Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2005. / Includes bibliographical references (leaves 75-79). Also available in electronic version.

In this thesis, we solve for (equivariant) Gromov-Witten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2-bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g-2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of non-toric threefolds. Secondly, we compute the C-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Grornov-Witten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the Gromov-Witten theory of Nakamura's G- Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the Cromov-Witten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)].

In this thesis, we solve for (equivariant) Gromov-Witten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2-bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g-2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of non-toric threefolds. Secondly, we compute the C-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Grornov-Witten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the Gromov-Witten theory of Nakamura's G- Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the Cromov-Witten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)].

In this thesis, we solve for (equivariant) Gromov-Witten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2-bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g-2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of non-toric threefolds. Secondly, we compute the C-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Grornov-Witten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the Gromov-Witten theory of Nakamura's G- Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the Cromov-Witten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)]. / Science, Faculty of / Mathematics, Department of / Graduate

Although it was invented by Mikhail Gromov, in 1987, to describe some family of groups[1], the notion of Gromov hyperbolicity has many applications and interpretations in different fields. It has applications in Biology, Networking, Graph Theory, and many other areas of research. The Gromov hyperbolicity constant of several families of graphs and geometric spaces has been determined. However, so far, the only known algorithm for calculating the Gromov hyperbolicity constant δ of a discrete metric space is the brute force algorithm with running time O (n4) using the four-point condition. In this thesis, we first introduce an approximation algorithm which calculates a O (log n)-approximation of the hyperbolicity constant δ, based on a layering approach, in time O(n2), where n is the number of points in the metric space. We also calculate the fixed base point hyperbolicity constant δr for a fixed point r using a (max, min)−matrix multiplication algorithm by Duan in time O(n2.688)[2]. We use this result to present a 2-approximation algorithm for calculating the hyper-bolicity constant in time O(n2.688). We also provide an exact algorithm to compute the hyperbolicity constant δ in time O(n3.688) for a discrete metric space. We then present some partial results we obtained for designing some approximation algorithms to compute the hyperbolicity constant δ.

Ce mémoire revient sur l'invariant de Gromov-Witten dans le contexte de topologie symplectique. D'abord, on présente un survol des notions nécessaires de la topologie symplectique, qui inclut les espaces vectoriels symplectiques, les variétés symplectiques, les structures presque complexes et la première classe de Chern. Ensuite, on présente une définition de l'invariant de Gromov-Witten, qui utilise les courbes pseudoholomorphes, les espaces de modules ainsi que les applications d'évaluation. Finalement, on donne quelques exemples de calcul d'invariant à la fin de ce mémoire. / The present work reviews the Gromov-Witten invariant in the context of symplectic topology. First, we showcase the basic concepts required for the understanding of the matter, which includes symplectic vector spaces, symplectic manifolds, almost complex structures and the first Chern class. Then, we provide a definition of the Gromov-Witten invariant, after studying pseudoholomorphic curves, moduli spaces and evaluation maps. In the end, we present some examples of Gromov-Witten invariant calculations.

We defined a new type of open Gromov-Witten invariants on hyperK\"aher manifolds with holomorphic / Mathematics

SANTOS, Renan da Silva. Geometria enumerativa via invariantes de Gromov-Witten e mapas estáveis. 2015. 78 f. Dissertação (mestrado) - Universidade Federal do Ceará, Centro de Ciências, Programa de Pós-Graduação em Matemática, Fortaleza-Ce, 2015 / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-05-29T18:19:53Z No. of bitstreams: 1 2015_dis_rssantos.pdf: 870583 bytes, checksum: f5ebc0c90f1e8aaca61f2be5057d0448 (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-06-01T10:53:48Z (GMT) No. of bitstreams: 1 2015_dis_rssantos.pdf: 870583 bytes, checksum: f5ebc0c90f1e8aaca61f2be5057d0448 (MD5) / Made available in DSpace on 2015-06-01T10:53:49Z (GMT). No. of bitstreams: 1 2015_dis_rssantos.pdf: 870583 bytes, checksum: f5ebc0c90f1e8aaca61f2be5057d0448 (MD5) Previous issue date: 2015 / In this work, I present the Gromov-Witten theory, quantum cohomology and stable maps and use these tools to obtain some enumerative results. In particular, I proof the Kontsevich formula to projective rational plane curves of degree d. I do an introductory study of Mumford-Knudsen spaces and construct the Kontsevich spaces in order to define gromov-witten invariants. These are used to define the quantum cohomology ring. Next, I apply the general theory to the case of the projective plane and, using the the associativity of the quantum product, I obtain the Kontsevich formula. I also study the boundary of the modulli space of stable maps and describe its Picard group. Following the ideas of Pandharipand, especially the algorithm he developed, I calculate some characteristic numbers of curves in the projective space. / Neste trabalho apresento a teoria de Gromov-Witten, cohomologia quântica e mapas estáveis e uso estas ferramentas para obter alguns resultados enumerativos. Em particular, provo a fórmula de Kontsevich para curvas racionais projetivas planas de grau d. Faço um estudo introdutório dos espaços de Mumford-Knudsen e construo os espaços de Kontsevich a fim de definir os invariantes de Gromov-Witten. Estes são usados para definir o anel de cohomologia quântica. Em seguida, aplico a teoria geral para o caso do plano projetivo e, usando a associatividade do produto quântico, obtenho a fórmula de Kontsevich. Também estudo a fronteira do espaço modulli de mapas estáveis e descrevo o grupo de Picard destes. Com isso, seguindo as ideias de Pandharipand, especialmente o algoritmo por este desenvolvido, calculo alguns números característicos de curvas no espaço projetivo.

Thompson’s groups F, T, and V represent crucial examples of groups in geometric group theory that bridge it with other areas of mathematics such as logic, computer science, analysis, and geometry. One of the ways to study these groups is by understanding the geometric meaning of their actions. In this thesis we deal with Thompson’s group T that acts naturally on the unit circle S1, that is identified with the segment [0, 1] with the end points glued together. The main result of this work is the explicit construction of the Schreier graph of T with respect to the action on the orbit of 1/2. This is done by careful examination of patterns in how the generators of T act on binary words. As a main application, the nonamenability of the action of T on S1 is proved by defining injections on the set of vertices of the constructed graph that satisfy Gromov’s doubling condition. This gives an alternative proof of the known fact that T is nonamenable.

In this thesis, we report on two projects applying representation theoretic techniques to solve enumerative and geometric problems, which were carried out by the author during his pursuit of Ph.D. at Columbia. We first study the relative Gromov-Witten theory on T*P¹ x P¹ and show that certain equivariant limits give relative invariants on P¹ x P¹. By formulating the quantum multiplications on Hilb(T*P¹) computed by Davesh Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion operator computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits. Brenti proves a non-recursive formula for the Kazhdan-Lusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation is given by Sophie Morel via weight truncation of perverse sheaves. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We demonstrate the result with essentially new examples using sl₃ and sl₄..