Thèses : « Geometry Mathematics »

1

Miller, Richard A. « Geometric algebra| An introduction with applications in Euclidean and conformal geometry ». Thesis, San Jose State University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=1552269.

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This thesis presents an introduction to geometric algebra for the uninitiated. It contains examples of how some of the more traditional topics of mathematics can be reexpressed in terms of geometric algebra along with proofs of several important theorems from geometry. We introduce the conformal model. This is a current topic among researchers in geometric algebra as it is finding wide applications in computer graphics and robotics. The appendices provide a list of some of the notational conventions used in the literature, a reference list of formulas and identities used in geometric algebra along with some of their derivations, and a glossary of terms.

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Lurie, Jacob 1977. « Derived algebraic geometry ». Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/30144.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (p. 191-193).
The purpose of this document is to establish the foundations for a theory of derived algebraic geometry based upon simplicial commutative rings. We define derived versions of schemes, algebraic spaces, and algebraic stacks. Our main result is a derived analogue of Artin's representability theorem, which provides a precise criteria for the representability of a moduli functor by geometric objects of these types.
by Jacob Lurie.
Ph.D.

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Bunch, Eric. « K-Theory in categorical geometry ». Diss., Kansas State University, 2015. http://hdl.handle.net/2097/20350.

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Doctor of Philosophy
Department of Mathematics
Zongzhu Lin
In the endeavor to study noncommutative algebraic geometry, Alex Rosenberg defined in [13] the spectrum of an Abelian category. This spectrum generalizes the prime spectrum of a commutative ring in the sense that the spectrum of the Abelian category R − mod is homeomorphic to the prime spectrum of R. This spectrum can be seen as the beginning of “categorical geometry”, and was used in [15] to study noncommutative algebriac geometry. In this thesis, we are concerned with geometries extending beyond traditional algebraic geometry coming from the algebraic structure of rings. We consider monoids in a monoidal category as the appropriate generalization of rings–rings being monoids in the monoidal category of Abelian groups. Drawing inspiration from the definition of the spectrum of an Abelian category in [13], and the exploration of it in [15], we define the spectrum of a monoidal category, which we will call the monoidal spectrum. We prove a descent condition which is the mathematical formalization of the statment “R − mod is the category of quasi-coherent sheaves on the monoidal spectrum of R − mod”. In addition, we prove a functoriality condidition for the spectrum, and show that for a commutative Noetherian ring, the monoidal spectrum of R − mod is homeomorphic to the prime spectrum of the ring R. In [1], Paul Balmer defined the prime tensor ideal spectrum of a tensor triangulated cat- gory; this can be thought of as the beginning of “tensor triangulated categorical geometry”. This definition is very transparent and digestible, and is the inspiration for the definition in this thesis of the prime tensor ideal spectrum of an monoidal Abelian category. It it shown that for a polynomial identity ring R such that the catgory R − mod is monoidal Abelian, the prime tensor ideal spectrum is homeomorphic to the prime ideal spectrum.

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Dodds, Peter Sheridan 1969. « Geometry of river networks ». Thesis, Massachusetts Institute of Technology, 2000. http://hdl.handle.net/1721.1/9177.

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Segarra, Escandón Jaime Rodrigo. « Pre-service teachers' mathematics teaching beliefs and mathematical content knowledge ». Doctoral thesis, Universitat Rovira i Virgili, 2021. http://hdl.handle.net/10803/671686.

