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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenSat, 13 Apr 2024 02:04:06 +0000On the Link Floer Homology of L-space Link
https://resolver.caltech.edu/CaltechTHESIS:05222015-133207861
Authors: {'items': [{'email': 'nakuldawra@gmail.com', 'id': 'Dawra-Nakul', 'name': {'family': 'Dawra', 'given': 'Nakul'}, 'show_email': 'NO'}]}
Year: 2015
DOI: 10.7907/Z9CZ353T
We will prove that, for a 2 or 3 component L-space link, HFL<sup>-</sup> is completely determined by the multi-variable Alexander polynomial of all the sub-links of L, as well as the pairwise linking numbers of all the components of L. We will also give some restrictions on the multi-variable Alexander polynomial of an L-space link. Finally, we use the methods in this paper to prove a conjecture of Yajing Liu classifying all 2-bridge L-space links.https://thesis.library.caltech.edu/id/eprint/8879The Kapustin-Witten Equations with Singular Boundary Conditions
https://resolver.caltech.edu/CaltechTHESIS:05092018-094640290
Authors: {'items': [{'email': 'siqihe@pku.edu.cn', 'id': 'He-Siqi', 'name': {'family': 'He', 'given': 'Siqi'}, 'orcid': '0000-0002-3690-7355', 'show_email': 'NO'}]}
Year: 2018
DOI: 10.7907/GMA0-9Z96
<p>Witten proposed a fasinating program interpreting the Jones polynomial of knots on a 3-manifold by counting solutions to the Kapustin-Witten equations with singular boundary conditions.</p>
<p>In Chapter 1, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact four-manifold with <i>S</i><sup>3</sup> or <i>T</i><sup>3</sup> boundary, there exists a Nahm pole solution to the obstruction perturbed Kapustin-Witten equations. This is also the case for a four-manifold with hyperbolic boundary under some topological assumptions.</p>
<p>In Chapter 2, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. The imaginary parts of the solutions are singular. By rescaling, we find some limit behavior for these singular solutions. In addition, for any integer <i>k</i>, we can construct a 5|<i>k</i>| dimensional family of <i>C</i><sup>1</sup> solutions to the Kapustin-Witten equations on Euclidean space, again with singular imaginary parts. Moreover, we get solutions to the Kapustin-Witten equation with Nahm pole boundary condition over <i>S</i><sup>3</sup> × (0, +∞).</p>
<p>In Chapter 3, we develop a Kobayashi-Hitchin type correspondence for the extended Bogomolny equations on Σ× with Nahm pole singularity at Σ × {0} and the Hitchin component of the stable <i>SL</i>(2, ℝ) Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable <i>SL</i>(2, ℝ) Higgs bundles. We also prove the existence and uniqueness of solutions with knot singularities on ℂ × ℝ<sup>+</sup>. This is joint a work with Rafe Mazzeo.</p>
<p>In Chapter 4, for a 3-manifold <i>Y</i>, we study the expansions of the Nahm pole solutions to the Kapustin-Witten equations over <i>Y</i> × (0, +∞). Let <i>y</i> be the coordinate of (0, +∞) and assume the solution convergence to a flat connection at <i>y</i> → ∞, we prove the sub-leading terms of the Nahm pole solution is <i>C</i><sup>1</sup> to the boundary at <i>y</i> → 0 if and only if <i>Y</i> is an Einstein 3-manifold. For <i>Y</i> non-Einstein, the sub-leading terms of the Nahm pole solutions behave as <i>y</i> log <i>y</i> to the boundary. This is a joint work with Victor Mikhaylov.</p>https://thesis.library.caltech.edu/id/eprint/10867Self-Gluing Formula of the Monopole Invariant and its Application on Symplectic Structures
https://resolver.caltech.edu/CaltechTHESIS:05252018-080955604
Authors: {'items': [{'email': 'ghjeong0717@gmail.com', 'id': 'Jeong-Gahye', 'name': {'family': 'Jeong', 'given': 'Gahye'}, 'orcid': '0000-0003-3273-7691', 'show_email': 'YES'}]}
Year: 2018
DOI: 10.7907/BH06-KS91
<p>Seiberg-Witten theory has been an important tool in studying a class of 4-manifolds. Moreover, the Seiberg-Witten invariants have been used to compute for simple structures of symplectic manifolds. The normal connected sum operation on 4- manifolds has been used to construct 4-manifolds. In this thesis, we demonstrate how to compute the Seiberg-Witten invariant of 4-manifolds obtained from the normal connected sum operation. In addition, we introduce the application of the formula on the existence of symplectic structures of manifolds given by the normal connected sum.</p>
<p>In Chapter 1, we study the Seiberg-Witten theory for various types of 3- and 4- manifolds. We review the Seiberg-Witten equation and invariants for 4-manifolds with cylindrical ends as well as closed and smooth 4-manifolds . Furthermore, we explain how to compute the Seiberg-Witten invariants for two types of 4-manifolds: the products of a circle and a 3-manifold and sympectic manifolds.</p>
<p>In Chapter 2, we prove that the Seiberg-Witten invariant of a new manifold obtained from the normal connected sum can be represented by the Seiberg-Witten invariant of the original manifolds. In [Tau01], the author has proved the case of the operation along tori. In [MST96], the authors have proved the case of the operation along surfaces with genus at least 2 when the product of the circle and the surface is separating in the ambient 4-manifold. In this thesis, we show the proof of the remaining case.</p>
<p>In Chapter 3, we prove the existence of certain symplectic structures on manifolds obtained from the normal connected sum of two 4-manifolds using the multiple gluing formula stated in Chapter 2. We explain how to construct covering spaces of the manifold and compute the Seiberg-Witten invariant of the covering spaces by the gluing formula. From the relation between the Seiberg-Witten invariants and symplectic structures, we prove the main application.</p>https://thesis.library.caltech.edu/id/eprint/10934