Área de Concentração: 45131

Concentration area: 45131

Criação: 03/03/2020

Creation: 03/03/2020

Ativação: 03/03/2020

Activation: 03/03/2020

Nr. de Créditos: 1

Credits: 1

Carga Horária:

Workload:

Docente Responsável:

Professor:

Cristian Andres Ortiz Gonzalez

Objetivos:

The geometric theory of characteristic clases developed by Chern and Weil, which relates topological information with intrinsic geometric quantities, is one of the fundamental tools in modern geometry. It has its origins in the Gauss-Bonnet theorem and allows for great generalizations such as the Chern-Gauss-Bonnet, Hirzebruch's signature theorem and the Atiyah- Singer index theorem. The course aims at introducing graduate students to Chern-Weil theory, rst by discussing classical results and then moving to recent results connecting Chern-Weil theory with Lie theory.

Objectives:

The geometric theory of characteristic clases developed by Chern and Weil, which relates topological information with intrinsic geometric quantities, is one of the fundamental tools in modern geometry. It has its origins in the Gauss-Bonnet theorem and allows for great generalizations such as the Chern-Gauss-Bonnet, Hirzebruch's signature theorem and the Atiyah- Singer index theorem. The course aims at introducing graduate students to Chern-Weil theory, rst by discussing classical results and then moving to recent results connecting Chern-Weil theory with Lie theory.

Justificativa:

Conteúdo:

The space of singular chains C(G) on a Lie group G has the structure of a dierential graded Hopf algebra. In [4], it is shown that, just as representations of the group G are determined by those of the Lie algebra, modules over C(G) admit innitesimal description. The goal of the course will be to present an introduction to Chern-Weil theory, to explain the relationship it has with the Lie theory for singular chains on Lie groups and with the theory of higher local system on classifying spaces. The course will consist of 4 lectures. The rst two lectures will be an introduction to Chern-Weil theory and some of the applications of the theory of characteristic classes in topology and geometry. The last two lectures will describe the relationship between Chern-Weil theory and some recent work on the Lie theory of singular chains on Lie groups. The contents of the lectures will be approximately as follows: Lecture 1: The Gauss-Bonnet theorem and Chern Weil theory Lecture 2: Applications: Chern-Gauss-Bonnet, Hirzebruch's signature theorem and Thom's classication of cobordism invariants Lecture 3: Lie theory for the algebra of singular chains on a Lie group Lecture 4: Chern-Weil theory for 1-local systems.

Content:

The space of singular chains C(G) on a Lie group G has the structure of a dierential graded Hopf algebra. In [4], it is shown that, just as representations of the group G are determined by those of the Lie algebra, modules over C(G) admit innitesimal description. The goal of the course will be to present an introduction to Chern-Weil theory, to explain the relationship it has with the Lie theory for singular chains on Lie groups and with the theory of higher local system on classifying spaces. The course will consist of 4 lectures. The rst two lectures will be an introduction to Chern-Weil theory and some of the applications of the theory of characteristic classes in topology and geometry. The last two lectures will describe the relationship between Chern-Weil theory and some recent work on the Lie theory of singular chains on Lie groups. The contents of the lectures will be approximately as follows: Lecture 1: The Gauss-Bonnet theorem and Chern Weil theory Lecture 2: Applications: Chern-Gauss-Bonnet, Hirzebruch's signature theorem and Thom's classication of cobordism invariants Lecture 3: Lie theory for the algebra of singular chains on a Lie group Lecture 4: Chern-Weil theory for 1-local systems.

Forma de Avaliação:

Observação:

There are many excellent references for the rst part of the course, some of which are mentioned below. The second part is based on joint work with Alexander Quintero.

Notes/Remarks:

There are many excellent references for the rst part of the course, some of which are mentioned below. The second part is based on joint work with Alexander Quintero.

Bibliografia:

[1] J. Milnor and J. Shtashe, Characteristic classes, Annals of Mathematics Studies, 1974. [2] L. Nicolaescu, Lectures on the geometry of manifolds, World Scientic, 1996. [3] F. Hirzebruch, Manifolds and modular forms, Aspects of Mathematics, 1992. [4] C. Arias Abad, Singular chains on Lie groups and the Cartan relations I, arXiv:1908.10460, submitted for publication. [5] C. Arias Abad and A. Quintero Velez, Singular chains on Lie groups and the Cartan relations II, available upon request.

Bibliography:

[1] J. Milnor and J. Shtashe, Characteristic classes, Annals of Mathematics Studies, 1974. [2] L. Nicolaescu, Lectures on the geometry of manifolds, World Scientic, 1996. [3] F. Hirzebruch, Manifolds and modular forms, Aspects of Mathematics, 1992. [4] C. Arias Abad, Singular chains on Lie groups and the Cartan relations I, arXiv:1908.10460, submitted for publication. [5] C. Arias Abad and A. Quintero Velez, Singular chains on Lie groups and the Cartan relations II, available upon request.

Tipo de oferecimento da disciplina:

Presencial

Class type:

Presencial