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L’estudi del coneixement matemàtic i les creences de l’eficàcia de l’ensenyament de les matemàtiques en la formació inicial dels futurs mestres és fonamental, ja que influencia el rendiment acadèmic dels seus estudiants. L’objectiu d’aquesta tesi és estudiar tant el coneixement matemàtic inicial dels futurs mestres com també les seves creences sobre l’eficàcia matemàtica i la seva actitud envers les matemàtiques. Per a complir amb l’objectiu, es realitzen vàries investigacions. Primer, s’estudien els coneixements inicials de nombres i geometria dels estudiants del primer curs del Grau d’Educació Primària a la Universitat Rovira i Virgili (URV). En segon lloc, s’estudien les creences de l’eficàcia de l’ensenyament de les matemàtiques dels futurs mestres durant el grau. En tercer lloc, en aquesta Tesi es compara l’autoeficàcia i l’expectativa de resultats de l’ensenyament de les matemàtiques de futurs mestres, mestres novells i mestres experimentats. En quart lloc, s’estudia la relació entre les creences de l’ensenyament de les matemàtiques, l’actitud envers les matemàtiques i el rendiment acadèmic dels futurs mestres. En cinquè lloc, s’estudia la influència dels factors experiència docent, nivell d’educació i nivell d’ensenyament sobre les creences de l’eficàcia de l’ensenyament de les matemàtiques en mestres en actiu. Finalment, es compara l’autoeficàcia de l’ensenyament de les matemàtiques entre els estudiants del quart any del grau de mestres a la Universitat del Azuay i a la URV. Els resultats d’aquesta Tesi ofereixen informació potencialment important sobre el coneixement matemàtic, les creences, l’autoeficàcia de l’ensenyament de les matemàtiques i l’actitud envers les matemàtiques dels futurs mestres i dels mestres en actiu. Aquests resultats poden ajudar a desenvolupar polítiques adients a l’hora de dissenyar plans d’estudis i també assessorar als professors dels graus de mestre en les institucions d’educació superior.
El estudio del conocimiento matemático y las creencias de la eficacia de la enseñanza de las matemáticas en la formación inicial de los futuros maestros es fundamental, ya que influye en el rendimiento académico de los estudiantes. El objetivo de esta tesis es estudiar tanto el conocimiento matemático inicial de los futuros maestros como sus creencias sobre la eficacia matemática y su actitud hacia las matemáticas. Para cumplir con el objetivo se realiza varias investigaciones. Primero, se estudia los conocimientos iniciales de números y geometría de los estudiantes de primer año del Grado de Educación Primaria en la Universidad Rovira y Virgili (URV). En segundo lugar, se estudia las creencias de la eficacia de la enseñanza de las matemáticas de los futuros maestros a lo largo del grado. Tercero, esta Tesis compara la autoeficacia y la expectativa de resultados de la enseñanza de las matemáticas de futuros maestros, maestros novatos y maestros experimentados. Cuarto, se estudia la relación entre las creencias de la enseñanza de las matemáticas, la actitud hacia las matemáticas y su rendimiento académico. Quinto, se estudia la influencia de los factores experiencia docente, nivel de educación y nivel de enseñanza, sobre las creencias de la eficacia de la enseñanza de las matemáticas en maestros en servicio. Finalmente, se compara la autoeficacia de la enseñanza de las matemáticas entre los estudiantes de cuarto año del grado de maestro en la Universidad del Azuay y en la URV. Los resultados de esta Tesis ofrecen información potencialmente importante sobre el conocimiento matemático, las creencias, la autoeficacia de la enseñanza de las matemáticas y la actitud hacia las matemáticas de los futuros maestros y maestros en servicio. Estos resultados pueden ayudar a desarrollar políticas adecuadas para diseñar planes de estudios y también asesorar a los profesores de los grados de maestro en las instituciones de educación superior.
The study of mathematical content knowledge and teachers’ mathematics teaching beliefs of the pre-service teachers is fundamental, since it influences the academic performance of students. The objective of this Thesis is to study the initial mathematical knowledge of pre-service teachers and also their teachers’ mathematics teaching beliefs and their attitude towards mathematics. To meet the objective, various investigations are carried out. First, the initial knowledge of numbers and geometry of first-year students of the primary education degree at the Rovira and Virgili University (URV) is studied. Second, pre-service teachers’ mathematics teaching beliefs are studied throughout the grade. Third, this Thesis compares the self-efficacy and the expectation of results of the teaching of mathematics of pre-service teachers, novice in-service teachers and experienced in-service teachers. Fourth, the relationship between the teachers’ mathematics teaching beliefs, the attitude towards mathematics and their academic performance is studied. Fifth, the influence of the factors teaching level factor and level of training on the teachers’ mathematics teaching beliefs of in-service teachers is studied. Finally, the self-efficacy of mathematics teaching of fourth-year students at the Azuay University and at the URV is compared. The results of this Thesis offer potentially important information on the mathematical knowledge, beliefs, self-efficacy of mathematics teaching and the attitude towards mathematics of pre-service teachers and in-service teachers. These results can help develop policies for curriculum developers and teaching professors at institutes of higher education.

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Armstrong, John. « Almost-Kahler geometry ». Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.268139.

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Markham, Sarah. « Hypercomplex hyperbolic geometry ». Thesis, Durham University, 2003. http://etheses.dur.ac.uk/3698/.

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The rank one symmetric spaces of non-compact type are the real, complex, quaternionic and octonionic hyperbolic spaces. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still in its infancy. The purpose of this thesis is to investigate the conditions for discrete group action in quaternionic and octonionic hyperbolic 2-spaces and their geometric consequences, in the octonionic case, in terms of lower bounds on the volumes of non-compact manifolds. We will also explore the eigenvalue problem for the 3 x 3 octonionic matrices germane to the Jordan algebra model of the octonionic hyperbolic plane. In Chapters One and Two we concentrate on discreteness conditions in quaternionic hyperbolic 2-space. In Chapter One we develop a quaternionic Jørgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. In Chapter Two we give a generalisation of Shimizu's Lemma to groups of isometries of quaternionic hyperbolic 2-space containing a screw-parabolic element. In Chapter Three we present the Jordan algebra model of the octonionic hyperbolic plane and develop a generalisation of Shimizu's Lemma to groups of isometries of octonionic hyperbolic 2-space containing a parabolic map. We use this result to determine estimates of lower bounds on the volumes of non-compact closed octonionic 2-manifolds. In Chapter Four we construct an octonionic Jørgensen's inequality for non-elementary groups of isometries of octonionic hyperbolic 2-space generated by two elements, one of which is loxodromic. In Chapter Five we solve the real eigenvalue problem Xv = λv, for the 3 x 3 ɸ-Hermitian matrices, X, of the Jordan algebra model of the octonionic hyperbolic plane. Finally, in Chapter Six we consider the embedding of collars about real geodesies in complex hyperbohc 2-space, quaternionic hyperbolic 2-space and octonionic hyperbolic 2-space.

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Marshall, T. H. (Timothy Hamilton). « Hyperbolic Geometry and Reflection Groups ». Thesis, University of Auckland, 1994. http://hdl.handle.net/2292/2140.

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The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ≤ k ≤ n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length γ we then derive power series for the volume, both in u = sinγ/2, and in γ itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and pεC). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all π/2, and the remaining dihedral angles are all sub-multiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g≥2, which we conjecture to be of least volume among such manifolds.

