Dissertations / Theses: 'Geometry, Algebraic'

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Miscione, Steven. "Loop algebras and algebraic geometry." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=116115.

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This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.

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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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Balchin, Scott Lewis. "Augmented homotopical algebraic geometry." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/40623.

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In this thesis we are interested in extending the theory of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. We introduce the concept of augmentation categories, which are a class of generalised Reedy categories. An augmentation category is a category which has enough structure that we can mirror the simplicial constructions which make up the theory of homotopical algebraic geometry. In particular, we construct a Quillen model structure on their presheaf categories, and introduce the concept of augmented hypercovers to define a local model structure on augmented presheaves. As an application, we show that a crossed simplicial group is an example of an augmentation category. The resulting augmented geometric theory can be thought of as being equivariant. Using this, we define equivariant cohomology theories as special mapping spaces in the category of equivariant stacks. We also define the SO(2)-equivariant derived stack of local systems by using a twisted nerve construction. Moreover, we prove that the category of planar rooted trees appearing in the theory of dendroidal sets is also an augmentation category. The augmented geometry over this setting should be thought of as being stable in the spectral sense of the word. Finally, we show that we can combine the two main examples presented using a categorical amalgamation construction.

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Rennie, Adam Charles. "Noncommutative spin geometry." Title page, contents and introduction only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phr4163.pdf.

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Dos, Santos João Pedro Pinto. "Fundamental groups in algebraic geometry." Thesis, University of Cambridge, 2006. https://www.repository.cam.ac.uk/handle/1810/252015.

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Slaatsveen, Anna Aarstrand. "Decoding of Algebraic Geometry Codes." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for fysikk, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-13729.

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Codes derived from algebraic curves are called algebraic geometry (AG) codes. They provide a way to correct errors which occur during transmission of information. This paper will concentrate on the decoding of algebraic geometry codes, in other words, how to find errors. We begin with a brief overview of some classical result in algebra as well as the definition of algebraic geometry codes. Then the theory of cyclic codes and BCH codes will be presented. We discuss the problem of finding the shortest linear feedback shift register (LFSR) which generates a given finite sequence. A decoding algorithm for BCH codes is the Berlekamp-Massey algorithm. This algorithm has complexity O(n^2) and provides a general solution to the problem of finding the shortest LFSR that generates a given sequence (which usually has running time O(n^3)). This algorithm may also be used for AG codes. Further we proceed with algorithms for decoding AG codes. The first algorithm for decoding algebraic geometry codes which we discuss is the so called basic decoding algorithm. This algorithm depends on the choice of a suitable divisor F. By creating a linear system of equation from the bases of spaces with prescribed zeroes and allowed poles we can find an error-locator function which contains all the error positions among its zeros. We find that this algorithm can correct up to (d* - 1 - g)/2 errors and have a running time of O(n^3). From this algorithm two other algorithms which improve on the error correcting capability are developed. The first algorithm developed from the basic algorithm is the modified algorithm. This algorithm depends on a restriction on the divisors which are used to build the code and an increasing sequence of divisors F1, ... , Fs. This gives rise to an algorithm which can correct up to (d*-1)/2 -S(H) errors and have a complexity of O(n^4). The correction rate of this algorithm is larger than the rate for the basic algorithm but it runs slower. The extended modified algorithm is created by the use of what we refer to as special divisors. We choose the divisors in the sequence of the modified algorithm to have certain properties so that the algorithm runs faster. When s(E) is the Clifford's defect of a set E of special divisor, the extended modified algorithm corrects up to (d*-1)/2 -s(E) which is an improvement from the basic algorithm. The running time of the algorithm is O(n^3). The last algorithm we present is the Sudan-Guruswami list decoding algorithm. This algorithm searches for all possible code words within a certain distance from the received word. We show that AG codes are (e,b)-decodable and that the algorithm in most cases has a a higher correction rate than the other algorithms presented here.

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Birkar, Caucher. "Topics in modern algebraic geometry." Thesis, University of Nottingham, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.421475.

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Lundman, Anders. "Topics in Combinatorial Algebraic Geometry." Doctoral thesis, KTH, Matematik (Avd.), 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-176878.

