# Graduate Student Topology Seminar

## in the PIMS library (ESB 4133)

Introduction to Kahler Geometry and a Discussion on the Calabi Conjecture

A talk by John Ma

Thursday April 17th at 10 am

Following Atsushi's talk, I will (1) describe the use of Hodge theory in the study of Kahler manifolds, ultimately outlining a proof of Kodaira Embedding theorem which describes when a Kahler manifolds is projective, and (2) discuss the Calabi conjecture, some of its consequences and outline a proof given by S.T. Yau.

(Dis)continuity of the Jordan-Chevalley Decomposition

A talk by Maxime Bergeron

Friday April 11th at 1 pm

The Jordan-Chevalley decomposition is a crucial tool in the study of algebraic groups (e.g. general linear groups, orthogonal groups, symplectic groups…) which allows us to write any group element uniquely as a product of its semisimple and unipotent parts. It is a rather unlucky fact that the maps sending an element to its semisimple and unipotent part are not continuous; this makes the decomposition difficult to use if we want to understand algebraic groups from a topological point of view.

In this talk, I will outline one way around this issue by constructing a continuous approximation of the Jordan-Chevalley Decomposition. This so-called *Approximate Jordan Decomposition* was first used by Alexandra Pettet and Juan Souto to show that the space of commuting tuples in a reductive complex algebraic group (e.g. a complex general linear group) is homotopy equivalent to the space of commuting tuples in any of its maximal compact subgroups (e.g. the subgroup of unitary matrices).

Reference:

A. Pettet and J. Souto, Commuting Tuples in Reductive Groups and their Maximal Compact Subgroups, Geometry & Topology 17 (2013).

Topological Spherical Space Forms

A talk by Ian Hambleton (McMaster)

Friday April 4th at 1 pm

Free actions of finite groups on spheres give rise to topological spherical
space forms. The existence and classication problems for space forms have a long
history in the geometry and topology of manifolds. We present a survey
of some of the main results and a guide to the literature.

Kahler Geometry

A talk by Atsushi Kanazawa

Friday March 28th at 12:30 pm

A Kahler manifold is a Riemannian, complex and symplectic manifold in a compatible way.
Kahler manifolds enjoy very rich properties and have played a significant role in complex geometry.
In this talk, I will review some basics of Kahler geometry.
My talk also serves as a gentle introduction to John's talk on Calabi conjecture in April.

Model Categories and Rational Homotopy Theory of Topological Spaces (III/III)

A mini-course by Man Chuen Cheng

Friday February 7th at 1:30 pm
Homotopy theory has been proved to be a useful tool in the studies of topological spaces, simplicial sets and chain complexes.
Quillen generalized this concept and invented model categories as the categorical framework for doing homotopy theory. The
language of model categories also allows us to compare the homotopy theories of different categories. For instance, one can show
that simplicial sets and topological spaces have the same homotopy theory. The work of Quillen and Sullivan also showed that the
rational homotopy theory of simply connected spaces can be modeled by a certain subcategory of commutative differential graded
algebra. In the talks I will introduce the language of model categories, look at different examples and explain the result of
Quillen and Sullivan on rational homotopy types.

Model Categories and Rational Homotopy Theory of Topological Spaces (II/III)

A mini-course by Man Chuen Cheng

Friday January 24 at 1:30 pm

Homotopy theory has been proved to be a useful tool in the studies of topological spaces, simplicial sets and chain complexes. Quillen generalized this concept and invented model categories as the categorical framework for doing homotopy theory. The language of model categories also allows us to compare the homotopy theories of different categories. For instance, one can show that simplicial sets and topological spaces have the same homotopy theory. The work of Quillen and Sullivan also showed that the rational homotopy theory of simply connected spaces can be modeled by a certain subcategory of commutative differential graded algebra. In the talks I will introduce the language of model categories, look at different examples and explain the result of Quillen and Sullivan on rational homotopy types.

Model Categories and Rational Homotopy Theory of Topological Spaces (I/III)

A mini-course by Man Chuen Cheng

Friday January 17 at 1:30 pm

Homotopy theory has been proved to be a useful tool in the studies of topological spaces, simplicial sets and chain complexes. Quillen generalized this concept and invented model categories as the categorical framework for doing homotopy theory. The language of model categories also allows us to compare the homotopy theories of different categories. For instance, one can show that simplicial sets and topological spaces have the same homotopy theory. The work of Quillen and Sullivan also showed that the rational homotopy theory of simply connected spaces can be modeled by a certain subcategory of commutative differential graded algebra. In the talks I will introduce the language of model categories, look at different examples and explain the result of Quillen and Sullivan on rational homotopy types.

Supersymmetric Quantum Mechanics on Differential Manifolds

A talk by Atsushi Kanazawa

Friday November 8 at 1 pm

Supersymmetry is the largest possible symmetry of our space-time.
In this talk, I will discuss supersymmetric quantum mechanics on differentiable manifolds.
The goal is to introduce Witten's work, its connection to Hodge-de Rham theory and Morse theory.
I assume basic differential geometry, but no physics background is necessary.

Periodic Complexes and Group Actions (II)

A talk by Cihan Okay

Friday November 1 at 1 pm

I will talk about some applications of a theorem by A. Adem and J. Smith on spaces with periodic cohomology to finite group actions. There are applications of this theorem to produce free group actions on a finite dimensional CW-complex homotopy equivalent to a product of two spheres.

Cohomology of Lie Groups Made Discrete (II)

A talk by Bernardo Villarreal-Herrera

Friday October 25 at 1 pm

We will review Pascual Gainza's paper on the Friedlander-Milnor conjecture which states that the identity homomorphism between a Lie group with the discrete topology and the Lie group with its usual topology induces isomorphisms in the homology and cohomology with finite coefficients of their classifying spaces. In this talk we will see the rational coefficients case and some of Milnor's general results on the conjecture.

Reference: P. Gainza, "Cohomolgy of Lie groups made discrete", Publicacions Matemàtiques, Vol. 34 (1990) 151
-174.

Cohomology of Lie Groups Made Discrete (I)

A talk by Bernardo Villarreal-Herrera

Friday October 18 at 1 pm

We will review Pascual Gainza's paper on the Friedlander-Milnor conjecture which states that the identity homomorphism between a Lie group with the discrete topology and the Lie group with its usual topology induces isomorphisms in the homology and cohomology with finite coefficients of their classifying spaces. In this talk we will see the rational coefficients case and some of Milnor's general results on the conjecture.

Reference: P. Gainza, "Cohomolgy of Lie groups made discrete", Publicacions Matemàtiques, Vol. 34 (1990) 151-174.

An Introduction to Contact Geometry

A talk by Galo Higuera Rojo

Friday October 4 at 2 pm

This will be an introductory talk on contact topology concentrating on 3-manifolds. We will introduce all the required definitions and talk about the relation between contact structures and open book decompositions of 3-manifolds. We will also touch on the dichotomy between overtwisted and tight contact 3-manifolds. This talk should be accessible to anyone with basic knowledge of differential geometry

Along with his lecture notes

Periodic Complexes and Group Actions

A talk by Cihan Okay

Friday September 27 at 2 pm

I will talk about a theorem of A. Adem and J. Smith on spaces with periodic cohomology. There are applications of this theorem to produce free group actions on a finite dimensional CW-complex homotopy equivalent to a product of two spheres.

Archived talks