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Baer, Lawrence H. « Numerical aspects of computational geometry ». Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=22507.

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This thesis is concerned with the numerical issues resulting from the implementation of geometric algorithms on finite precision digital computers. From an examination of the general problem and a survey of previous research, it appears that the central problem of numerical computational geometry is how to deal with degenerate and nearly degenerate input. For some applications, such as solid modeling, degeneracy is often intended but we cannot always ascertain its existence using finite precision. For other applications, degenerate input is unwanted but nearly degenerate input is unavoidable. Near degeneracy is associated with ill-conditioning of the input and can lead to a serious loss of accuracy and program failure. These observations lead us to a discussion of problem condition in the context of computational geometry. We use the Voronoi diagram construction problem as a case study and show that problem condition can also play a role in algorithm design.

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Sam, Steven V. « Free resolutions, combinatorics, and geometry ». Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/73178.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student submitted PDF version of thesis.
Includes bibliographical references (p. 71-72).
Boij-Söderberg theory is the study of two cones: the first is the cone of graded Betti tables over a polynomial ring, and the second is the cone of cohomology tables of coherent sheaves over projective space. Each cone has a triangulation induced from a certain partial order. Our first result gives a module-theoretic interpretation of this poset structure. The study of the cone of cohomology tables over an arbitrary polarized projective variety is closely related to the existence of an Ulrich sheaf, and our second result shows that such sheaves exist on the class of Schubert degeneracy loci. Finally, we consider the problem of classifying the possible ranks of Betti numbers for modules over a regular local ring.
by Steven V Sam.
Ph.D.

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Yang, Xiaochun 1971. « Geometry of cone-beam reconstruction ». Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/8338.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.
Includes bibliographical references (p. 89-91).
Geometry is the synthetic tool we use to unify all existing analytical cone-beam reconstruction methods. These reconstructions are based on formulae derived by Tuy [Tuy, 1983], Smith [Smith, 1985] and Grangeat [Grangeat, 1991] which explicitly link the cone-beam data to some intermediate functions in the Radon transform domain. However, the essential step towards final reconstruction, that is, differential-backprojection, has not yet achieved desired efficiency. A new inversion formula is obtained directly from the 3D Radon inverse [Radon, 1917, Helgason, 1999]. It incorporates the cone-beam scanning geometry and allows the theoretical work mentioned above to be reduced to exact and frugal implementations. Extensions can be easily carried out to 2D fan-beam reconstruction as well as other scanning modalities such as parallel scans by allowing more abstract geometric description on the embedding subspace of the Radon manifold. The new approach provides a canonical inverse procedure for computerized tomography in general, with applications ranging from diagnostic medical imaging to industrial testing, such as X-ray CT, Emission CT, Ultrasound CT, etc. It also suggests a principled frame for approaching other 3D reconstruction problems related to the Radon transform. The idea is simple: as was spelled out by Helgason on the opening page of his book, The Radon Transform [Helgason, 1999] - a remarkable duality characterizes the Radon transform and its inverse. Our study shows that the dual space, the so-called Radon space, can be geometrically decomposed according to the specified scanning modality.
(cont.) In cone-beam X-ray reconstruction, for example, each cone-beam projection is seen as a 2D projective subspace embedded in the Radon manifold. Besides the duality in the space relation, the symbiosis played between algebra and geometry, integration and differentiation is another striking feature in the tomographic reconstruction. Simply put, * Geometry and algebra: the two play fundamentally different roles during the inverse. Algebraic transforms carry cone-beam data into the Radon domain, whereas, the geometric decomposition of the dual space determines how the differential-backprojection operator should be systematically performed. The reason that different algorithms in cone-beam X-ray reconstruction share structural similarity is that the dual space decomposition is intrinsic to the specified scanning geometry. The differences in the algorithms lie in the appearance of algebra on the projection submanifold. The algebraic transforms initiate diverse reconstruction methods varying in terms of computational cost and stability. Equipped with this viewpoint, we are able to simplify mathematical analysis and develop algorithms that are easy to implement. Integration and differentiation: forward projection is the integral along straight lines (or planes) in the Euclidean space. During the reconstruction, differentiation is performed over the parallel planes in the projective Radon space, a manifold with clear differential structure. It is important to learn about this differential structure to ensure that correct differentiation can be carried out with respect to the parameters governing the scanning process during the reconstruction ...
by Xiaochun Yang.
Ph.D.

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Postnikov, Alexander. « Enumeration in algebra and geometry ». Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42693.

