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 Title
 Longtime convergence of harmonic map heat flows from surfaces into Riemannian manifolds
 Creator
 Choi, Kwangho
 Date
 2011
 Collection
 Electronic Theses & Dissertations
 Description

We study the longtime convergence of harmonic map heat flows from a closed Riemann surface into a compact Riemannian manifold. P. Topping constructed an example of a flow that does not converge in the infinitetime limit. Motivated by the observation that Topping's flow has accumulation points at which the Hessian of the energy function is degenerate, we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive definite, and (b) no...
Show moreWe study the longtime convergence of harmonic map heat flows from a closed Riemann surface into a compact Riemannian manifold. P. Topping constructed an example of a flow that does not converge in the infinitetime limit. Motivated by the observation that Topping's flow has accumulation points at which the Hessian of the energy function is degenerate, we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive definite, and (b) no bubbling occurs at infinite time. In addition, we present examples of heat flows for geodesics which show that the convexity of the energy function and convergence at infinite time may not hold even for 1dimensional harmonic map heat flows.
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 Title
 Jonestype link invariants and applications to 3manifold topology
 Creator
 Lee, Christine Ruey Shan
 Date
 2015
 Collection
 Electronic Theses & Dissertations
 Description

"It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semiadequate link has a welldefined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semiadequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new...
Show more"It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semiadequate link has a welldefined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semiadequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new family of pretzel knots which are not adequate. We also study the relationship between the Jones polynomial and the topology of the knot complement by relating its coefficients to the nonorientable genus of an alternating knot. The twosided bound we obtain can often determine the nonorientable genus. Lastly, we develop the connection between the Jones polynomial and a knot invariant coming from Heegaard Floer homology by using it to classify 3braids which are Lspace knots."  Abstract.
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 Title
 Statistical properties of some almost Anosov systems
 Creator
 Zhang, Xu
 Date
 2016
 Collection
 Electronic Theses & Dissertations
 Description

We investigate the polynomial lower and upper bounds for decay of correlations of a class of twodimensional almost Anosov diffeomorphisms with respect to their SinaiRuelleBowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the...
Show moreWe investigate the polynomial lower and upper bounds for decay of correlations of a class of twodimensional almost Anosov diffeomorphisms with respect to their SinaiRuelleBowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and can be expressed by using coefficients of the third order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed points.We discuss the relationship between the existence of SRB measures and the differentia bility of some almost Anosov diffeomorphisms near the indifferent fixed points in dimensions bigger than one. The eigenvalue of Jacobian matrix at the indifferent fixed point along the onedimensional contraction subspace is less than one, while the other eigenvalues along the expansion subspaces are equal to one. As a consequence, there are twicedifferentiable al most Anosov diffeomorphisms that admit infinite SRB measures in two or threedimensional spaces; there exist twicedifferentiable almost Anosov diffeomorphisms with SRB measures in dimensions bigger than three. Further, we obtain the polynomial lower and upper bounds for the correlation functions of these almost Anosov maps that admit SRB measures.
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 Title
 A combinatorial approach to knot theory : volume bounds for hyperbolic semiadequate link complements
 Creator
 Giambrone, Adam Joseph
 Date
 2014
 Collection
 Electronic Theses & Dissertations
 Description

An interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semiadequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semiadequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed...
Show moreAn interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semiadequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semiadequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed braids, and families of links that have both plat and closed braid aspects. By more closely studying each of these families, we can often improve the lower bounds on volume provided by the main result. Furthermore, we show that the bounds on volume can be expressed in terms of a single stable coefficient of the colored Jones polynomial. By doing this, we provide new collections of links that satisfy a Coarse Volume Conjecture. The main approach of this entire work is to use a combinatorial perspective to study the connections among knot theory, hyperbolic geometry, and graph theory.
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 Title
 Topics in knot theory : on generalized crossing changes and the additivity of the Turaev genus
 Creator
 Balm, Cheryl Lyn Jaeger
 Date
 2013
 Collection
 Electronic Theses & Dissertations
 Description