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This thesis consists of six papers in algebraic geometry –all of which have close connections to combinatorics. In Paper A we consider complete smooth toric embeddings X ↪ P^N such that for a fixed positive integer k the t-th osculating space at every point has maximal dimension if and only if t ≤ k. Our main result is that this assumption is equivalent to that X ↪ P^N is associated to a Cayley polytope of order k having every edge of length at least k. This result generalizes an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalizing a result of Atsushi Ito. In Paper B we introduce H-constants that measure the negativity of curves on blow-ups of surfaces. We relate these constants to the bounded negativity conjecture. Moreover we provide bounds on H-constants when restricting to curves which are a union of lines in the real or complex projective plane. In Paper C we study Gauss maps of order k for k > 1, which maps a point on a variety to its k-th osculating space at that point. Our main result is that as in the case k = 1, the higher order Gauss maps are finite on smooth varieties whose k-th osculating space is full-dimensional everywhere. Furthermore we provide convex geometric descriptions of these maps in the toric setting. In Paper D we classify fat point schemes on Hirzebruch surfaces whose initial sequence are of maximal or close to maximal length. The initial degree and initial sequence of such schemes are closely related to the famous Nagata conjecture. In Paper E we introduce the package LatticePolytopes for Macaulay2. The package extends the functionality of Macaulay2 for compuations in toric geometry and convex geometry. In Paper F we compute the Seshadri constant at a general point on smooth toric surfaces satisfying certain convex geometric assumptions on the associated polygons. Our computations relate the Seshadri constant at the general point with the jet seperation and unnormalised spectral values of the surfaces at hand.
Den här avhandlingen utgörs av sex artiklar inom algebraisk geometri som är nära kopplade till kombinatorik. I artikel A betraktar vi kompletta inbäddningar av glatta toriska variteter X ↪ PN sådana att för något fixt heltal k är det t-te oskulerande rummet i varje punkt av maximal dimension om och endast om t ≤ k. Vårt huvudresultat är att detta antagande är ekvivalent med att den polytop som motsvarar inbäddningen är en Cayleypolytop av ordning k, vars samtliga kanter har längd åtminstonde k. Detta resultat generaliserar en tidigare känd karaktärisering av David Perkinson. Vi visar även att ovanstående antagande är ekvivalent med antagandet att Seshadri- konstanten är lika med k i varje punkt i X. Därmed generaliserar vårt resultat ett tidigare resultat av Atsushi Ito. I artikel B introducerar vi H-konstanter, vilka mäter negativiteten av kurvor på uppblåsningar av ytor. Vi relaterar dessa konstanter till den begränsade negativitetsförmodan. Vidare erhåller vi begränsningar för konstanterna när vi enbart betraktar unioner av linjer i det reella och komplexa projektiva planet. I artikel C studerar vi Gaussavbildningen av ordning k, för k > 1, som avbildar en punkt i en varitet på det k-te oskulerande rummet i samma punkt. Vårt huvudresultat är att, i likhet med fallet k = 1, är dessa högre ordningens Gaussavbildningar ändliga på glatta variteter vars k-te oskulerande rum är fulldimensionellt överallt. Vidare ger vi konvexgeometriska beskrivningar av dessa avbildningar för toriska variteter. I artikel D klassificerar vi scheman av tjocka punkter på Hirzebruchytor vars initalsekvenser är av maximal eller nära maximal längd. Intitialgraden och initialsekvensen för sådana scheman är nära relaterade till den välkända Nagata- förmodan. I artikel E introducerar vi paketet LatticePolytopes till Macaulay2. Detta paket utökar funktionaliteten i Macaulay2 för beräkningar inom torisk och konvex geometri. I artikel F beräknar vi Seshadrikonstanten i generella punkter på glatta toriska ytor som uppfyller vissa konvexgeometriska villkor på de associerade polygonerna. Våra beräkningar koppplar samman Seshadrikonstanten i en generell punkt med jetsepareringen och det icke-normaliserade spektralvärdet hos ytorna.

QC 20151112

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Hu, Jiawei. "Partial actions in algebraic geometry." Doctoral thesis, Universite Libre de Bruxelles, 2018. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/273459.