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Svaldi, Roberto. « Log geometry and extremal contractions ». Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/99064.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 77-82).
The Minimal Model Program (in short, MMP) aims at classifying projective algebraic varieties from a birational point of view. That means that starting from a projective algebraic variety X, [Delta] it is allowed to change the variety under scrutiny as long as its field of rational functions remains the same. In this thesis we study two problems that are inspired by the techniques developed in the last 30 years by various mathematicians in an attempt to realize the Minimal Model Program for varieties of any dimension. In the first part of the thesis, we prove a result about the existence and distribution of rational curves in projective algebraic varieties. We consider projective log canonical pairs (X,[Delta] A) where the locus Nklt(X,[Delta] A) of maximal singularities of the pair (X,[Delta] A) is nonempty. We show that if Kx[Delta]+ A is not nef then there exists an algebraic curves C, whose normalization is isomorphic to A1, contained either in X \ Nklt(X,[Delta] A) or in certain locally closed varieties that stratify Nklt(X,[Delta] A). This result implies a strengthening of the Cone Theorem for log canonical pairs. In the second part, we study certain varieties that naturally arise as possible outcomes of the classification algorithm proposed by the MMP. These are called Mori fibre spaces. A Mori fibre space is a variety X with log canonical singularities together with a morphism f : X --> Y such that the general fiber of f is a positive dimensional Fano variety and the monodromy of f is as large as possible. We show that being the general fiber of a Mori fiber space is a very restrictive condition for Fano varieties with terminal Q-factorial singularities. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realized as a fiber of a Mori fiber space. We apply our criteria to figure out what Fano varieties satisfy this property up to dimension three and to study the case of certain homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.
by Roberto Svaldi.
Ph. D.

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Murray, Marilee Anne. « Hyperbolic Geometry and Coxeter Groups ». Bowling Green State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1343040882.

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Kileel, Joseph David. « Algebraic Geometry for Computer Vision ». Thesis, University of California, Berkeley, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10282753.

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This thesis uses tools from algebraic geometry to solve problems about three-dimensional scene reconstruction. 3D reconstruction is a fundamental task in multiview geometry, a field of computer vision. Given images of a world scene, taken by cameras in unknown positions, how can we best build a 3D model for the scene? Novel results are obtained for various challenging minimal problems, which are important algorithmic routines in Random Sampling Consensus pipelines for reconstruction. These routines reduce overfitting when outliers are present in image data.

Our approach throughout is to formulate inverse problems as structured systems of polynomial equations, and then to exploit underlying geometry. We apply numerical algebraic geometry, commutative algebra and tropical geometry, and we derive new mathematical results in these fields. We present simulations on image data as well as an implementation of general-purpose homotopy-continuation software for implicitization in computational algebraic geometry.

Chapter 1 introduces some relevant computer vision. Chapters 2 and 3 are devoted to the recovery of camera positions from images. We resolve an open problem concerning two calibrated cameras raised by Sameer Agarwal, a vision expert at Google Research, by using the algebraic theory of Ulrich sheaves. This gives a robust test for identifying outliers in terms of spectral gaps. Next, we quantify the algebraic complexity for notorious poorly understood cases for three calibrated cameras. This is achieved by formulating in terms of structured linear sections of an explicit moduli space and then computing via homotopy-continuation. In Chapter 4, a new framework for modeling image distortion is proposed, based on lifting algebraic varieties in projective space to varieties in other toric varieties. We check that our formulation leads to faster and more stable solvers than the state of the art. Lastly, Chapter 5 concludes by studying possible pictures of simple objects, as varieties inside products of projective planes. In particular, this dissertation exhibits that algebro-geometric methods can actually be useful in practical settings.

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Saksida, Pavle. « Geometry of integrable systems ». Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.308545.

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Whiteway, L. « Topics in differential geometry ». Thesis, University of Oxford, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.379896.

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Mavra, Boris. « Bounded geometry index theory ». Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318820.

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Aghasi, Mansour. « Geometry of arithmetic surfaces ». Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5270/.

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In this thesis my emphasis is on the resolution of the singularities of fibre products of Arithmetic Surfaces. In chapter one as an introduction to my thesis some elementary concepts related to regular and singular points are reviewed and the concept of tangent cone is defined for schemes over a discrete valuation ring. The concept of arithmetic surfaces is introduced briefly in the end of this chapter. In chapter 2 my new procedures namely the procedure of Mojgan(_1) and the procedure of Mahtab(_2) and a new operator called Moje are introduced. Also the concept of tangent space is defined for schemes over a discrete valuation ring. In chapter 3 the singularities of schemes which are the fibre products of some surfaces with ordinary double points are resolved. It is done in two different methods. The results from both methods are consistent. In chapter 4, I have tried to resolve the singularities of a special class of arithmetic three-folds, namely those which are the fibre product of two arithmetic surfaces, which were very helpful to achieve my final results about the resolution of singularities of fibre products of the minimal regular models of Tate. Chapter 5 includes my final results which are about the resolution of singularities of the fibre product of two minimal regular models of Tate.

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Khukhro, Anastasia. « Coarse geometry and groups ». Thesis, University of Southampton, 2012. https://eprints.soton.ac.uk/341780/.

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The central idea of coarse geometry is to focus on the properties of metric spaces which survive under deformations that change distances in a controlled way. These large scale properties, although too coarse to determine what happens locally, are nevertheless often able to capture the most important information about the structure of a space or a group. The relevant notions from coarse geometry and group theory are described in the beginning of this thesis. An overview of the cohomological characterisation of property A of Brodzki, Niblo and Wright is given, together with a proof that the cohomology theories used to detect property A are coarse invariants. The cohomological characterisation is used alongside a symmetrisation result for functions defining property A to give a new direct, more geometric proof that expanders do not have property A, making the connection between the two properties explicit. This is based on the observation that both the expander condition and property A can be expressed in terms of a coboundary operator which measures the size of the (co)boundary of a set of vertices. The rest of the thesis is devoted to the study of box spaces, including a description of the connections between analytic properties of groups and coarse geometric properties of box spaces. The construction of Arzhantseva, Guentner and Spakula of a box space of a finitely generated free group which coarsely embeds into Hilbert is the first example of a bounded geometry metric space which coarsely embeds into Hilbert space but does not have property A. This example is generalised here to box spaces of a large class of groups via a stability result for box spaces.

