries. To do so we make use of the work of Gabai et. al. from a forthcoming paper which parameterizes measurements of the cusp, then uses a rigorous computer aided search of the space to classify all hyperbolic 3-manifolds up to a specified cusp size. Their approach hinges on the discreteness of manifold points in the parameter space, an assumption which cannot be made if the manifolds have infinite volume. In this paper we also show that infinite-volume manifolds, which must be Free Bicuspid, can have cusp volume as low as 3.159. As such, these manifolds are a concern for any future expansion of the approach of Gabai et. al. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis investigates the structure and properties of hyperbolic 3-manifold groups (particularly knot and link groups) and arithmetic Kleinian groups. In Chapter 2, we establish a stronger version of a conjecture of A. Reid and others in the arithmetic case: if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic (and hence peripheral). In Chapter 3, we identify all Kleinian groups that can be generated by two elements for which equality holds in Jørgensen’s Inequality in two cases: torsion-free Kleinian groups and non-cocompact arithmetic Kleinian groups. / text

Thesis advisor: George R. Meyerhoff / Recent advances in normal surface algorithms enable the determination by computer of the hyperbolicity of compact orientable 3-manifolds with zero Euler characteristic and nonempty boundary. Recent advances in hyperbolic geometry enable the determination by computer of the Dehn paternity relation between two orientable compact hyperbolic 3-manifolds. Presented here is an exposition of these developments, along with prototype implementations of one of these determinations in software. These have applications to two questions about Mom technology. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis presents an algorithm for producing 4–manifold triangulations with boundary an arbitrary orientable, closed, triangulated 3–manifold. The research is an extension of Costantino and Thurston’s work on determining upper bounds on the number of 4–dimensional simplices necessary to construct such a triangulation. Our first step in this bordism construction is the geometric partitioning of an initial 3–manifold M using smooth singularity theory. This partition provides handle attachment sites on the 4–manifold Mx[0,1] and the ensuing handle attachments eliminate one of the boundary components of Mx[0,1], yielding a 4-manifold with boundary exactly M. We first present the construction in the smooth case before extending the smooth singularity theory to triangulated 3–manifolds. / Graduate

Le résultat principal est une caractérisation homotopique des orbifolds de dimension 3 qui sont fibrés de Seifert : si O est un orbifold de dimension 3 fermé, orientable et petit dont le groupe fondamental admet un sous-groupe infini cyclique normal, alors O est de Seifert. Ce théorème généralise un résultat de Scott, Mess, Tukia, Gabai et Casson-Jungreis pour les variétés. Il repose sur une caractérisation des groupes de surfaces virtuels comme groupes quasi-isométriques à un plan riemannien complet. D'autres résultats sur les quasi-isométries entre groupes et surfaces sont obtenus.

abstract: Continuous monitoring of sensor data from smart phones to identify human activities and gestures, puts a heavy load on the smart phone's power consumption. In this research study, the non-Euclidean geometry of the rich sensor data obtained from the user's smart phone is utilized to perform compressive analysis and efficient classification of human activities by employing machine learning techniques. We are interested in the generalization of classical tools for signal approximation to newer spaces, such as rotation data, which is best studied in a non-Euclidean setting, and its application to activity analysis. Attributing to the non-linear nature of the rotation data space, which involve a heavy overload on the smart phone's processor and memory as opposed to feature extraction on the Euclidean space, indexing and compaction of the acquired sensor data is performed prior to feature extraction, to reduce CPU overhead and thereby increase the lifetime of the battery with a little loss in recognition accuracy of the activities. The sensor data represented as unit quaternions, is a more intrinsic representation of the orientation of smart phone compared to Euler angles (which suffers from Gimbal lock problem) or the computationally intensive rotation matrices. Classification algorithms are employed to classify these manifold sequences in the non-Euclidean space. By performing customized indexing (using K-means algorithm) of the evolved manifold sequences before feature extraction, considerable energy savings is achieved in terms of smart phone's battery life. / Dissertation/Thesis / M.S. Electrical Engineering 2014

We study the relationship between subdivision rules, 3-dimensional manifolds, and circle packings. We find explicit subdivision rules for closed right-angled hyperbolic manifolds, a large family of hyperbolic manifolds with boundary, and all 3-manifolds of the E^3,H^2 x R, S^2 x R, SL_2(R), and S^3 geometries (up to finite covers). We define subdivision rules in all dimensions and find explicit subdivision rules for the n-dimensional torus as an example in each dimension. We define a graph and space at infinity for all subdivision rules, and use that to show that all subdivision rules for non-hyperbolic manifolds have mesh not going to 0. We provide an alternate proof of the Combinatorial Riemann Mapping Theorem using circle packings (although this has been done before). We provide a new definition of conformal for subdivision rules of unbounded valence, show that the subdivision rules for the Borromean rings complement are conformal and show that barycentric subdivision is almost conformal. Finally, we show that subdivision rules can be degenerate on a dense set, while still having convergent circle packings.

