
SCIENTIFIC PROGRAMS AND ACTIVITIES 

May 3, 2016  
July 24August 4, 2006
Organizing Committee: Ragnar Buchweitz (Toronto), Greg Smith (Queen's),
Alexander Yong (Fields/Minnesota)

Monday July 24 
Tuesday July 25 
Wednesday July 26 
Thursday July 27 
Friday July 28  
Intro/Monomial ideals  Resolutions  Sheaf cohomology  Hyperplane arrangements  Schubert/special varieties  




 
Abstracts below are listed alphabetically by speaker (as they become available)
 The classification of smooth nongeneraltype 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 DeckerEinSchreyer 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".  
 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 CohenMacaulay). 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.  
 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 OrlikSolomon algebra (skewcommutative 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.  
 In this talk, we study complex hyperplane arrangements via OrlikSolomon algebras. After giving the basic definitions and properties of hyperplane arrangements and OrlikSolomon algebras, we examine the OrlikSolomon 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 StanleyReisner ideals.  
 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 pforms. 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! Divisors After giving the definition and important examples, we consider: canonical divisor intersection numbers (compute using RiemannRoch!) 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".  
 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, fvectors, pure, KruskalKatona'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.  
 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 (CohenMacaulay) 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}).  
 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 ReisnerStanley
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. 