Category Archives: Algebraic Topology

Monodromy and Moduli

Posted on March 2, 2010 by Charles Siegel

Today, we’re going to prove a BIG theorem, but only in the characteristic zero case (we’ll be working over as usual). The theorem is rather tough, and to do it in positive characteristics it’s best done through stacks. Specifically, we’ll … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Big Theorems, Curves, Deformation Theory, Moduli of Curves | 3 Comments

Monodromy Representations

Posted on February 25, 2010 by Charles Siegel

A departure from directly working with varieties, we’re going to do something that’s strictly topological (at first glance) but which really has deep and important connections with Hodge theory. We’re going to talk about monodromy and monodromy representations. Let be … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Hodge Theory, Vector Bundles | 9 Comments

Applications of the Schubert Calculus

Posted on May 12, 2009 by Charles Siegel

Ok, this is going to be my last post in enumerative geometry for a while, as I’m kind of drifting away from the subject.  However, this one will be fun.  We’ve already established the structure of the cohomology ring for … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | Leave a comment

Pieri and Giambelli Formulas

Posted on April 20, 2009 by Charles Siegel

It’s been a few weeks, but now I’m back and today we’ll talk about the multiplication in the cohomology ring of Grassmannians.  Though we won’t talk about the Littlewood-Richardson rule in its full glory, we will howver discuss the special … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | 1 Comment

Schubert Classes and Cellular Cohomology

Posted on March 27, 2009 by Charles Siegel

So, as of the last post in the series, we defined Schubert cells.  We’re going to use them to discuss the Cohomology of the Grassmannian, and to write down an explicit basis.  With an eye looking forward, next time, we’ll … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | 6 Comments

Schubert Varieties

Posted on March 20, 2009 by Charles Siegel

It’s been awhile since the last post, but Spring Break happened.  Anyway, back to Schubert Calculus! Last time, we discussed Grassmannians, this time, we’re going to talk about their most important subvarieties, the Schubert Varieties.

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory | 5 Comments

Grassmannians, Redux

Posted on February 24, 2009 by Charles Siegel

Ok, today we start our march towards Schubert Calculus.  Before we start, we’ll review the Grassmannian variety itself, because it’s central to the story.  A lot of this will consist of setting up notation, and there will be two different … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | 1 Comment

The Hilbert Polynomial Explained

Posted on October 9, 2008 by Charles Siegel

So, today I discovered something rather nice, that I think could easily get more time in books and the like, but doesn’t. We’ll first have need of a theorem that I don’t want to prove: Theorem: Let be a projective … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Differential Geometry, Intersection Theory | 5 Comments

The Twenty-Seven Lines on the Cubic Surface

Posted on June 18, 2008 by Charles Siegel

Ok, I just came across this proof, and it’s so cool I have to blog about it. This is not related to my ongoing series, nor is this anything to do with the conference I’m at (I won’t be blogging … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Combinatorics, Enumerative Geometry | 14 Comments