Category Archives: Computational Methods

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

Kontsevich’s Formula

Posted on December 23, 2008 by Charles Siegel

This is my last post of 2008, so happy holidays to everyone!. This one shouldn’t take too long, it’s just applying the lessons of the last few posts to compute some numbers. We start by talking like physicists. We’ll write … Continue reading →

Posted in Algebraic Geometry, Big Theorems, Cohomology, Computational Methods, Curves, Enumerative Geometry | 6 Comments

Quantum Cohomology

Posted on December 18, 2008 by Charles Siegel

Now that we know what a Gromov-Witten invariant is (at least, for nice spaces…we’re avoiding stacks for this series, and assuming that everything behaves nicely and that these actually count things…), we can start talking about Quantum Cohomology, which organizes … Continue reading →

Posted in Algebraic Geometry, Cohomology, Computational Methods, Curves, Enumerative Geometry | 2 Comments

Gromov-Witten Invariants

Posted on December 12, 2008 by Charles Siegel

Ok, sorry this took so long, but I’m back. Hopefully will be more regular now. Anyway, last time we talked about Stable Maps, and in the meantime, there’s been a post at the Secret Blogging Seminar talking about Gromov-Witten invariants … Continue reading →

Posted in Algebraic Geometry, Cohomology, Computational Methods, Curves, Enumerative Geometry | 3 Comments

Request: Projective Elimination Theory

Posted on August 14, 2008 by Charles Siegel

We talked before about elimination theory, doing it entirely in the affine case. The question was asked about how to do it projectively. There are a couple of subtleties to it, but the idea is simple: we eliminate in each … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Computational Methods | 2 Comments

Computing Hilbert Functions

Posted on July 30, 2008 by Charles Siegel

Today, we’ll link the computational thread back to the thread involving Hilbert schemes, by working out how to compute the Hilbert function (and thus polynomial) for any ideal in the ring . The trick involves Groebner bases and flat families, … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Computational Methods | 1 Comment