We first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev...
Show moreWe first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev genus one.
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 Title
 The Homology Polynomial and the Burau Representation for PseudoAnosov Braids
 Creator
 Shultz, Warren Michael
 Date
 2021
 Collection
 Electronic Theses & Dissertations
 Description

The homology polynomial is an invariant for pseudoAnosov mapping classes \cite{Birman2010}. We study the homology polynomial as an invariant for pseudoAnosov braids and its connection to the Burau representation. Given a pseudoAnosov braid \(\beta\in B_n\), we determine necessary and sufficient conditions under which the homology polynomial of \(\beta\) is equal to the the characteristic polynomial of the image of \(\beta\) under the Burau representation. In particular, we build upon \cite...
Show moreThe homology polynomial is an invariant for pseudoAnosov mapping classes \cite{Birman2010}. We study the homology polynomial as an invariant for pseudoAnosov braids and its connection to the Burau representation. Given a pseudoAnosov braid \(\beta\in B_n\), we determine necessary and sufficient conditions under which the homology polynomial of \(\beta\) is equal to the the characteristic polynomial of the image of \(\beta\) under the Burau representation. In particular, we build upon \cite{Band2007} and show that the orientation cover associated to a pseudoAnosov braid is equivalent to a quotient to the Burau cover when the measured foliations associated to \(\beta\) have oddordered singularities at each puncture and any singularity that occurs in the interior of \(D_n\) is evenordered. We next construct an algorithm which allows us to determine the homology polynomial from the Burau representation for an arbitrary pseudoAnosov braid. As an application, we show how to easily determine the homology polynomial for large family of pseudoAnosov braids.
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 Title
 Three Variations on JohnsonLindenstrauss Maps for Submanifolds of Euclidean Space via Reach
 Creator
 Tavakoli, Arman
 Date
 2021
 Collection
 Electronic Theses & Dissertations
 Description

In this thesis we investigate 3 variations of the classical JohnsonLindenstrauss (JL) maps. In one direction we build on the earlier work of Wakin and Eftekhari (2015), by considering generalizations to manifolds with boundary. In a second direction we extend the work of Noga Alon (2003) for lower bounds for the final embedding dimension in JL maps. In the third direction, we consider matrices with fast matrixvector multiply and improve the runtime in the earlier work of Oymak, Recht and...
Show moreIn this thesis we investigate 3 variations of the classical JohnsonLindenstrauss (JL) maps. In one direction we build on the earlier work of Wakin and Eftekhari (2015), by considering generalizations to manifolds with boundary. In a second direction we extend the work of Noga Alon (2003) for lower bounds for the final embedding dimension in JL maps. In the third direction, we consider matrices with fast matrixvector multiply and improve the runtime in the earlier work of Oymak, Recht and Soltanolkotabi (2018), and Ailon and Liberty (2009).This thesis is organized into 6 chapters. The three variations are discussed in chapters 4, 5 and 6. The variation for manifolds with boundary is presented in chapter 4. The lower bound problem is discussed in chapter 5, and chapter 6 is regarding the runtime improvements. The first chapter is an introduction to JohnsonLindenstrauss maps. The second chapter is about a regularity parameter called reach and geometrical estimates for manifolds. The third chapter is regarding two geometry questions about reach that arise from the discussions in chapter 2.
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 Title
 Some computations and applications of Heegaard Floer correction terms
 Creator
 Park, Kyungbae
 Date
 2014
 Collection
 Electronic Theses & Dissertations
 Description

In this dissertation we study some computations and applications of Heegaard Floer correction terms. In particular we explore the correction terms for the double covers of the threesphere branched along the Whitehead doubles of knots. As a consequence we show that Whitehead double and iterated double of some classes of knots are independent in the smooth knot concordance group. We also compute the correction terms of nontrivial circle bundles over oriented surfaces and discuss how they can...
Show moreIn this dissertation we study some computations and applications of Heegaard Floer correction terms. In particular we explore the correction terms for the double covers of the threesphere branched along the Whitehead doubles of knots. As a consequence we show that Whitehead double and iterated double of some classes of knots are independent in the smooth knot concordance group. We also compute the correction terms of nontrivial circle bundles over oriented surfaces and discuss how they can be applied to fourdimensional topology.
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 Title
 Discrete de RhamHodge Theory
 Creator
 Zhao, Rundong
 Date
 2020
 Collection
 Electronic Theses & Dissertations
 Description