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We introduce geometrically partial comodules over coalgebras in monoidal categories, as an alternative notion to the notion of partial action and coaction of Hopf algebras introduced by Caenepeel and Janssen. We show that our new notion suits better if one wants to describe phenomena of partial actions in algebraic geometry. We show that under mild conditions, the category of geometric partial comodules is complete and cocomplete and the category of partial comodules over a Hopf algebra is lax monoidal. We develop a Hopf-Galois theory for geometric partial coactions to illustrate that our new notion might be a useful additional tool in Hopf algebra theory.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

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Garcia-Puente, Luis David. "Algebraic Geometry of Bayesian Networks." Diss., Virginia Tech, 2004. http://hdl.handle.net/10919/11133.

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We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This allows us to create an algebraic geometry--statistics dictionary. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification, in terms of primary decomposition of polynomial ideals, is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. Moreover, a complete algebraic classification, in terms of generating sets of polynomial ideals, is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed.
Ph. D.

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Mikami, Ryota. "Tropical geometry and algebraic cycles." Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263437.

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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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Waelder, Robert. "Elliptic genera in algebraic geometry." Diss., Restricted to subscribing institutions, 2008. http://proquest.umi.com/pqdweb?did=1619148881&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.

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Milione, Piermarco. "Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/402209.

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The main purpose of this dissertation is to introduce Shimura curves from the non-Archimedean point of view, paying special attention to those aspects that can make this theory amenable for computations. Despite the fact that the theory of p-adic uniformization of Shimura curves goes back to the 1960s with the results of Cerednik and Drinfeld, only in the last years explicit examples related to these uniformizations have been computed. The structure of this dissertation is as follows. In Chapter 1 we introduce Shimura curves starting from an indefinite quaternion algebra H over a totally real field F. This is done mostly following the fundamental paper of Shimura [Shi67]. We also give the definitions using the adelic approach of [Shi70b] and [Shi70c]. The point of view we adopt is the arithmetical one, since we try to make clear the link connecting Shimura curves to the arithmetic of quaternion algebras. In this sense, we give evidence of why Shimura curves have to be considered a geometric interpretation of most arithmetical phenomena in quaternion orders. Chapter 2 has the aim of introducing those non-Archimedean objects which appear later in the statements of the theorems of Cerednik and Drinfeld. In Chapter 3 we start the study of fundamental domains in Hp for the action of discrete and cocompact subgroups of PGL2(Qp) arising in the p-adic uniformization of Shimura curves. In Chapter 4 we associate to the p-adic uniformization of the Shimura curve X(DH;N) certain parameters in Hp(Cp) analogous to the complex multiplication parameters in H: we refer to them by p-imaginary multiplication paramters, since they are defined over the unramified quadratic extension of Qp. In the study of these parameters, we follow the p-adic analog of the line adopted in [AB04]. Specifically, we are able to recover these parameters as zeros of certain binary quadratic forms with p-adic coefficients.

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Eklund, David. "Topics in computation, numerical methods and algebraic geometry." Doctoral thesis, KTH, Matematik (Avd.), 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-25941.

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This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers. In Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes. In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism. Paper IV is a study of curves on certain special quartic surfaces in projective 3-space. The surfaces are invariant under the action of a finite group called the level (2,2) Heisenberg group. In the paper, we determine the Picard group of a very general member of this family of quartics. We have found that the general Heisenberg invariant quartic contains 320 smooth conics and we prove that in the very general case, this collection of conics generates the Picard group.
QC 20101115

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Björklund, Johan. "Knots and Surfaces in Real Algebraic and Contact Geometry." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-156908.

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This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure. In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces. In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.

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Heier, Gordon. "Some effective results in algebraic geometry." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=965086631.

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De, Zeeuw Frank. "An algebraic view of discrete geometry." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/38158.

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This thesis includes three papers and one expository chapter as background for one of the papers. These papers have in common that they combine algebra with discrete geometry, mostly by using algebraic tools to prove statements from discrete geometry. Algebraic curves and number theory also recur throughout the proofs and results. In Chapter 1, we will detail these common threads. In Chapter 2, we prove that an infinite set of points in R² such that all pairwise distances are rational cannot be contained in an algebraic curve, except if that curve is a line or a circle, in which case at most 4 respectively 3 points of the set can be outside the line or circle. In the proof we use the classification of curves by their genus, and Faltings' Theorem. In Chapter 3, we informally present an elementary method for computing the genus of a planar algebraic curve, illustrating some of the techniques in Chapter 2. In Chapter 4, we prove a bound on the number of unit distances that can occur between points of a finite set in R², under the restriction that the line segments corresponding to these distances make a rational angle with the horizontal axis. In the proof we use graph theory and an algebraic theorem of Mann. In Chapter 5, we give an upper bound on the length of a simultaneous arithmetic progression (a two-dimensional generalization of an arithmetic progression) on an elliptic curve, as well as for more general curves. We give a simple proof using a theorem of Jarnik, and another proof using the Crossing Inequality and some bounds from elementary algebraic geometry, which gives better explicit bounds.

