Wednesday, November 25, 2009
Hilbert polynomials and cohomology
Wed, 11/25/2009 - 1:00pm, Skiles 255
Matt Baker, School of Mathematics, Georgia Tech Organizer: Matt Baker
We will state Serre's fundamental finiteness and vanishing results for the cohomology of coherent sheaves on a projective algebraic variety. As an application, we'll prove that the constant term of the Hilbert Polynomial does not depend on the projective embedding, a fact which is hard to understand using classical (non-cohomological) methods.
Wednesday, November 18, 2009
Grothendieck Topologies
Wed, 11/18/2009 - 1:00pm, Skiles 255
Doug Ulmer, Ga Tech Organizer: John Etnyre
In the 60s, Grothendieck had the remarkable idea of introducing a new kind of topology where open coverings of X are no longer collections of subsets of X, but rather certain maps from other spaces to X. I will give some examples to show why this is reasonable and what one can do with it.
Wednesday, November 11, 2009
Derived functors and Cech cohomology
Wed, 11/11/2009 - 1:00pm, Skiles 255
Farbod Shokrieh, Ga Tech Organizer: John Etnyre
We will show that the construction of derived functors satisfy the required universal property.I will then show that, for any ringed space, the abelian category of all sheaves of Modules has enough injectives. We achieve this by first characterizing injective abelian groups (Baer's theorem).The relation with Cech cohomology will also be studied. In particular, I will show that the first Cech and Grothendieck sheaf cohomology groups are isomorphic for any topological space (without using spectral sequences).
Wednesday, November 4, 2009
Derived functors and sheaf cohomology
Wed, 11/04/2009 - 1:00pm, Skiles 255
Farbod Shokrieh, Ga Tech Organizer: John Etnyre
We will continue the study of derived functors between abelian categories. I will show why injective objects are needed for the construction. I will then show that, for any ringed space, the abelian category of all sheaves of Modules has enough injectives. The relation with Cech cohomology will also be studied.
Wednesday, October 21, 2009
The Grothendieck definition of sheaf cohomology
Wed, 10/21/2009 - 1:00pm, Skiles 255
Farbod Shokrieh, Ga Tech Organizer: John Etnyre
As we have seen already, the global section functor is left exact. To get a long exact sequence, I will first give the construction of derived functors in the more general setting of abelian categories withenough injectives. If time permits, I will then show that for any ringed space the category of all sheaves of Modules is an abelian category with enough injectives, and so we can construct sheaf cohomology as the right derived functors of the global section functor. The relation with Cech cohomology will be studied in a subsequent talk.
Wednesday, October 14, 2009
Cech cohomology of a sheaf
Wed, 10/14/2009 - 1:00pm, Skiles 255
John Etnyre, Ga Tech Organizer: John Etnyre
We will briefly review the definition of the Cech cohomology groups of a sheaf (so if you missed last weeks talk, you should still be able to follow this weeks), discuss some basic properties of the Cech construction and give some computations that shows how the theory connects to other things (like ordinary cohomology and line bundles).
Wednesday, October 7, 2009
Cech Cohomology of Sheaves
Wed, 10/07/2009 - 1:00pm, Skiles 255
Matt Baker, School of Mathematics, Georgia Tech Organizer: John Etnyre
We will define the Cech cohomology groups of a sheaf and discuss some basic properties of the Cech construction.
Wednesday, September 30, 2009
Introduction to Sheaf Cohomology
Wed, 09/30/2009 - 1:00pm, Skiles 255
Matt Baker, School of Mathematics, Georgia Tech Organizer: John Etnyre
After a few remarks to tie up some loose ends from last week's talk on locally ringed spaces, I will discuss exact sequences of sheaves and give some natural examples coming from real, complex, and algebraic geometry. In the context of these examples, we'll see that a surjective map of sheaves (meaning a morphism of sheaves which is surjective on the level of stalks) need not be surjective on global sections. This observation will be used to motivate the need for "sheaf cohomology" (which will be discussed in detail in subsequent talks).
Wednesday, September 23, 2009
Locally ringed spaces
Wed, 09/23/2009 - 1:00pm, Skiles 269
Matt Baker, School of Mathematics, Georgia Tech Organizer: John Etnyre
I will discuss how various geometric categories (e.g. smooth manifolds, complex manifolds) can be be described in terms of locally ringed spaces. (A locally ringed space is a topological spaces endowed with a sheaf of rings whose stalks are local rings.) As an application of the notion of locally ringed space, I'll define what a scheme is.
Wednesday, September 16, 2009
Introduction to Sheaf Theory
Wed, 09/16/2009 - 1:00pm, Skiles 255
John Etnyre, Ga Tech Organizer: John Etnyre
In these talks we will introduced the basic definitions and examples of presheaves, sheaves and sheaf spaces. We will also explore various constructions and properties of these objects.
Wednesday, September 9, 2009
Introduction to Sheaf Theory
Wed, 09/09/2009 - 1:00pm, Skiles 269
John Etnyre, Ga Tech Organizer: John Etnyre
In these talks we will introduced the basic definitions and examples of presheaves, sheaves and sheaf spaces. We will also explore various constructions and properties of these objects.
Wednesday, September 2, 2009
Introduction to Sheaf Theory
Wed, 09/02/2009 - 1:00pm, Skiles 255
John Etnyre, Ga Tech Organizer: John Etnyre
In these talks we will introduced the basic definitions and examples of presheaves, sheaves and sheaf spaces. We will also explore various constructions and properties of these objects.