Tony J. Puthenpurakal (IIT Bombay)

A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory

Let (A,m) be a Gorenstein local ring of dimension d ≥ 1. Let CM(A) be the stable category of maximal Cohen-Macaulay A-modules and let ICM(A) denote the set of isomorphism classes in CM(A). We define a function ξ: ICM(A) → Z which behaves well with respect to exact triangles in CM(A). We then apply this to (Gorenstein) liason theory. We prove that if dimA ≥ 2 and A is not regular then the even liason classes of mn;n ≥ 1 is an infinite set. We also prove that if A is an complete equi-characteristic simple singularity with A/m uncountable then for each m ≥ 1 the set Cm = {I | I is a codim 2 CM-ideal with e0(A/I) ≤ m} is contained in finitely many even liason classes L1, . . . , Lr (here r may depend on m).

Eleonore Faber (U. Toronto)

Non-commutative desingularizations and the global spectrum

This talk is about non-commutative analogs of resolutions of singularities. In short, non-commutative desingularizations of a commutative ring R are endomorphism rings of certain R-modules of finite global dimension. After some motivating examples, we discuss several approaches to non-commutative resolutions. Then we study their relevance for non-normal rings, in particular the question, under which conditions a non-commutative resolution exists. This leads us to consider the so-called global spectrum of a ring, that is, the set of all possible global dimensions of endomorphism rings of MCM-modules. This is joint work with H. Dao and C. Ingalls.

Branden Stone (Bard C.)

Super-stretched and graded countable Cohen-Macaulay type

In order to classify the graded rings of graded finite Cohen-Macaulay type, D. Eisenbud and J. Herzog showed that rings with graded finite type are stretched in the sense of J. Sally. In an attempt to classify the graded Cohen-Macaulay rings of graded countable type, the more general notion of super-stretched was defined and proved useful. In this talk we will define what it means to be super-stretched and discuss the implications of this condition for graded rings. Further, we will generalize the notion to the local case and show that the associated graded ring of a super-stretched local ring is Cohen-Macaulay.

Hara Charalambous (Aristotle U. of Thessaloniki)

Markov fibers and toric ideals

Let L⊂Zⁿ be a lattice, k a field, R=k[x₁,...,x_n] and I_L = ⟨xu−xv : u−v∈L⟩ the corresponding lattice ideal. We partition the set of monomials of R into fibers: xu, xv are in the same fiber if u−v ∈ L. We generalize the fiber graph construction, from the case where the intersection of L with Nⁿ is 0 and thus all fibers are finite, to all lattices. We use the fiber graphs to characterize minimal generating sets of I_L of minimal cardinality and to give invariants for these generating sets. As an application we characterize all binomial complete intersection lattice ideals.

Anna Bertiger (Waterloo U.)

The action of Sp_n on Flags(C²ⁿ)

I will present the seemingly geometric problem of understanding the orbits of the action of the symplectic group on the flag manifold and a very combinatorial and algebraic plan of attack for studying this problem. Along the way we will meet such combinatorial objects as involutions, permutations matrix Schubert varieties, and pipedreams and also investigate unions of matrix Schubert varieties. I intend for this talk to be friendly with all of the appropriate notions defined.

Kuei-Nuan Lin (Smith C.)

The blowup algebras of truncations of complete intersections

This is joint work with Claudia Polini. Let k be a field and R be a polynomial ring over k. We find the defining equations of blowup algebras of direct sum of power of homogeneous maximal ideal of R and show they are normal domain. Then we describe the blowup algebras of truncations of complete intersections in term of quotients of those normal domains.

Afshan Sadiq (Superior U., Lahore)

An Algorithm to Compute a Primary Decomposition of Modules in Polynomial Rings over the Integers