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Thaddeus, Michael. "Algebraic geometry and the Verlinde formula." Thesis, University of Oxford, 1992. http://ora.ox.ac.uk/objects/uuid:12af7dda-26f7-44ec-b335-74193ce1c538.

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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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Kotschick, Dieter. "On the geometry of certain 4 - manifolds." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236179.

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Li, Shiyue. "Tropical Derivation of Cohomology Ring of Heavy/Light Hassett Spaces." Scholarship @ Claremont, 2017. http://scholarship.claremont.edu/hmc_theses/104.

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The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of $\calm_{0, w}$, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of $\overline{\calm}_{0,w}$ as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology.

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Lundkvist, Christian. "Moduli spaces of zero-dimensional geometric objects." Doctoral thesis, Stockholm : Matematik, Kungliga Tekniska högskolan, 2009. http://www.diva-portal.org/smash/record.jsf?searchId=1&pid=diva2:223079.

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Sarmiento-Lopez, X. I. "Algebraic problems in matroid theory." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298554.

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Lewis, Matthew. "Error correction of generalised algebraic-geometry codes." Thesis, Imperial College London, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.407473.

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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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Shifler, Ryan M. "Computational Algebraic Geometry Applied to Invariant Theory." Thesis, Virginia Tech, 2013. http://hdl.handle.net/10919/23154.

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Commutative algebra finds its roots in invariant theory and the connection is drawn from a modern standpoint. The Hilbert Basis Theorem and the Nullstellenstatz were considered lemmas for classical invariant theory. The Groebner basis is a modern tool used and is implemented with the computer algebra system Mathematica. Number 14 of Hilbert\'s 23 problems is discussed along with the notion of invariance under a group action of GLn(C). Computational difficulties are also discussed in reference to Groebner bases and Invariant theory.The straitening law is presented from a Groebner basis point of view and is motivated as being a key piece of machinery in proving First Fundamental Theorem of Invariant Theory.
Master of Science

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Massarenti, Alex. "Biregular and Birational Geometry of Algebraic Varieties." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4679.

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Every area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory.

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Redman, Lynn. "Algebraic Methods for Proving Geometric Theorems." CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/923.

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Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal without changing the variety is a powerful tool in determining a variety. In general, the quotient and remainder on division of polynomials in more than one variable are not unique. One property of a Groebner basis is that it yields a unique remainder on division. To prove geometric theorems algebraically, we first express the hypotheses and conclusions as polynomials. Then, with the aid of a computer, apply the Groebner Basis Algorithm to determine if the conclusion polynomial(s) vanish on the same variety as the hypotheses.

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30

Hammes, Emily. "Unifications of Pythagorean Triple Schema." Digital Commons @ East Tennessee State University, 2019. https://dc.etsu.edu/honors/502.

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Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.

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Miadantsoa, Rakoto. "Groupes finis d'automorphismes des varietes abeliennes de dimension deux." Toulouse 3, 1988. http://www.theses.fr/1988TOU30051.

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On etablit dans un premier chapitre la liste des sous-groupes finis g dits effectifs, contenus dans sl(2, c), qui peuvent apparaitre comme sous-groupes finis d'automorphismes d'une surface abelienne. Les deuxieme et troisieme chapitres sont consacres a la description et classification des g-surfaces: ce sont les surfaces abeliennes polarisees associees a g, dont la polarisation est g-invariante. Le cas ou g est contenu dans gl(2, c) est examine dans le dernier chapitre

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32

MOUSSA, OUSMANE. "Theoremes des zeros centraux en geometrie analytique reelle." Rennes 1, 1989. http://www.theses.fr/1989REN10062.