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Helfgott, Michel. « A Sojourn Through Geometry and Algebra ». Digital Commons @ East Tennessee State University, 2013. http://amzn.com/1492798894.

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This textbook is intended for college juniors or seniors majoring in mathematics, who plan to become high school teachers. It seeks to provide a deeper perspective on secondary mathematics, showing the interplay between plane geometry and algebra. A distinctive characteristic of the book is the frequent discussion of multiple paths to the solution of a problem or the proof of a theorem. Practically none of the topics covered in the book overlap with the content of courses taken by mathematics majors, say real analysis, abstract algebra, differential equations, combinatorics, probability and statistics, number theory, etc. These courses, and several others, provide indispensable mathematical maturity but are rather distant from the core of high school mathematics. Precisely, one of our main objectives is to bridge the gap between the latter and college-level mathematics.
https://dc.etsu.edu/etsu_books/1084/thumbnail.jpg

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Eriksson, Emil. « An Introduction to Orthogonal Geometry ». Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-184316.

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In this thesis, we use algebraic methods to study geometry. We give an introduction to orthogonal geometry, and prove a connection to the Clifford algebra.
I denna uppsats använder vi algebraiska metoder för att studera geometri. Vi ger en introduktion till ortogonal geometri och visar en koppling till Cliffordalgebran.

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Francis, John (John Nathan Kirkpatrick). « Derived algebraic geometry over En̳-rings ». Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43792.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
In title on t.p., double underscored "n" appears as subscript.
Includes bibliographical references (p. 55-56).
We develop a theory of less commutative algebraic geometry where the role of commutative rings is assumed by En-rings, that is, rings with multiplication parametrized by configuration spaces of points in Rn. As n increases, these theories converge to the derived algebraic geometry of Tobn-Vezzosi and Lurie. The class of spaces obtained by gluing En-rings form a geometric counterpart to En-categories, which are higher topological variants of braided monoidal categories. These spaces further provide a geometric language for the deformation theory of general E, structures. A version of the cotangent complex governs such deformation theories, and we relate its values to E&-Hochschild cohomology. In the affine case, this establishes a claim made by Kontsevich. Other applications include a geometric description of higher Drinfeld centers of SE-categories, explored in work with Ben-Zvi and Nadler.
by John Francis.
Ph.D.

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Ho, Chiu-chi. « The use of computer software in geometry learning ». Hong Kong : University of Hong Kong, 1998. http://sunzi.lib.hku.hk/hkuto/record.jsp?B20135944.

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Nivens, Ryan Andrew, Tara Carver Peters et Jesse Nivens. « Views of Isometric Geometry ». Digital Commons @ East Tennessee State University, 2012. https://dc.etsu.edu/etsu-works/293.

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Buttler, Michael. « The geometry of CR manifolds ». Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.312247.

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Martin, Shaun K. « Symplectic geometry and gauge theory ». Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389209.

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Kobak, Piotr Z. « Quaternionic geometry and harmonic maps ». Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357422.

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Smith, Ivan. « Symplectic geometry of Lefschetz fibrations ». Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299234.

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30

Lenssen, Mark. « A topic in differential geometry ». Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314920.

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31

Guo, Guang-Yuan. « Differential geometry of holomorphic bundles ». Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239283.

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Hale, Mark. « Developments in noncommutative differential geometry ». Thesis, Durham University, 2002. http://etheses.dur.ac.uk/3948/.

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One of the great outstanding problems of theoretical physics is the quantisation of gravity, and an associated description of quantum spacetime. It is often argued that, at short distances, the manifold structure of spacetime breaks down and is replaced by some sort of algebraic structure. Noncommutative geometry is a possible candidate for the mathematics of this structure. However, physical theories on noncommutative spaces are still essentially classical and need to be quantised. We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries (the two-point space and the matrix geometry M(_2)(C)) and a circle. In each case, we start with the partition function and calculate the graviton propagator and Greens functions. The expectation values of distances are also evaluated. We find on the finite noncommutative geometries, distances shrink with increasing graviton excitations, while on a circle, they grow. A comparison is made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We also briefly discuss the quantisation of a general Riemannian manifold. Included, is a comprehensive overview of the homological aspects of noncommutative geometry. In particular, we cover the index pairing between K-theory and K-homology, KK-theory, cyclic homology/cohomology, the Chern character and the index theorem. We also review the various field theories on noncommutative geometries.

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Wilkinson, T. C. « The geometry of folding maps ». Thesis, University of Newcastle Upon Tyne, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.315692.

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Bartocci, C. « Foundations of graded differential geometry ». Thesis, University of Warwick, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386972.

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Roberts, Kieran. « Lie algebras and incidence geometry ». Thesis, University of Birmingham, 2012. http://etheses.bham.ac.uk//id/eprint/3483/.

