Category Archives: Computational Methods

Applications of the Schubert Calculus

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

Pieri and Giambelli Formulas

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

Schubert Classes and Cellular Cohomology

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

Schubert Varieties

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.

Grassmannians, Redux

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

Kontsevich’s Formula

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

Quantum Cohomology

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

Gromov-Witten Invariants

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

Request: Projective Elimination Theory

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

Computing Hilbert Functions

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