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RICHARD PAUL GONZALES VILCARROMERO

RICHARD PAUL GONZALES VILCARROMERO

RICHARD PAUL GONZALES VILCARROMERO

DOCTOR OF PHILOSOPHY, MATHEMATICS, UNIVERSIDAD DE WESTERN ONTARIO

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Master of Science, Mathematics (UNIVERSIDAD DE WESTERN ONTARIO)

Licenciado en Matemática
DOCENTE ORDINARIO - ASOCIADO
Docente a tiempo completo (DTC)
Departamento Académico de Ciencias - Sección Matemáticas

Investigaciones

Se encontraron 5 investigaciones

2019 - 2020

Irregular Sasakian Structures and Torus actions

Understand a moduli of Sasakian manifolds that allows the variation of the associated CR-structures. The natural object of study is a set that gives a configuration of different CR-structures compatible with a contact structure. We think that the notion of the bouquet of Sasaki cones given in Boyer [B] encodes important information that can be useful to understand the connectedness of a moduli of Sasakian structures on a manifold. Please see ANEXO for more details.

Participantes:

Instituciones participantes:

  • PONTIFICIA UNIVERSIDAD CATOLICA DEL PERU - Dirección de Gestión de la Investigación (DGI) (Financiadora)
2018 - 2019

Equivariant intersection cohomology of projective group embeddings

The purpose of this project is to investigate and describe the equivariant intersection cohomology of possibly singular projective equivariant embeddings of reductive groups. Such a description is only available in the case of equivariant compactifications of tori, i.e. toric varieties, by work of Barthel, Brasselet, Fieseler and Kaup. Our goal is to extend their results to more general group embeddings, and to write down, explicitly and combinatorially, the equivariant intersection cohomology of possibly singular projective group embeddings, without placing any assumptions on the singular locus. Crucial to our description is the fact that all of these embeddings can be realized as projectivizations of affine monoids. In this setting, we anticipate that a suitable extension of GKM theory for the equivariant intersection cohomology of singular varieties with torus actions could lead to our main results. This is a further step in the program started in [G2]. The outcome should reveal certain aspects of the geometry of projective group embeddings which are not captured by equivariant bivariant theories.

Participantes:

Instituciones participantes:

  • PONTIFICIA UNIVERSIDAD CATOLICA DEL PERU - Dirección de Gestión de la Investigación (DGI) (Financiadora)
2018 - 2019

Geometry of horospherical varieties of Picard rank one

Proyecto conjunto con Nicolas Perrin (Versailles Saint-Quentin, Francia), Alexander Samokhin (Institute for Information Transmission Problems, Rusia) y Clelia Pech (University of Kent, Reino Unido). Basados en la clasificación de variedades horoesféricas con número de Picard 1 hecha por B. Pasquier, buscamos: (1) describir los grupos de cohomología cuántica de dichas variedades y (2) construir colecciones excepcionales completas en las respectivas categorías derivadas de haces coherentes. Nuestro proyecto busca verificar la conjetura de Dubrovin.

Participantes:

Instituciones participantes:

  • GERMAN RESEARCH FOUNDATION - - (Financiadora)
  • Institute for InfOrmation Transmission Problems - departamento de matematicas (Financiadora)
  • PONTIFICIA UNIVERSIDAD CATOLICA DEL PERU - Departamento Académico de Ciencias (Financiadora)
  • UNIVERSITE VERSAILLES SAINT-QUENTIN-EN-YVELINES - departamento de Matematicas (Financiadora)
  • University of DÜsseldorf - Mathematics Department (Financiadora)
  • UNIVERSITY OF KENT - departamento de matematicas (Financiadora)
  • UNIVERSITY OF KENT - MATHEMATICS DEPARTMENT (Financiadora)
2017 - 2019

Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X whose equivariant K-theory of vector bundles does not surject onto its ordinary K-theory, and describe the operational K-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational K-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic K-theory of relatively perfect complexes to bivariant operational K-theory.

Participantes:

Instituciones participantes:

  • national science foundation - - (Financiadora)
  • PONTIFICIA UNIVERSIDAD CATOLICA DEL PERU - Departamento Académico de Ciencias (Financiadora)
  • PONTIFICIA UNIVERSIDAD CATOLICA DEL PERU - Dirección de Gestión de la Investigación (DGI) (Financiadora)
  • THE OHIO STATE UNIVERSITY - Department of mathematics (Financiadora)
  • UNIVERSIDAD DE YALE - Department of mathematics (Financiadora)
2017

Group embeddings and Cox rings

The purpose of this project is to investigate the cone of projective group embeddings with the same Cox ring R(X). The ring R(X) is a unique factorization domain. Q-factorial projective varieties with a finitely generated Cox ring are characterized using Mori theory. If X is a spherical variety, then R(X) is a finitely generated algebra, and it carries a natural grading by the classes of effective divisors. The Cox ring of spherical varieties has been described by Brion and Gagliardi. In the case when X is a G-embedding, we can associate a certain cone H, in the class group Cl(X), so that any element y of H defines a group embedding X_y satisfying R(X_y) = R(X). It seems that H corresponds to the movable cone of X. The purpose of this project is to calculate H explicitely, using the knowledge of NE(X), the cone of effective curves on X, and nef(X), the cone of numerically effective divisors on X, and certain monoid data. Moreover, we would like to understand, and quantify, how the cones H and H', associated to different G-embeddings, are related.

Participantes:

Instituciones participantes:

  • PONTIFICIA UNIVERSIDAD CATOLICA DEL PERU - Dirección de Gestión de la Investigación (DGI) (Financiadora)