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On etudie dans ce travail les theoremes des zeros centraux de la geometrie analytique reelle: il s'agit de degager une caracterisation algebrique des ideaux de fonctions analytiques s'annulant sur un ensemble de points contenus dans la partie de dimension maximale (c'est-a-dire les points centraux) d'un ensemble (ou d'un germe d'ensemble) analytique reel. Ce probleme est traite dans le cas local, puis dans le cas relatif aux ensembles c-analytiques compacts. La methode de demonstration, dans les deux cas, utilise la theorie du spectre reel pour construire une bonne correspondance - l'operation tilda - entre les situations algebriques et celles de type geometrique. On prouve egalement une forme globale, puis une forme locale du positivstellensatz et de ses variantes

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Le, Stum Bernard. "Cohomologie rigide et varietes abeliennes." Rennes 1, 1985. http://www.theses.fr/1985REN10007.

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On decrit la cohomologie de de rham d'une variete abelienne a reduction semi-stable sur un corps local (de caracteristique 0) a l'aide d'une theorie cohomologique recente : la cohomologie rigide des varietes de caracteristique p0. Comme grothendieck l'avait montre pour le module de tate, le premier espace de cohomologie de de rham d'une telle variete abelienne est naturellement muni d'une structure d'extension panachee. Cette extension panachee est construite autour du premier espace de cohomologie rigide de la fibre speciale du modele de neron, lequel est naturellement muni d'une structure de module de dieudonne filtre. Il en resulte en fait que le premier espace de cohomologie de de rham de la variete abelienne est une extension panachee d'une suite exacte de modules de dieudonne filtres par une autre suite exacte de modules de dieudonne filtres dans la categorie des espaces vectoriels filtres. La theorie de fontaine permet de retrouver cette structure lorsque l'on connait la structure d'extension panachee sur le module de tate de la variete abelienne. L'operation inverse n'est malheureusement pas possible ; cependant, la connaissance de la cohomologie rigide de la fibre speciale du modele de neron (connexe) est suffisante pour retrouver la partie fixe du module de tate de la variete abelienne

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34

Jadda, Zoubida. "Constructions de places reelles et geometrie semi-algebrique." Rennes 1, 1986. http://www.theses.fr/1986REN10102.

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Cette these a pour objet de montrer l'existence d'une chaine d'anneaux de valuations reels d'un corps de fonctions r(v) d'une variete algebrique affine irreductible v, qui sont convexes pour un meme ordre sur r(v) et dont le centre, la dimension, le rang et le rang rationnel, verifiant certaines conditions, sont donnes. La technique de demonstration est un pur produit de la geometrie algebrique reelle. Elle utilise le spectre et la triangulation par un homeomorphisme semi-algebrique

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35

Fekak, Azzeddine. "Sur les exposants de Lojasiewicz." Grenoble 2 : ANRT, 1986. http://catalogue.bnf.fr/ark:/12148/cb37597565q.

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36

Deshpande, D. V. "Topological methods in algebraic geometry : cohomology rings, algebraic cobordism and higher Chow groups." Thesis, University of Cambridge, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.598515.

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This thesis is divided into three chapters. The first chapter is about the cohomology ring of the space of rotational functions. In the second chapter, we define algebraic cobordism of classifying spaces, Ω*(BG) and G-equivariant algebraic cobordism Ω*_G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^j(Ω*(-)); which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We compute Ω*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n + 1). We also compute Ω*(BG) when G is a finite abelian group. A finite non-abelian group for which we compute Ω*(BG) is the quaternion group of order 8. In all the above cases we check that Ω*(BG) is isomorphic to MU*(BG). The third chapter is work-in-progress on Steenrod operations on higher Chow groups. Voevodsky has defined motivic Steenrod operations on motivic cohomology and used them in his proof of the Minor Conjecture.

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37

Gong, Shengjun, and 龔勝軍. "Linear coordinates, test elements, retracts and automorphic orbits." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40988065.

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Gong, Shengjun. "Linear coordinates, test elements, retracts and automorphic orbits." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40988065.

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39

Nyman, Adam. "The geometry of points on quantum projectivizations /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/5727.

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40

Abbott, Kevin Toney. "Applications of algebraic geometry to object/image recognition." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-1935.