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An element \(\char{cmti10}{0x78}\) of a Lie algebra \(\char{cmmi10}{0x4c}\) over the field \(\char{cmmi10}{0x46}\) is extremal if [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] \(\subseteq\)\(\char{cmmi10}{0x46}\)\(\char{cmti10}{0x78}\). One can define the extremal geometry of \(\char{cmmi10}{0x4c}\) whose points \(\char{cmsy10}{0x45}\) are the projective points of extremal elements and lines \(\char{cmsy10}{0x46}\) are projective lines all of whose points belong to \(\char{cmsy10}{0x45}\). We prove that any finite dimensional simple Lie algebra \(\char{cmmi10}{0x4c}\) is a classical Lie algebra of type A\(_n\) if it satisfies the following properties: \(\char{cmmi10}{0x4c}\) contains no elements \(\char{cmti10}{0x78}\) such that [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] = 0, \(\char{cmmi10}{0x4c}\) is generated by its extremal elements and the extremal geometry \(\char{cmsy10}{0x45}\) of \(\char{cmmi10}{0x4c}\) is a root shadow space of type A\(_{n,(1,n)}\).

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Boalch, Philip Paul. « Symplectic geometry and isomonodromic deformations ». Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301848.

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Finn-Sell, Martin. « Inverse semigroups in coarse geometry ». Thesis, University of Southampton, 2013. https://eprints.soton.ac.uk/361324/.

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Inverse semigroups provide a natural way to encode combinatorial data from geometric settings. Examples of this occur in both geometry and topology, where the data comes in the form of partial bijections that preserve the topology, and operator algebras, where the partial bijections encode *-subsemigroups of partial isometries of Hilbert space. In this thesis we explore the connections between these two pictures within the backdrop of coarse geometry. The first collection of results is concerned primarily with inverse semigroups and their C*-algebras. We give a construction of a six term sequence of C*-algebras connecting the semigroup C*-algebra to that of a naturally associated group C*-algebra. This result is a generalisation of the ideas of Pimsner and Voiculescu, who were concerned with computing K-theory groups associated to actions of groups. We outline how to connect this picture, via groupoids, to that of a partial translation algebra of Brodzki, Niblo andWright, and further consider applications of these sequences to computations of certain K-groups associated with group and semigroup C*-algebras. Secondly, we give an account of the coarse Baum-Connes conjecture associated to a uniformly discrete bounded geometry metric space and rephrase the conjecture in terms of groupoids and their C*-algebras that can naturally be associated to a metric space. We then consider the well known counterexamples to this conjecture, giving a unifying framework for their study in terms of groupoids and a new conjecture for metric spaces that we call the boundary coarse Baum-Connes conjecture. Generalising a result of Willett and Yu we prove this conjecture for certain classes of expanders including those of large girth by constructing a partial action of a discrete group on such spaces.

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Marsh, Duncan. « On the geometry of mechanisms ». Thesis, University of Liverpool, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.328177.

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Webb, Richard Charles Henry. « Effective geometry of curve graphs ». Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/64122/.

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We show that curve graphs and arc graphs are uniformly hyperbolic i.e. all of these graphs are hyperbolic with some fixed hyperbolicity constant. This is joint work with Sebastian Hensel and Piotr Przytycki. Note that curve graphs were shown to be hyperbolic by Masur and Minsky, arc graphs were shown to be hyperbolic by Masur and Schleimer, and curve graphs were shown to be uniformly hyperbolic by Aougab, Bowditch and Clay-Rafi-Schleimer independently. We show that if every vertex of a fixed geodesic in the curve graph cuts some fixed subsurface then the image of that geodesic under the corresponding subsurface projection has its diameter bounded from above by 62 (or 50) if the subsurface is an annulus (or otherwise). This is an effective version of the bounded geodesic image theorem of Masur and Minsky, indeed, there is a universal constant that can be taken as a bound. In terms of the complexity of a surface we provide an exponential upper bound on the so-called slices of tight geodesics. This is an effective version of a theorem of Bowditch. Using this we provide algorithms to compute tight geodesic axes of pseudo-Anosovs in the curve graph. All of our proofs are combinatorial in nature. In particular, we do not use any geometric limiting arguments.

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Coffey, Michael R. « Ricci flow and metric geometry ». Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/67924/.

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This thesis considers two separate problems in the field of Ricci flow on surfaces. Firstly, we examine the situation of the Ricci flow on Alexandrov surfaces, which are a class of metric spaces equipped with a notion of curvature. We extend the existence and uniqueness results of Thomas Richard in the closed case to the setting of non-compact Alexandrov surfaces that are uniformly non-collapsed. We complement these results with an extensive survey that collects together, for the first time, the essential topics in the metric geometry of Alexandrov spaces due to a variety of authors. Secondly, we consider a problem in the well-posedness theory of the Ricci flow on surfaces. We show that given an appropriate initial Riemannian surface, we may construct a smooth, complete, immortal Ricci flow that takes on the initial surface in a geometric sense, in contrast to the traditional analytic notions of initial condition. In this way, we challenge the contemporary understanding of well-posedness for geometric equations.

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Davis, Robert Tucker. « Geometric Build-up Solutions for Protein Determination via Distance Geometry ». TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/102.

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Proteins carry out an almost innumerable amount of biological processes that are absolutely necessary to life and as a result proteins and their structures are very often the objects of study in research. As such, this thesis will begin with a description of protein function and structure, followed by brief discussions of the two major experimental structure determination methods. Another problem that often arises in molecular modeling is referred to as the Molecular Distance Geometry Problem (MDGP). This problem seeks to find coordinates for the atoms of a protein or molecule when given only a set of pair-wise distances between atoms. To introduce the complexities of the MDGP we begin at its origins in distance geometry and progress to the specific sub-problems and some of the solutions that have been developed. This is all in preparation for a discussion of what is known as the Geometric Build-up (GBU) Solution. This solution has lead to the development of several algorithms and continues to be modified to account for more and different complexities. The culmination of this thesis, then, is a new algorithm, the Revised Updated Geometric Build-up, that is faster than previous GBU’s while maintaining the accuracy of the resulting structure.

