April 17, 2014

July 24-August 4, 2006
Week 1

Workshop on Computational and Combinatorial Commutative Algebra
The Fields Institute, Toronto

Organizing Committee: Ragnar Buchweitz (Toronto), Greg Smith (Queen's), Alexander Yong (Fields/Minnesota)
July 24
July 25
July 26
July 27
July 28
Intro/Monomial ideals Resolutions Sheaf cohomology Hyperplane arrangements Schubert/special varieties
9:00-10:30: A. Yong
10:45-12:15: A. Van Tuyl
2:00-5:00: A. Yong and A. Van Tuyl
9:00-10:30: G.Smith
10:45-12:15: R. Buchweitz
2:00-5:00: G. Smith and R. Buchweitz
9:00-10:30: M. Stillman
10:45-12:15: M. Stillman
2:00-5:00: M. Stillman
9:00-10:30: G. Smith
10:45-12:15: G. Denham
2:00-5:00: G. Smith and G. Denham
9:00-10:30: A. Woo
10:45-12:15: H. Abo
2:00-5:00: A. Woo and H. Abo
The sessions will be held at the Fields Institute library. An LCD projector and chalkboard will be available. The afternoon sessions will use laptops. There will be an informational session to discuss installing Macaulay 2.

Abstracts below are listed alphabetically by speaker (as they become available)

Theory and Computation in the Search for Special Surfaces
Hirotachi Abo
(Colorado State U.)
The classification of smooth nongeneral-type surfaces in projective fourspace is motivated by the theorem of Ellingsrud and Peskine, which says the degrees of such surfaces are bounded. For constructing smooth surfaces in projective fourspace, Decker, Ein and Schreyer developed a powerful method, which can be implemented in Macaulay 2. With a few exceptions, the known surfaces of degree greater than or equal to 11 have been constructed in a systematic way as an application of this method.

In this talk, I will discuss the systematic study of smooth surfaces in projective fourspace based on the Decker-Ein-Schreyer method. I also want to a random search over a finite field employing a ``needles in a haystack" approach, if possible.

This talk is closely related to the sessions "Resolutions (over both polynomial rings and exterior algebras)" and "Sheaf Cohomology".

Here is the afternoon session tutorial.

Free resolutions in algebra and geometry
Ragnar-Olaf Buchweitz and Gregory G. Smith
(U. Toronto/Queen's U.)
We'll survey some properties and applications of the minimal free resolutions for modules over a polynomial ring. The first talk will review the basic invariants associated to a free resolution and examine related algebra conditions (i.e. flatness and being Cohen-Macaulay). The second talk will investigate some the features in the free resolutions for the ideal of a projective variety, concentrating on points and curves. Here is the afternoon session tutorial.
Orlik-Solomon Algebras II
Graham Denham
(Western U.)
I will continue the themes of Greg Smith's talk. The singular variety for a module over an exterior algebra coincides with the idea of resonance for a topological space. The systematic study of resonance began with the complements of complex hyperplane arrangements in the early 90's. The ideas have since found application elsewhere, but the story for arrangements is particularly nice. Via BGG duality, it is useful to think about resonance alternately in terms of the Orlik-Solomon algebra (skew-commutative algebra) and in terms of certain modules over a (commutative) polynomial ring. I will end by describing some recent results about resonance and other algebraic invariants of arrangements that began as conjectures motivated by Macaulay2 experiments. This will leave a place for some open problems in the afternoon session.
Orlik-Solomon Algebras
Greg Smith
(Queen's University)
In this talk, we study complex hyperplane arrangements via Orlik-Solomon algebras. After giving the basic definitions and properties of hyperplane arrangements and Orlik-Solomon algebras, we examine the Orlik-Solomon algebra as a finite module over the exterior algebra. In particular, we look at singular varieties (an invariant of a module over the exterior algebra), resolutions over the exterior algebra, and some connections with Stanley-Reisner ideals.

Here is the afternoon session tutorial.

Computing with sheaves and cohomology
Mike Stillman
(Cornell University)
The purpose of the Wednesday morning session is twofold: to show how to compute with coherent sheaves and their cohomology, and to apply this to some important applications: the Hodge diamond of a projective variety, and computing with divisors on curves and surfaces.

Part I.

We use a 'working persons' definition of sheaf on projective space and an algebraic version of the Cech complex to define cohomology of a (coherent) sheaf on projective space. We also consider the related notion of local cohomology (with support in a maximal ideal).

We state the very useful theorems of Serre on computing sheaf cohomology and local cohomology. The proofs are doable given the definitions and are mostly left as an exercise for the afternoon.

Part II. Examples and applications

Differentials and the Hodge diamond.

We first show how to compute the sheaf of differentials on a projective variety (or scheme), and then also the sheaf of differential p-forms. We apply Serre to finding the Hodge diamond of a variety. A computational challenge: write a Macaulay2 routine to compute the Hodge diamond. Fastest one wins!


After giving the definition and important examples, we consider: canonical divisor intersection numbers (compute using Riemann-Roch!) Castelnuovo's rationality criterion for surfaces When are two divisors linearly equivalent? the map corresponding to a divisor We apply these techniques to a "mystery surface".

Here is the mystery.pdf file.

Here are the lecture notes for the morning talk.

Here are the notes for the afternoon tutorial.

Monomial ideals
Adam Van Tuyl
(Lakehead University)
We pick up where the first lecture left off. We begin by introducing some of the standard terminology associated to simplicial complexes, e.g., faces, facets, f-vectors, pure, Kruskal-Katona's theorem, and dimension; we then describe how one can use Macaulay 2 to study simplicial complexes. We also set the ground work for upcoming lectures by describing how to find the minimal free resolution of an ideal using Macaulay 2, and how to read the Betti diagram of an ideal. We introduce the notions of regularity, projective dimension, and ideals with linear resolutions. A quick introduction to generic initial ideals rounds out the lecture. The topics covered in the lecture form the background for the afternoon tutorials on resolutions of edge ideals and componentwise linear ideals. Find the afternoon tutorial here.
Singularities of Schubert Varieties
Alexander Woo
(UC Davis)
This lecture will apply some of the techniques discussed during the week to a specific group of mostly open problems, namely calculating local invariants of singularities of Schubert varieties. I will concentrate on multiplicity and (Cohen-Macaulay) type.

I will begin by explaining what Schubert varieties are and how to get equations for local charts on them. Then I will explain how multiplicity and type can be calculated (in general).

I will also introduce interval pattern avoidance, a combinatorial idea which gives local isomorphisms between certain charts on Schubert varieties; this allows us to extrapolate information about one Schubert variety to certain other Schubert varieties.

I will focus on the usual flag variety (type A/SL_n), though if time permits I will say a few things about type D (SO_{2n}).

Here is the tutorial.

Computational and Combinatorial Commutative Algebra
Alexander Yong
(U. Minnesota and U. Toronto)
We begin with a brief overview of the objectives and themes of the workshop. In this lecture we'll explore Gr\"{o}bner degeneration, Gr\"{o}bner (``enough'') bases, monomial ideals and the Reisner-Stanley simplicial complex. We also introduce some basics of computation and coding in Macaulay 2 in relation to these topics. We then explain a research theme combining these ideas: combinatorial formulae for Hilbert series of (generalized) determinantal ideals. This connects to, and is motivated by, symmetric function theory from algebraic combinatorics, as well as the geometry of flag varieties and degeneracy loci of vector bundles. I'll discuss some of the related background that will be used in studying related (open) questions during the afternoon session. Here is the afternoon tutorial.
Here are the slides from the morning.

Back to top