Notes, etc.


Equivariant Intersection Theory

This link takes you to notes from William Fulton's Eilenberg lectures on "Equivariant Cohomology in Algebraic Geometry":

Eilenberg lectures, Columbia University, Spring 2007

Our book will soon be available here:

Equivariant Cohomology in Algebraic Geometry

Other notes

"Gillet descent for connective K-theory" PDF

This technical note establishes a descent sequence for connective K-theory, which lets one extend results from the category of quasi-projective schemes to the category of all schemes.  The technique is due to H. Gillet. (July 2020)

"Introduction to equivariant cohomology in algebraic geometry" PDF

Lectures at the July 2010 IMPANGA summer school (to appear in "Impanga Lecture Notes," volume 3).  They give a quick, example-based introduction to the subject.  (April 2011)

"Linear algebraic groups: a crash course in three lectures" PDF

Just the bare basics of the theory, with a few examples.  For a course on GIT by Zinovy Reichstein.  (January 2011)

"Positivity in the cohomology of flag bundles (after Graham)" PDF

William Graham showed that the structure constants of the equivariant cohomology of a (generalized) flag variety are positive in the roots.  This note gives a short, geometric proof, based on a transversality argument.  (April 2007)

"Double Schubert polynomials and double Schubert varieties" PDF

This note gives a geometric explanation for a certain polynomial identity relating double Schubert polynomials to single ones, together with some generalizations to other Lie types. As Allen Knutson kindly pointed out to me, experts were already familiar with the main results, but I do not know a good reference in the literature.  (June 2006)

Some of the following might eventually be polished enough to be posted.  In the meantime, feel free to ask me if you're interested.

"Singularities of Schubert varieties"

Notes from lectures by me, Steve Mitchell, and Monty McGovern, geared toward a proof of Peterson's "ADE" theorem.  (April 2010)

"Notes on moment maps (for equivariant cohomology in algebraic geometry)"

These are notes from an expository talk given at Columbia, to accompany Bill Fulton's lecture series.  (April 2007)  (Comments welcome on the current draft!)

"Notes on real Schubert calculus (mod 2)"

In the cohomology of the complex flag manifold, Schubert classes have particularly nice polynomial representatives, called Schubert polynomials.  It turns out that the same is true in the manifold of real flags, when you take mod 2 coefficients; these notes explain why.  (April 2005)

"Perverse Sheaves and Intersection (Co)homology"

From a talk given at the Student Algebraic Geometry Seminar on November 18, 2005.

"Tropical linear spaces and tropical Grassmannians"

From a talk given at the Tropical Geometry Seminar on October 6, 2005.

"Introduction to codes from algebraic curves"

From a talk given at the Student Algebraic Geometry Seminar on February 8, 2005.

"Sheaves in geometry: an introduction" 

Notes from a talk given at the Student Topology Seminar on November 19, 2004.

"Path integrals in quantum mechanics"

I spent the summer of 2001 under the mistaken impression that I wanted to be a physicist.  The time wasn't completely wasted, though: I learned something about Feynman integrals, and played some incredible Capture-the-Flag.