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41

Dindyal, Jaguthsing Presmeg Norma C. "Algebraic thinking in geometry at high school level." Normal, Ill. Illinois State University, 2003. http://wwwlib.umi.com/cr/ilstu/fullcit?p3087865.

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Thesis (Ph. D.)--Illinois State University, 2003.
Title from title page screen, viewed November 15, 2005. Dissertation Committee: Norma C. Presmeg (chair), Nerida F. Ellerton, Beverly S. Rich, Sharon S. McCrone. Includes bibliographical references (leaves 208-219) and abstract. Also available in print.

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42

Hampton, III Earl Ravi M. "A PRIMER FOR THE FOUNDATIONS OF ALGEBRAIC GEOMETRY." [Greenville, N.C.] : East Carolina University, 2010. http://hdl.handle.net/10342/2797.

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43

Tang, Xin. "Applications of noncommutative algebraic geometry to representation theory /." Search for this dissertation online, 2006. http://wwwlib.umi.com/cr/ksu/main.

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44

Jost, Christine. "Topics in Computational Algebraic Geometry and Deformation Quantization." Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-87399.

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This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group. In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically. In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm. Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes. In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Accepted.

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45

Riccomagno, Eva M. "Algebraic geometry in experimental design and related fields." Thesis, University of Warwick, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.263314.

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46

Di, Natale Carmelo. "Grassmannians and period mappings in derived algebraic geometry." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709191.

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47

Anderson, William Erik 1976. "Applications of algebraic geometry to coding & cryptography." Thesis, Massachusetts Institute of Technology, 2001. http://hdl.handle.net/1721.1/86656.

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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.
Includes bibliographical references (p. 73-74).
by William Erik Anderson.
S.M.

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48

Abou-Rached, John. "Sheaves and schemes: an introduction to algebraic geometry." Kansas State University, 2016. http://hdl.handle.net/2097/32608.

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Master of Science
Department of Mathematics
Roman Fedorov
The purpose of this report is to serve as an introduction to the language of sheaves and schemes via algebraic geometry. The main objective is to use examples from algebraic geometry to motivate the utility of the perspective from sheaf and scheme theory. Basic facts and definitions will be provided, and a categorical approach will be frequently incorporated when appropriate.

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Berardini, Elena. "Algebraic geometry codes from surfaces over finite fields." Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0170.

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Nous proposons, dans cette thèse, une étude théorique des codes géométriques algébriques construits à partir de surfaces définies sur les corps finis. Nous prouvons des bornes inférieures pour la distance minimale des codes sur des surfaces dont le diviseur canonique est soit nef soit anti-strictement nef et sur des surfaces sans courbes irréductibles de petit genre. Nous améliorons ces bornes inférieures dans le cas des surfaces dont le nombre de Picard arithmétique est égal à un, des surfaces sans courbes de petite auto-intersection et des surfaces fibrées. Ensuite, nous appliquons ces bornes aux surfaces plongées dans P3. Une attention particulière est accordée aux codes construits à partir des surfaces abéliennes. Dans ce contexte, nous donnons une borne générale sur la distance minimale et nous démontrons que cette estimation peut être améliorée en supposant que la surface abélienne ne contient pas de courbes absolument irréductibles de petit genre. Dans cette optique nous caractérisons toutes les surfaces abéliennes qui ne contiennent pas de courbes absolument irréductibles de genre inférieur ou égal à 2. Cette approche nous conduit naturellement à considérer les restrictions de Weil de courbes elliptiques et les surfaces abéliennes qui n'admettent pas de polarisation principale
In this thesis we provide a theoretical study of algebraic geometry codes from surfaces defined over finite fields. We prove lower bounds for the minimum distance of codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Then we apply these bounds to surfaces embedded in P3. A special attention is given to codes constructed from abelian surfaces. In this context we give a general bound on the minimum distance and we prove that this estimation can be sharpened under the assumption that the abelian surface does not contain absolutely irreducible curves of small genus. In this perspective we characterize all abelian surfaces which do not contain absolutely irreducible curves of genus up to 2. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization

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Drake, Nathan. "Decoding of multipoint algebraic geometry codes via lists." Connect to this title online, 2009. http://etd.lib.clemson.edu/documents/1263409538/.

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Dissertations / Theses on the topic 'Geometry, Algebraic'

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