A Macaulay 2 Conference,

Cornell University,

March 16-19, 2008

Conference's Homepage

Abstract: A simplified version of string theory, known as topological field theory, can be understood in terms of commutative algebra. The ends of open strings can be described in terms of matrix factorizations. The algorithm, due to Avramov and Grayson, which is built into Macaulay 2 for computing Ext groups turns out to be uncannily well-suited to computations in this context.

Abstract: We will discuss some methods, following Bermejo and Gimenez, for calculating the Castelnuovo-Mumford regularity of a homogeneous ideal without actually computing its minimal graded free resolution. These techniques rely on the fact that the extremal Betti numbers of a homogeneous ideal $I \subset R=K[X_1,\dots,X_n]$ and its revlex initial ideal are the same, provided that $X_n,\dots,X_1$ form a filter regular sequence for $R/I$. The main drawback of these methods is that they require the calculation of a Grobner basis after a random change of coordinates, which is usually a computationally-expensive task. We propose a strategy, which can be implemented in M2, to avoid this problem. We show how to search for a reasonably sparse change of coordinates in order to obtain the above condition on $X_n, \dots,X_1$ and in particular we prove that the sparsity of a filter regular sequence for $R/I$ is bounded above by the one of $R/in(I)$.

Abstract: I will discuss ongoing work with Dan Cohen, Michael Falk and Alexander Varchenko that examines the critical points of a product of (powers of) linear forms on $\C^\ell$, $\Phi_\lambda=\prod_{i=1}^n f_i^{\lambda_i}$. It is known that for generic choices of parameters $\{\lambda_i\}$, the critical set is isolated and nondegenerate. (For appropriate linear forms, the critical points index a basis of solutions to a certain physically significant system of differential equations.) The genericity condition on the parameters $\{\lambda_i\}$ is closely related to a condition for certain local system cohomology groups to vanish. In order to make the relationship precise, we construct a suitable ``universal'' complex variety for the problem. The obvious choice is smooth but noncompact; its naive closure is interesting but singular; the ``wonderful models'' of de Concini and Procesi provide an informative smooth compactification.

The commutative algebra in this project is explicit and nicely illuminated by {\em Macaulay 2} calculations which, in fact, were responsible for pointing us in the right direction more than once.

Abstract: A group of conjectures by Boij and Soederberg have led, through their work and work of mine with Frank Schreyer, Jerzy Weyman and Gunnar Floeystad, to a remarkable and quick expansion of our understanding of free resolutions, on the one hand, and of the cohomology of vector bundles on projective space on the other. In fact the cones generated by the Betti tables of modules and by the cohomology tables of vector bundles are now completely described. The extremal rays of correspond to special modules and vector bundles that are easily described (but not so easily constructed.) And the two cones are ``almost'' dual to each other, in the sense of convex geometry. I'll describe the duality that underlies this progress, and some of the problems that remain.

Abstract: Algorithmic desingularization has long been considered a purely theoretical subject, until Bodnar and Schicho showed that at least Villamayor's algorithmic approach can be implemented. The main focus of computational resolution of singularities has since been on this algorithm and more efficient implementations have been achieved - all of them of course using the weak transform. The use of the strict transform for a desingularization implies the use of the notion of normal flatness and of the Hilbert-Samuel stratum, which is notoriously expensive to compute. Hence the approach of Bierstone and Milman has (up to now) never been considered as a practial alternative. In this talk I shall introduce a hybrid approach using some ideas of Villamayor's algorithm and modifying the Bierstone-Milman approach in a way which allows the use of the strict transform and the Hilbert-Samuel function at significantly lower cost.

Abstract: We discuss some recent progress in theoretical physics, focusing on string theory and supersymmetric quantum field theory, in which Macaulay2 has served as an indispensable tool and guide. The interaction between computational algebraic geometry and gauge theories is becoming increasingly intimate. We draw a list of some computational problems which string theorists deeply wish could be imminently addressed by Macaulay 2.

Abstract: The research in this talk is motivated by the following question: Given a normal toric ideal (which is CM by Hochster's Theorem), does it possess an initial monomial ideal that is also CM? I"ll survey what is known about this question and prove that every normal toric ideal of codimension two is minimally generated by a Gr\"obner basis with squarefree initial monomials. I will also present a polynomial time algorithm for checking whether a toric ideal of fixed codimension is normal.