Generalized geometry of type B_n is the study of geometric structures in T+T^*+1, the sum of the tangent and cotangent bundles of a manifold and a trivial rank 1 bundle. The symmetries of this theory include, apart from B-fields, the novel A-fields. The relation between B_n-geometry and usual generalized geometry is stated via generalized reduction. We show that it is possible to twist T+T^*+1 by choosing a closed 2-form F and a 3-form H such that dH+F²=0. This motivates the definition of an odd exact Courant algebroid. When twisting, the differential on forms gets twisted by d+Fτ+H. We compute the cohomology of this differential, give some examples, and state its relation with T-duality when F is integral. We define B_n-generalized complex structures (B_n-gcs), which exist both in even and odd dimensional manifolds. We show that complex, symplectic, cosymplectic and normal almost contact structures are examples of B_n-gcs. A B_n-gcs is equivalent to a decomposition (T+T^*+1)_ℂ= L + L + U. We show that there is a differential operator on the exterior bundle of L+U, which turns L+U into a Lie algebroid by considering the derived bracket. We state and prove the Maurer-Cartan equation for a B_n-gcs. We then work on surfaces. By the irreducibility of the spinor representations for signature (n+1,n), there is no distinction between even and odd B_n-gcs, so the type change phenomenon already occurs on surfaces. We deal with normal forms and L+U-cohomology. We finish by defining G²₂-structures on 3-manifolds, a structure with no analogue in usual generalized geometry. We prove an analogue of the Moser argument and describe the cone of G²₂-structures in cohomology.

Dans cette thèse, nous considérons quatre cas de conjectures de Boyd pour la mesure de Mahler de polynômes. Le premier cas concerne un polynôme associé à une courbe de genre 1, deux autres cas couvrent des courbes de genre 2, et le dernier cas traite d’une courbe de genre 3. Pour le cas de la courbe de genre 1, nous étudions une identité conjecturée par Boyd et prouvée par Boyd et Rodriguez-Villegas. On trouve un expression de la mesure de Mahler donnée par une combinaison linéaire de certaines valeurs du dilogarithme de Bloch-Wigner. En combinant cela avec le résultat prouvé par Boyd et Rodriguez-Villegas, nous pouvons établir certaines identités entre différentes valeurs du dilogarithme de Bloch-Wigner. Pour les problèmes liés aux courbes de genre 2, nous utilisons le régulateur elliptique pour récupérer des identités entre les mesures de Mahler des certaines familles de courbes de genre 2 qui ont ́eté conjecturées par Boyd et prouvèes par Bertin et Zudilin en différenciant le paramètre des formules de la mesure de Mahler et en utilisant des identités hypergéométriques. Pour le cas impliquant la courbe de genre 3, nous utilisons le régulateur elliptique pour prouver une identité entièrement nouvelle entre les mesures de Mahler d’une famille polynomiale de genre 3 et d’une famille polynomiale de genre 1 qui à été initialement conjectur ́ee par Liu et Qin. Comme nos preuves pour les cas des courbes des genres 2 et 3 impliquent le régulateur, elles éclairent la relation des mesures de Mahler des familles des genres 2 ou 3 avec des valeurs spéciales des fonctions L associées aux familles de genre 1. / In this dissertation, we consider four cases of Boyd’s conjectures for the Mahler measure of polynomials. The first case involves a polyno- mial defining a genus 1 curve, two other cases cover genus 2 curves, and the final case deals with a genus 3 curve. For the case of the genus 1 curve, we study an identity conjectured by Boyd and proven by Boyd and Rodriguez-Villegas. We find an expression of the Mahler measure given by a linear combination of some values of the Bloch-Wigner dilogarithm. Combining this with the result proven by Boyd and Rodriguez-Villegas, we can establish some identities among different values of the Bloch-Wigner dilogarithm. For the problems related to the genus 2 curves, we use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by differentiating the parameter in the Mahler measure formulas and using hypergeometric identities. For the case involving the genus 3 curve, we use the elliptic regulator to prove an entirely new identity between the Mahler measures of a genus 3 polynomial family and of a genus 1 polynomial family that was initially conjectured by Liu and Qin. Since our proofs for the cases of genus 2 and 3 curves involve the regulator, they yield light into the relation of the Mahler measures of the genus 2 or 3 families with special values of the L-functions associ- ated to the genus 1 families.

We study two topics on surface diffeomorphisms, their mapping classes and dynamics. For the mapping classes of a punctured disc, we study the $\ZxZ$ subgroups of the fundamental groups of the corresponding mapping tori. An application is the proof of the fact that a satellite knot with braid pattern is prime. For the mapping classes of the disc minus a Cantor set, we study a special type of reducible mapping class. This has direct application on the embeddings of solenoids in $\mathbb{S}^3$. We also give some examples of other types of mapping classes of the disc minus a Cantor set. For the dynamics of surface diffeomorphisms, we prove three formulas for computing the topological pressure of a $C^1$-generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For $C^1$ generic conservative diffeomorphisms on compact surfaces with no dominated splitting and $\phi_m(x):=-\frac{1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb{N}$, we show that there exist equilibrium states with zero entropy and there exists a transition point $t_0$ for the one parameter family $\lbrace t \phi_m\rbrace_{t\geq 0}$, such that there is no equilibrium states for $ t \in [0, t_0)$ and there is an equilibrium state for $t \in [t_0,+\infty)$.