We present a systematic treatment to 3D shape analysis based on the wellestablished de RhamHodge theory in differential geometry and topology. The computational tools we developed are widely applicable to research areas such as computer graphics, computer vision, and computational biology. We extensively tested it in the context of 3D structure analysis of biological macromolecules to demonstrate the efficacy and efficiency of our method in potential applications. Our contributions are...
Show moreWe present a systematic treatment to 3D shape analysis based on the wellestablished de RhamHodge theory in differential geometry and topology. The computational tools we developed are widely applicable to research areas such as computer graphics, computer vision, and computational biology. We extensively tested it in the context of 3D structure analysis of biological macromolecules to demonstrate the efficacy and efficiency of our method in potential applications. Our contributions are summarized in the following aspects. First, we present a compendium of discrete Hodge decompositions of vector fields, which provides the primary building block of the de RhamHodge theory for computations performed on the commonly used tetrahedral meshes embedded in the 3D Euclidean space. Second, we present a realworld application of the above computational tool to 3D shape analysis on biological macromolecules. Finally, we extend the above method to an evolutionary de RhamHodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. Our work on the decomposition of vector fields, spectral shape analysis on static shapes, and evolving shapes has already shown its effectiveness in biomolecular applications and will lead to a rich set of features for machine learningbased shape analysis currently under development.
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 Title
 Bubble tree construction for Harmonic maps using DeligneMumford moduli space
 Creator
 Park, Woongbae
 Date
 2021
 Collection
 Electronic Theses & Dissertations
 Description

We formulate and prove a general compactness theorem for harmonic maps.Convergence is defined using DeligneMumford space and families of curves. Given a sequence of harmonic maps from a sequence of closed Riemann surfaces to a compact Riemannian manifold with uniformly bounded energy, the main theorem shows that there is a family of curves and a subsequence such that both the domains and the maps converge off the set of ``nonregular'' nodes. This convergence result extends existing bubble...
Show moreWe formulate and prove a general compactness theorem for harmonic maps.Convergence is defined using DeligneMumford space and families of curves. Given a sequence of harmonic maps from a sequence of closed Riemann surfaces to a compact Riemannian manifold with uniformly bounded energy, the main theorem shows that there is a family of curves and a subsequence such that both the domains and the maps converge off the set of ``nonregular'' nodes. This convergence result extends existing bubbletree construction to the case of varying domains.
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 Title
 Volumes, determinants, and meridian lengths of hyperbolic links
 Creator
 Burton, Stephan D., 1987
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

We study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of nonalternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi...
Show moreWe study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of nonalternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi log det(K)/c(K). We prove that the closure of the set of determinant densities contains the set [0, v_8]. We examine the conjecture, due to Champanerkar, Kofman, and Purcell that vol(K) < 2 pi log det (K) for alternating hyperbolic links, where vol(K) = vol(S^3\ K) is the hyperbolic volume and det(K) is the determinant of K. We prove that the conjecture holds for 2bridge links, alternating 3braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.We derive bounds on the length of the meridian and the cusp volumeof hyperbolic knots in terms of the topology of essential surfaces spanned by the knot.We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound.As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots withmeridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.
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 Title
 Topics in link homology
 Creator
 Jaeger, Thomas Constantin
 Date
 2011
 Collection
 Electronic Theses & Dissertations
 Description

We prove two results about mutation invariance of link homology theories: Weshow that Khovanov's universal sl(2) homology is invariant under mutationand that the reduced sl(n) homology defined by Khovanov and Rozansky isinvariant under componentpreserving positive mutation when n is odd. Wealso give a relationship between the Khovanov homology of a closed positive3braid and the Khovanov homology of the braid after adding a number of fulltwists.
 Title
 Asymptotics of the TuraevViro invariants and their connections in lowdimensional topology
 Creator
 Kumar, Sanjay Lakshman
 Date
 2021
 Collection
 Electronic Theses & Dissertations
 Description