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Murugan, Jeffrey. « Études on fuzzy geometry and cosmology ». Doctoral thesis, University of Cape Town, 2007. http://hdl.handle.net/11427/19023.

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We investigate various aspects of noncommutative geometry and fuzzy field theory and their relations to string theory. In particular, we study the BPS and non-BPS solutions of the CJPN nonlinear sigma model on the noncommutative plane in some detail and show among other things that a class of its solitonic excitations may be built from bound states of noncommutative scalar solitons. We then go on to construct a fuzzy extension of the semilocal SU(N)a x U(l)L Yang-Mills-Riggs model. We find that not only does this noncommutative model support a large class of BPS vortex solutions but, unlike in the commutative model, these are exact solutions of the BPS equations. We also study the large coupling limit of the semilocal model and demonstrate conclusively the metamorphosis of the semilocal vortex to an appropriate degree instanton of the fuzzy CJPN model. In the second part of this work, we study the perpendicular intersection of Dl- and D7-branes in type liB string theory and the fuzzy 6-sphere that resolves the singularity of the intersection. We demonstrate the equivalence of the D7 and dual D-string descriptions by computing the energy, charge and radial profiles of the solution in each description. We conclude the thesis with a foray into cosmology by constructing a realisation of a recently proposed singularity-free inflating universe. We discuss the basic characteristics of this model and show that none are at odds with current observations.

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Lozano, Guadalupe I. « Poisson geometry of the Ablowitz-Ladik equations ». Diss., The University of Arizona, 2004. http://hdl.handle.net/10150/290120.

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This research seeks to understand the Poisson Geometry of the Ablowitz-Ladik equations (AL), an integrable discretization of the Non-linear Schrodinger equation (NLS) first proposed by Ablowitz and Ladik in the 70's. More specifically, to argue that the AL hierarchy (an integrable hierarchy of equations which comprises AL) can be explicitly viewed as a hierarchy of commuting flows which: (1) are Hamiltonian with respect to both a (known) Poisson operator J, and a (new) non-local, skew, almost Poisson operator K, on the appropriate space; (2) can be recursively generated from an operator R = KJ⁻¹. This thesis also clarifies the geometric framework that underlies a certain class of evolving geodesic linkages related to the AL hierarchy via the evolution for their "discrete" geodesic curvature. In this regard, our results include a geometric interpretation of a compatibility condition associated to a Lax pair for AL and, consequently, a bijective correspondence between AL flows and linkage flows.

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Portegies, Jacobus W. « Spectral geometry with applications to data analysis ». Thesis, New York University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3635292.

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This thesis consists of two parts, in both of which eigenfunctions and eigenvalues of the Laplace operator on Riemannian manifolds play an important role. The first part of the thesis studies embeddings of manifolds into Euclidean space by heat kernels and eigenfunctions of the Laplacian, whereas the second part analyzes the behavior of eigenvalues as manifolds converge to a limit space.

To understand structure or the most important features in data gathered in large experiments or for the purpose of machine learning, classically linear methods such as Principal Component Analysis are used. However, when the data satisfies nonlinear constraints, or lies on a manifold, nonlinear methods are required to pick up the relevant structure. Several such algorithms in nonlinear data analysis use the eigenfunctions of the Laplace operator on the data graph to embed the data in a lower-dimensional Euclidean space. The first part of the thesis is devoted to developing an understanding of the continuum versions of these algorithms. In particular, we bound the complexity of such algorithms in terms of geometric information. That is, we show that the number of eigenfunctions or heat kernels needed can be bounded in terms of the dimension, the volume, the injectivity radius, a lower bound on the Ricci curvature, and a tolerance for the dilatation.

In the second part of the thesis, we study the behavior of the spectrum of the Laplace operator on (generalizations of) Riemannian manifolds, as these manifolds converge to a limit space. Previous results by Fukaya and by Cheeger and Colding show that under certain bounds on the curvature of the manifolds, the eigenvalues of the Laplace operator are continuous with respect to measured Gromov-Hausdorff convergence. We are interested in the behavior of the spectrum under flat and intrinsic flat convergence. First, we show the semicontinuity of eigenvalues of the Laplace operator on Riemannian manifolds under flat convergence in Euclidean space, if the volume is conserved. In the more general case of intrinsic flat convergence, we can find analogous results for min-max values that arise from a variational problem involving a Dirichlet energy. In the case of closed Riemannian manifolds, these min-max values correspond to the eigenvalues of the Laplace operator.

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Zong, Hong R. « Topics in birational geometry of algebraic varieties ». Thesis, Princeton University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3665359.

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Various questions related to birational properties of algebraic varieties are concerned.

Rationally connected varieties are recognized as the buildings blocks of all varieties by the Minimal Model theory. We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. As a consequence, we solve a question of Professor Burt Totaro on integral Hodge classes on rationally connected 3-folds. And by a result of Professor Claire Voisin, the general case will be a consequence of the Tate conjecture for surfaces over finite fields.