Abstract: We have implemented an algorithm in Macaulay 2 for computing versal deformations of MCM modules on hypersurfaces. Its main applications involve the construction of moduli spaces of such modules. For example, we have used it to classify graded rank-1 maximal Cohen-Macaulay modules on the classical discriminant. In this talk, we describe the algorithm and its implementation, and we briefly discuss its use in such classification problems.

Abstract: I will discuss a new algebraic result on the relationship between the socle of a graded Buchsbaum module and its local cohomology modules. I will then present several combinatorial applications of this result to the f-vectors of simplicial manifolds, among them the proof of the K\"uhnel conjecture on the Euler characteristic and the proof of Kalai's conjecture providing a lower bound on the number of edges. This is joint work with Ed Swartz.

Abstract: The slice algorithm is a new algorithm for computing irreducible decompositions of monomial ideals. It proceeds by recursively carving up the staircase of the monomial ideal into simpler pieces. This basic structure can be improved in a number of ways, such as a way of replacing a monomial ideal with a simpler one without changing the output. The algorithm performs well in practice, and there are no barriers to implementing it in Macaulay 2.

Abstract: I'll discuss two open problems in discrete geometry. The first comes from approximation theory: given a simplicial complex embedded in the plane, find a formula for the space of splines (of fixed degree and smoothness). This turns into a question about the regularity of a certain vector bundle on $P^2$. I'll discuss the problem in detail, and describe progress on a related question, when the complex is polyhedral, rather than simplicial. The second problem comes from topology, in particular,the topology of the complement $X$ of a divisor $D \subseteq P^n$. When $D$ has normal crossings, Grothendieck's algebraic de Rham theorem relates $H^*(X)$ to the cohomology of the bundle of logarithmic one-forms (with pole along $D$). When $D$ is a union of hyperplanes, many things simplify. First, even if $D=V(F)$ does not have normal crossings, there is a purely combinatorial description of $H^*(X)$, due to Orlik and Solomon. In special cases, a theorem of Terao relates $H^*(X)$ to the syzygy module of the Jacobian ideal $J_F$ of $F$. Question: what properties of $J_F$ depend only on the combinatorics of $D$? Terao's conjecture is that the freeness of $syz(J_F)$ is such a property. I'll discuss this in terms of the regularity of vector bundles on $P^2$, and discuss some generalizations to arrangements of smooth rational curves in $P^2$.

Both parts of the talk will have lots of examples, and lots of open questions.

Abstract: We discuss some results on Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated primes. The main focus will be on some challenging examples for Macaulay 2.

Abstract: The circuit ideal of a vector configuration A is a subideal of the toric ideal of A, generated by the circuits (minimal integral dependencies) of A. In many instances the two ideals are equal which has practical significance since circuits can be computed relatively easily. We give a characterization of when equality occurs in terms of the existence of certain polytopes. If time permits, I will also present some results on the primary decomposition of the circuit ideal and its Groebner fan.

Joint work with Tristram Bogart and Anders Jensen

Abstract: A classical way of studying GKZ A-hypergeometric systems is via Newton gradings. In the case of a normal (but perhaps inhomogeneous) A, the associated Newton filtration yields Cohen--Macaulay rings and can be used to show the absence of jump parameters. An interesting feature of the Newton filtration is that it is in general not a weight filtration, so that if CC[x_1,...,x_n] surjects on the given toric ring S_A, it may not surject on the Newton-graded ring of S_A. In this talk we hint at an algorithm to compute a presentation for the associated graded ring that is based on lead-term inspections in toric ideals. We also present examples that show that in general the Newton-grading of toric rings may destroy the CM property.

Abstract: In this talk I will discuss three constructions of pure resolutions with arbitrary degree shifts. Two first constructions (due to Eisenbud, Floystad and myself) are equivariant, but work only over a field of characteristic zero. The third one (due to Eisenbud and Schreyer) works in all characteristics.

I will also discuss relations between these constructions and some conjectures on existence of pure resolutions with given ranks of modules.