We study the TuraevViro invariants of 3manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the TuraevViro invariants for hyperbolic oncepunctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the TuraevViro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic...
Show moreWe study the TuraevViro invariants of 3manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the TuraevViro invariants for hyperbolic oncepunctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the TuraevViro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic links in the 3sphere which satisfy the conjecture. Additionally, we show a general method for augmenting a link such that the resulting link has hyperbolic complement which satisfies the conjecture. As an application of the constructed links in the 3sphere, we extend the class of known examples which satisfy a conjecture by Andersen, Masbaum, and Ueno relating the quantum representations of surface mapping class groups to its NielsenThurston classification. Most notably, we provide explicit elements in the mapping class group for a genus zero surface with n boundary components, for n > 3, as well as elements in the mapping class group for any genus g surface with four boundary components such that the obtained elements satisfy the conjecture. This is accomplished by utilizing the intrinsic relationship between the TuraevViro invariants and the quantum representations shown by Detcherry and Kalfagianni.
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 Title
 ON A FAMILY OF INTEGRAL OPERATORS ON THE BALL
 Creator
 Tian, Wenchuan
 Date
 2021
 Collection
 Electronic Theses & Dissertations
 Description

In this dissertation, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and identify the extremal functions using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally...
Show moreIn this dissertation, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and identify the extremal functions using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.
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 Title
 Diagrammatic and Geometric Invariants of Hyperbolic Weakly Generalized Alternating Knots
 Creator
 Bavier, Brandon
 Date
 2021
 Collection
 Electronic Theses & Dissertations
 Description

We study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to...
Show moreWe study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to ensure the diagram is interesting. A cusp of a hyperbolic knot $K$ is a neighborhood of $K$ in $M\setminus K$, and the cusp volume is the Euclidean volume of a maximal cusp. We show that the cusp volume of a weakly generalized alternating knot, with some additional conditions, is bounded both above and below based on the twist number of $\pi(K)$ and $\chi(F)$. This is done by constructing a new essential surface for $K$ that has nice properties, including having Euler characteristic based on the twist number as opposed to the crossing number. Our bound on cusp volume leads to interesting bounds on other geometric properties of $K$, including slope length and volumes of Dehn surgery. The volume}of a hyperbolic link $L$ in a manifold $M$ is the hyperbolic volume of the complement $M\setminus L$. We can show that volume is also bounded below by the twist number of $\pi(L)$ and $\chi(F)$. We do this by generalizing the Jones polynomial to weakly generalized alternating links, and showing that there is a relation between this polynomial and the twist number, and this polynomial and the volume. Through the course of proving this bound, we also get relations between the guts of the checkerboard surfaces of $L$ and this generalized Jones polynomial. In addition, if we are working inside a thickened surface, the twist number becomes an isotopy invariant of $L$.
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 Title
 Heegaard Floer homology and Lspace knots
 Creator
 Vafaee, Faramarz
 Date
 2014
 Collection
 Electronic Theses & Dissertations
 Description

Heegaard Floer theory consists of a set of invariants of three and fourdimensional manifolds. Threemanifolds with the simplest Heegaard Floer invariants are called Lspaces, and the name stems from the fact that lens spaces are Lspaces. The overarching goal of the dissertation is to understand Lspaces better. More specifically, this dissertation could be considered as a step towards finding topological characterizations of Lspaces and Lspace knots without referencing Heegaard Floer...
Show moreHeegaard Floer theory consists of a set of invariants of three and fourdimensional manifolds. Threemanifolds with the simplest Heegaard Floer invariants are called Lspaces, and the name stems from the fact that lens spaces are Lspaces. The overarching goal of the dissertation is to understand Lspaces better. More specifically, this dissertation could be considered as a step towards finding topological characterizations of Lspaces and Lspace knots without referencing Heegaard Floer homology. We study knots in $S^3$ that admit positive Lspace Dehn surgeries. In particular, we give new examples of knots in $S^3$ within both the families of hyperbolic and satellite knots admitting Lspace surgeries. It should be pointed out that for satellite knot examples, we use BergeGabai knots (i.e. knots in $S^1 \times D^2$ with nontrivial solid torus Dehn surgeries) as the pattern. Moreover, we study the relationship between satellite knots and Lspace surgeries in the general setting, i.e. when the pattern is an arbitrary knot in $S^1 \times D^2$.
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 Title
 The reduced knot Floer complex
 Creator
 Krcatovich, David Thaddeus
 Date
 2014
 Collection
 Electronic Theses & Dissertations
 Description

