We find formulas for the Segre-MacPherson classes of Schubert cells in T-equivariant cohomology and the motivic Segre classes of Schubert cells in T-equivariant K-theory. In doing so we look at the pushforward of the projection map from the Bott-Samelson (Kempf-Laksov) desingularization to the Grassmannian. We find that the Segre-MacPherson classes are stable under pullbacks of maps embedding a Grassmannian into a bigger Grassmannian. We also express these formulas using certain Demazure-Lusztig operators that have previously been used to study these classes. / Doctor of Philosophy / Schubert calculus was first introduced in the nineteenth century as a way to answer certain questions in enumerative geometry. These computations relied on the multiplication of Schubert classes in the cohomology ring of Grassmannians, which parameterize k-dimensional linear subspaces of a vector space. More recently Schubert calculus has been broadened to refer to computations in generalized cohomology theories, such as (equivariant) K-theory. In this dissertation, we study Segre-MacPherson classes and motivic Segre classes of Schubert cells in Grassmannians. Segre-MacPherson classes are related to Chern-Schwartz-MacPherson classes, which are a generalization to singular spaces of the total Chern class of the tangent bundle. Motivic Segre classes are similarly related to motivic Chern classes, which are a K-theory analogue of Chern-Schwartz-MacPherson classes. This dissertation also studies the relationship between Schubert varieties and their Bott-Samelson desingularizations, specifically their (T-equivariant) cohomology and K-theory rings. Since equivariant cohomology (or K-theory) classes can be represented by polynomials, we can represent the Segre-MacPherson (or motivic Segre) classes as rational functions. Furthermore, we use certain operators that act on such polynomials (or rational functions) to find formulas for the rational function representatives of the aforementioned classes.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T14:32:26Z No. of bitstreams: 1 juanantoniopachecocruz.pdf: 674238 bytes, checksum: 5fbe428a7cb6ca56e7ceb6582082376f (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-26T15:14:36Z (GMT) No. of bitstreams: 1 juanantoniopachecocruz.pdf: 674238 bytes, checksum: 5fbe428a7cb6ca56e7ceb6582082376f (MD5) / Made available in DSpace on 2017-05-26T15:14:36Z (GMT). No. of bitstreams: 1 juanantoniopachecocruz.pdf: 674238 bytes, checksum: 5fbe428a7cb6ca56e7ceb6582082376f (MD5) Previous issue date: 2015-11-19 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Um complexo de retas quadrático, ou simplesmente um complexo quadrático, é um conjunto de retas do espaço projetivo Pn (n = 3, no nosso caso) que satisfazem uma equação quadrática. Um complexo quadrático também pode ser considerado como um feixe de quádricas e portanto tem um símbolo de Segre bem definido. Sabe-se que as retas de um dado complexo, passando por um ponto p ∈P3, formam em geral um cone quadrático. Os pontos nos quais esses cones são a união de dois planos formam uma superfície em P3, chamada Superfície Singular do complexo. O objetivo desse trabalho é, fixado um símbolo de Segre, construir o espaço de Moduli dos complexos quadráticos, o espaço de Moduli das superfícies singulares desses complexos e então estudar a relação entre esse espaços. / A quadratic line complex, or a quadratic complex, is by definition a set of lines in a projective space Pn (n = 3, in our case) which satisfy a given quadratic equation. A quadratic complex can also be considered as a pencil of quadrics. Hence, it has a well defined Segre symbol. It is a classical fact that lines of a given complex through any point p ∈P3 form in general a quadratic cone. The points such that theses cones break up into two planes form a surface, the Singular Surface of the complex. The objective of this work is, for a fixed Segre symbol, to construct the Moduli space of quadratic complex, the Moduli space of corresponding singular surfaces and to study the relation between them.