Using the same philosophy looking for degenerated rational components through forgetful maps between moduli spaces of curves, we prove Weak Approximation conjecture to Prof. Hassett and Prof. Tschinkel for isotrivial families of rationally connected varieties. Theory of Twisted Stable maps is essentially used, with an alternative proof where some notion from Derived Algebraic Geometry is applied. It is remarkable that technics and ideas developed in this part, shed light upon and essentially led to the final solution to weak approximation of Cubic Surfaces, which is a problem concerned by Number Theorists for many years, and this is currently the best known result in this subject.

Then we turn to Minimal Model theory in both zero and positive characteristics. Firstly, projective globally F-regular threefolds of characteristic p ≥ 11, are shown to be rationally chain connected, and back to characteristic zero, we use hard-core technics of Minimal Model program, esp. finite generate of canonical rings due to Professor Hacon, Professor McKernan et al. to characterize Toric varieties and geometric rational varieties as log canonical log-Calabi Yau varieties with "large" boundary, where the specific meanings of "large" are originated from some notion of "charges" from String theory, and hence is related to Mirror Symmetry. This part of works also answered a Conjecture due to Prof. Shokurov.

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Adamowicz, Tomasz. « On the geometry of p-harmonic mappings ». Related electronic resource : Current Research at SU : database of SU dissertations, recent titles available full text, 2008. http://wwwlib.umi.com/cr/syr/main.

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Usher, Michael Joseph 1978. « Relative Hilbert scheme methods in pseudoholomorphic geometry ». Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/16631.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (p. 103-104).
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
This thesis takes up a program initiated by S. Donaldson and I. Smith aimed at using symplectic Lefschetz fibration techniques to obtain information about pseudoholomorphic curves in symplectic 4-manifolds. Donaldson and Smith introduced an invariant DS which counts holomorphic sections of a relative Hilbert scheme constructed from a symplectic Lefschetz fibration, and a number of considerations, including a duality relation for DS proven by Smith, led to the conjecture that DS agrees with the Gromov invariant G[gamma] earlier defined by C. Taubes in his study of Seiberg-Witten theory on symplectic manifolds. Our central result is a proof of this conjecture, which thus makes available new proofs of some results concerning pseudoholomorphic curves which had previously only been accessible via gauge theory. The crucial technical ingredient in the proof is an argument which allows us to work with curves C in the total space of the Lefschetz fibration that are made holomorphic by an almost complex structure which is integrable near C and with respect to which the fibration is a pseudoholomorphic map. We also introduce certain refinements of DS and show that these refinements are equal to Gromov invariants which count pseudoholomorphic subvarieties of symplectic 4-manifolds with a prescribed decomposition into reducible components. We prove a vanishing result for some of these invariants which might bear on the question of the uniqueness of the decomposition of the canonical class of a symplectic 4-manifold into classes with nontrivial Gromov-Witten invariants.
by Michael Joseph Usher.
Ph.D.

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Minton, Gregory T. (Gregory Thomas). « Computer-assisted proofs in geometry and physics ». Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/84405.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references.
In this dissertation we apply computer-assisted proof techniques to two problems, one in discrete geometry and one in celestial mechanics. Our main tool is an effective inverse function theorem which shows that, in favorable conditions, the existence of an approximate solution to a system of equations implies the existence of an exact solution nearby. This allows us to leverage approximate computational techniques for finding solutions into rigorous computational techniques for proving the existence of solutions. Our first application is to tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence of many hitherto unknown tight regular simplices in quaternionic projective spaces and in the octonionic projective plane. We also consider regular simplices in real Grassmannians. The second application is to gravitational choreographies, i.e., periodic trajectories of point particles under Newtonian gravity such that all of the particles follow the same curve. Many numerical examples of choreographies, but few existence proofs, were previously known. We present a method for computer-assisted proof of existence and demonstrate its effectiveness by applying it to a wide-ranging set of choreographies.
by Gregory T. Minton.
Ph.D.

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Tievsky, Aaron M. « Analogues of Kähler geometry on Sasakian manifolds ». Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/45349.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
Includes bibliographical references (p. 53-54).
A Sasakian manifold S is equipped with a unit-length, Killing vector field ( which generates a one-dimensional foliation with a transverse Kihler structure. A differential form a on S is called basic with respect to the foliation if it satisfies [iota][epsilon][alpha] = [iota][epsilon]d[alpha] = 0. If a compact Sasakian manifold S is regular, i.e. a circle bundle over a compact Kähler manifold, the results of Hodge theory in the Kahler case apply to basic forms on S. Even in the absence of a Kähler base, there is a basic version of Hodge theory due to El Kacimi-Alaoui. These results are useful in trying to imitate Kähler geometry on Sasakian manifolds; however, they have limitations. In the first part of this thesis, we will develop a "transverse Hodge theory" on a broader class of forms on S. When we restrict to basic forms, this will give us a simpler proof of some of El Kacimi-Alaoui's results, including the basic dd̄-lemma. In the second part, we will apply the basic dd̄-lemma and some results from our transverse Hodge theory to conclude (in the manner of Deligne, Griffiths, and Morgan) that the real homotopy type of a compact Sasakian manifold is a formal consequence of its basic cohomology ring and basic Kähler class.
by Aaron Michael Tievsky.
Ph.D.

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Hu, Yi. « The geometry and topology of quotient varieties ». Thesis, Massachusetts Institute of Technology, 1991. http://hdl.handle.net/1721.1/44268.

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