We define a “reduced” version of the knot Floer complex CFK
 We define a “reduced” version of the knot Floer complex CFK(K), and show that itbehaves well under connected sums and retains enough information to compute HeegaardFloer dinvariants of manifolds arising as surgeries on the knot K. As an application toconnected sums, we prove that if a knot in the threesphere admits an Lspace surgery, itmust be a prime knot. As an application to the computation of dinvariants, we show thatthe Alexander polynomial is a concordance...
Show more (K), and show that itbehaves well under connected sums and retains enough information to compute HeegaardFloer dinvariants of manifolds arising as surgeries on the knot K. As an application toconnected sums, we prove that if a knot in the threesphere admits an Lspace surgery, itmust be a prime knot. As an application to the computation of dinvariants, we show thatthe Alexander polynomial is a concordance invariant within the class of Lspace knots, andshow the fourgenus bound given by the dinvariant of +1surgery is independent of the genusbounds given by the OzsvathSzabo τ invariant, the knot signature and the Rasmussen sinvariant.
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 Title
 Knot theory of MorseBott critical loci
 Creator
 Ozsarfati, Metin
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

We give an alternative proof of that a critical knot of a MorseBott function f : S3 → R is a graph knot where the critical set of f is a link in S3. Our proof inducts on the number of index1 critical knots of f.  Abstract.
 Title
 Obstruction and existence for twisted KählerEinstein metrics and convexity
 Creator
 Rao, Ambar
 Date
 2013
 Collection
 Electronic Theses & Dissertations
 Description

Let $L \to X$ be an ample holomorphic line bundle over a compact K"{a}hler manifold $(X,omega_{0})$ so that $c_1(L)$ is represented by the K"{a}hler form $\omega_0$. Given a semipositive real $(1,1)$ form $\eta$ representing $c_{1}(K_{X}\otimes L)$, one can ask whether there exists a K"ahler metric $\omega\in c_{1}(L)$ that solves the equation $Ric(\omega) \omega=\eta$. We study this problem by twisting the K"ahlerRicci flow by $eta$ , that is evolve along the flow $\dot{\omega_t}=\omega_...
Show moreLet $L \to X$ be an ample holomorphic line bundle over a compact K"{a}hler manifold $(X,omega_{0})$ so that $c_1(L)$ is represented by the K"{a}hler form $\omega_0$. Given a semipositive real $(1,1)$ form $\eta$ representing $c_{1}(K_{X}\otimes L)$, one can ask whether there exists a K"ahler metric $\omega\in c_{1}(L)$ that solves the equation $Ric(\omega) \omega=\eta$. We study this problem by twisting the K"ahlerRicci flow by $eta$ , that is evolve along the flow $\dot{\omega_t}=\omega_{t}+\eta Ric(\omega_{t})$ starting at $\omega_{0}$. We prove that such a metric exists provided $\omega_{t}^{n}\geq K \omega^{n}_{0}$ for some $K>0$ and all $t \geq 0$. We also study a twisted version of Futaki's invariant, which we show is welldefined if $\eta$ is annihilated under the infinitesimal action of $\eta(X)$. Finally, using Chens $\epsilon$geodesics instead, we give another proof of the convexity of $\mathcal{L}_{\omega}$ along geodesics, which plays a central to Berman's proof of the uniqueness of critical points of $\mathcal{F}_{\omega}$.
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 Title
 Stability of the Almost Hermitian Curvature flow
 Creator
 Smith, D. J. (Daniel J.)
 Date
 2013
 Collection
 Electronic Theses & Dissertations
 Description

The Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusionreaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stable? We show that CalabiYau structures as well as KahlerEinstein structures with first Chern class negative are dynamically stable.