Le travail présenté dans cette thèse concerne l'analyticité et l'algébricité d'applications de Cauchy-Riemann (CR) lisse entre variétés CR analytiques ou algébriques réelles. Ce sujet a trait aux propriétés de prolongement d'applications et a récemment connu un regain d'activité. Notre contribution porte principalement sur l'étude du cas non équidimensionnel et sur le passage à la codimension supérieure à un. Dans la première partie de la thèse, nous considérons la question de l'algébricité d'une application holomorphe locale f envoyant une sous-variété algébrique réelle générique minimale M de Cn, n > 1, dans un sous-ensemble algébrique réel M' de Cn'. Ce problème a pour origine les travaux de Poincaré (1907), et plus récemment de Webster (1977). L'introduction de "variétés caractéristiques" associées à la fois aux ensembles M et M' et à l'application f nous permet de donner deux nouvelles conditions pour que f soit algébrique. Dans la deuxième partie de la thèse, nous étudions le problème de l'analyticité d'une application CR lisse f : M -> M' entre une sous-variété analytique réelle générique minimale M de Cn, n>1, et un sous-ensemble analytique réel M' de Cn'. Nous établissons une généralisation du principe de réflexion de Lewy-Pinchuk (1975-77) et prouvons que si la variété caractéristique est de dimension zéro, f est analytique réelle. Dans la troisième partie de la thèse, nous traitons la situation plus générale où la variété caractéristique est de dimension arbitraire. Nous démontrons que si M' ne contient pas de courbe complexe, f est analytique sur un ouvert dense de M. Plus généralement, nous établissons une estimation supérieure de l'analyticité partielle de f, en fonction de la dimension maximale des feuilletages holomorphes locaux contenus dans M'.

This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.

Due to the exponential growth of the dimension of the space of tensors V_(1)⊗• • •⊗V_(n), any naive method of representing these tensors is intractable on a computer. In practice, we consider feasible subspaces (subvarieties) which are defined to reduce the storage cost and the computational complexity. In this thesis, we study two such types of subvarieties: the third secant variety of the product of n projective spaces, and tensor network states. For the third secant variety of the product of n projective spaces, we determine set-theoretic defining equations, and give an upper bound of the degrees of these equations. For tensor network states, we answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops.

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. / <p>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.</p>

L'objet d'étude de ce mémoire est la géométrie des courbes holomorphes entières à valeurs dans le complémentaire d'hypersurfaces génériques de l'espace projectif complexe. Les conjectures célèbres de Kobayashi et de Green-Griffiths énoncent que pour de telles hypersurfaces, de grand degré, les images de ces courbes entières doivent satisfaire certaines contraintes algébriques. En adaptant les techniques de jets développées notamment par Bloch, Green-Griffiths, Demailly, Siu, Diverio-Merker-Rousseau, pour les courbes à valeurs dans une hypersurface projective (cas dit compact), nous obtenons la dégénérescence algébrique des courbes entières f : ℂ→Pⁿ∖Xd (cas dit logarithmique), pour les hypersurfaces génériques Xd de Pⁿ de degré d ≥ (5n)² nⁿ. Comme dans le cas compact, notre preuve repose essentiellement sur l'élimination algébrique de toutes les dérivées dans des équations différentielles qui sont vérifiées par toute courbe entière non constante. L'existence de telles équations différentielles est obtenue grâce aux inégalités de Morse holomorphes et à une variante simplifiée d'une formule de résidus originalement élaborée par Bérczi à partir de la formule de localisation équivariante d'Atiyah-Bott. La borne effective d ≥ (5n)² nⁿ est obtenue par réduction radicale d'un calcul de résidus itérés de très grande ampleur. Ensuite, la déformation de ces équations différentielles par dérivation le long de champs de vecteurs obliques, dont l'existence est ici généralisée et clarifiée, nous permet d'engendrer suffisamment de nouvelles équations pour réaliser l'élimination algébrique finale évoquée ci-dessus. / The topic of this memoir is the geometry of holomorphic entire curves with values in the complement of generic hypersurfaces of the complex projective space. The well-known conjectures of Kobayashi and of Green-Griffiths assert that for such hypersurfaces, having large degree, the images of these curves shall fulfill algebraic constraints. By adapting the jet techniques developed notably by Bloch, Green-Griffiths, Demailly, Siu, Diverio-Merker-Rousseau, in the case of curves with values in projective hypersurfaces (so-called compact case), we obtain the algebraic degeneracy of entire curves f : ℂ→Pⁿ∖Xd (so called logarithmic case), for generic hypersurfaces Xd in Pⁿ of degree d ≥ (5n)² nⁿ. As in the compact case, our proof essentially relies on the algebraic elimination of all derivatives in differential equations that are satisfied by every nonconstant entire curve. The existence of such differential equations is obtained thanks to the holomorphic Morse inequalities and a simplified variant of a residue formula firstly developed by Bérczi from the Atiyah-Bott equivariant localization formula. The effective lower bound d ≥ (5n)² nⁿ is obtained by radically simplifying a huge iterated residue computation. Next, the deformation of these differential equations by derivation along slanted vector fields, the existence of which is here generalized and clarified, allows us to generate sufficiently many new differential equations in order to realize the final algebraic elimination mentioned above.