# Browsing by Subject "Riemannian geometry"

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Item Open Access Complete noncompact g2-manifolds from asymptotically conical calabi-yau 3-folds(Duke Mathematical Journal, 2021-10-15) FOSCOLO, L; HASKINS, M; NORDSTRÖM, JWe develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7- manifolds .M;g/ whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger-Fukaya-Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M ! B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics g_ on M that collapses with bounded curvature as _ ! 0 to the original Calabi-Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.Item Open Access General and Efficient Bayesian Computation through Hamiltonian Monte Carlo Extensions(2017) Nishimura, AkihikoHamiltonian Monte Carlo (HMC) is a state-of-the-art sampling algorithm for Bayesian computation. Popular probabilistic programming languages Stan and PyMC rely on HMC’s generality and efficiency to provide automatic Bayesian inference platforms for practitioners. Despite its wide-spread use and numerous success stories, HMC has several well known pitfalls. This thesis presents extensions of HMC that overcome its two most prominent weaknesses: inability to handle discrete parameters and slow mixing on multi-modal target distributions.

Discontinuous HMC (DHMC) presented in Chapter 2 extends HMC to discontinuous target distributions – and hence to discrete parameter distributions through embedding them into continuous spaces — using an idea of event-driven Monte Carlo from the computational physics literature. DHMC is guaranteed to outperform a Metropolis-within-Gibbs algorithm since, as it turns out, the two algorithms coincide under a specific (and sub-optimal) implementation of DHMC. The theoretical justification of DHMC extends an existing theory of non-smooth Hamiltonian mechanics and of measure-valued differential inclusions.

Geometrically tempered HMC (GTHMC) presented in Chapter 3 improves HMC’s performance on multi-modal target distributions. The efficiency improvement is achieved through differential geometric techniques, relating a target distribution to

another distribution with less severe multi-modality. We establish a geometric theory behind Riemannian manifold HMC to motivate our geometric tempering methods. We then develop an explicit variable stepsize reversible integrator for simulating

Hamiltonian dynamics to overcome a stability issue of the usual Stormer-Verlet integrator. The integrator is of independent interest, being the first of its kind designed specifically for HMC variants.

In addition to the two extensions described above, Chapter 4 describes a variable trajectory length algorithm that generalizes the acceptance and rejection procedure of HMC — and in fact of any reversible dynamics based samplers — to allow for more flexible choices of trajectory lengths. The algorithm in particular enables an effective application of a variable stepsize integrator to HMC extensions, including GTHMC. The algorithm is widely applicable and provides a recipe for constructing valid dynamics based samplers beyond the known HMC variants. Chapter 5 concludes the thesis with a simple and practical algorithm to improve computational efficiencies of HMC and related algorithms over their traditional implementations.

Item Open Access Seven-Dimensional Geometries With Special Torsion(2019) Ball, GavinI use the methods of exterior differential systems and the moving frame to study two geometric structures in seven dimensions related to $G_2$-geometry, and linked by the idea of special torsion. The torsion tensor of a geometric structure is a basic first-order invariant of the structure, and both of the geometries I study have special torsion, meaning that the image of their torsion tensor is constrained to lie in a smaller than usual subset.

In part 1, I study quadratic closed $G_2$-structures. A closed $G_2$-structure on a 7-manifold $M$ is given by a closed nondegenerate 3-form $\varphi$, and the quadratic condition, introduced by Bryant, says that $\varphi$ satisfies one of a particular natural one-parameter family of second order equations. The torsion tensor associated to a closed $G_2$-structure $\varphi$ takes values in $\mathfrak{g}_2$, and I study the cases where the image of this map lies in an exceptional $G_2$-orbit. A closed $G_2$-structure $\varphi$ induces a metric, and I give a classification of closed $G_2$-structures with conformally flat induced metric.

In part 2, I study $G_2$-structures endowed with a distribution of calibrated planes. In this situation there is an induced $SO(4)$-structure, and I invesitigate the cases where the $G_2$-structure is torsion-free and the induced $SO(4)$-structure has torsion tensor taking values in an irreducible $SO(4)$-module. Additionally, I give a classification of $SO(4)$-structures with invariant torsion, meaning that their torsion tensor takes values in a direct sum of trivial $SO(4)$-modules.