Mathematics Theses and Dissertations

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https://hdl.handle.net/1794/2044

This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Mathematics Theses and Dissertations by Author "Botvinnik, Boris"

Now showing 1 - 4 of 4
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    Gluing manifolds with boundary and bordisms of positive scalar curvature metrics
    (University of Oregon, 2017-09-06) Kazaras, Demetre; Botvinnik, Boris
    This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.
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    Linking Forms, Singularities, and Homological Stability for Diffeomorphism Groups of Odd Dimensional Manifolds
    (University of Oregon, 2015-08-18) Perlmutter, Nathan; Botvinnik, Boris
    Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)-dimensional manifolds, with respect to forming the connected sum with (2n-1)-connected, (4n+1)-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest.
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    Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings
    (University of Oregon, 2019-09-18) Burdick, Bradley; Botvinnik, Boris
    The classification of simply connected manifolds admitting metrics of positive scalar curvature of initiated by Gromov-Lawson, at its core, relies on a careful geometric construction that preserves positive scalar curvature under surgery and, in particular, under connected sum. For simply connected manifolds admitting metrics of positive Ricci curvature, it is conjectured that a similar classification should be possible, and, in particular, there is no suspected obstruction to preserving positive Ricci curvature under connected sum. Yet there is no general construction known to take two Ricci-positive Riemannian manifolds and form a Ricci-positive metric on their connected sums. In this work, we utilize and extend Perelman’s construction of Ricci-positive metrics on connected sums of complex projective planes, to give an explicit construction of Ricci-positive metrics on connected sums given that the individual summands admit very specific Ricci- positive metrics, which we call core metrics. Working towards the new goal of constructing core metrics on manifolds known to support metrics of positive Ricci curvature: we show how to generalize Perelman’s construction to all projective spaces, we show that the existence of core metrics is preserved under iterated sphere bundles, and we construct core metrics on certain boundaries of plumbing disk bundles over spheres. These constructions come together to give many new examples of Ricci-positive connected sums, in particular on the connected sum of arbitrary products of spheres and on exotic projective spaces.
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    Scalar Curvature and Transfer Maps in Spin and Spin^c Bordism
    (University of Oregon, 2024-01-10) Granath, Elliot; Botvinnik, Boris
    In 1992, Stolz proved that, among simply connected Spin-manifolds of dimension5 or greater, the vanishing of a particular invariant α is necessary and sufficient for the existence of a metric of positive scalar curvature. More precisely, there is a map α: ΩSpin → ko (which may be realized as the index of a Dirac operator) ∗ which Hitchin established vanishes on bordism classes containing a manifold with a metric of positive scalar curvature. Stolz showed kerα is the image of a transfer map ΩSpinBPSp(3) → ΩSpin. In this paper we prove an analogous result for Spinc- ∗−8 ∗ manifolds and a related invariant αc : ΩSpinc → ku. We show that ker αc is the ∗ sum of the image of Stolz’s transfer ΩSpinBPSp(3) → ΩSpinc and an analogous map ∗−8 ∗ ΩSpinc BSU(3) → ΩSpinc . Finally, we expand on some details in Stolz’s original paper ∗−4 ∗ and provide alternate proofs for some parts.