37 research outputs found
Three flavors of extremal Betti tables
We discuss extremal Betti tables of resolutions in three different contexts.
We begin over the graded polynomial ring, where extremal Betti tables
correspond to pure resolutions. We then contrast this behavior with that of
extremal Betti tables over regular local rings and over a bigraded ring.Comment: 20 page
On the shape of a pure O-sequence
An order ideal is a finite poset X of (monic) monomials such that, whenever M
is in X and N divides M, then N is in X. If all, say t, maximal monomials of X
have the same degree, then X is pure (of type t). A pure O-sequence is the
vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree.
Equivalently, in the language of commutative algebra, pure O-sequences are the
h-vectors of monomial Artinian level algebras. Pure O-sequences had their
origin in one of Richard Stanley's early works in this area, and have since
played a significant role in at least three disciplines: the study of
simplicial complexes and their f-vectors, level algebras, and matroids. This
monograph is intended to be the first systematic study of the theory of pure
O-sequences. Our work, making an extensive use of algebraic and combinatorial
techniques, includes: (i) A characterization of the first half of a pure
O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel;
(ii) A study of (the failing of) the unimodality property; (iii) The problem of
enumerating pure O-sequences, including a proof that almost all O-sequences are
pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type
t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents
perhaps the strongest possible structural result short of an (impossible?)
characterization; (v) A pithy connection of the ICP with Stanley's matroid
h-vector conjecture; (vi) A specific study of pure O-sequences of type 2,
including a proof of the Weak Lefschetz Property in codimension 3 in
characteristic zero. As a corollary, pure O-sequences of codimension 3 and type
2 are unimodal (over any field); (vii) An analysis of the extent to which the
Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii)
Some observations about pure f-vectors, an important special case of pure
O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly
minor revisions with respect to last year's versio
On the Weak Lefschetz Property for Artinian Gorenstein algebras of codimension three
We study the problem of whether an arbitrary codimension three graded
artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this
problem to checking whether it holds for all compressed Gorenstein algebras of
odd socle degree. In the first open case, namely Hilbert function
(1,3,6,6,3,1), we give a complete answer in every characteristic by translating
the problem to one of studying geometric aspects of certain morphisms from
to , and Hesse configurations in .Comment: A few changes with respect to the previous version. 17 pages. To
appear in the J. of Algebr
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The minimal resolution conjecture on a general quartic surface in P3
Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links
Shapes of free resolutions over a local ring
We classify the possible shapes of minimal free resolutions over a regular
local ring. This illustrates the existence of free resolutions whose Betti
numbers behave in surprisingly pathological ways. We also give an asymptotic
characterization of the possible shapes of minimal free resolutions over
hypersurface rings. Our key new technique uses asymptotic arguments to study
formal Q-Betti sequences.Comment: 14 pages, 1 figure; v2: sections have been reorganized substantially
and exposition has been streamline
On the Weak Lefcchetz property for artinian Gorenstein algebras
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function , we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from to , and Hesse configurations in
Second trimester inflammatory and metabolic markers in women delivering preterm with and without preeclampsia.
ObjectiveInflammatory and metabolic pathways are implicated in preterm birth and preeclampsia. However, studies rarely compare second trimester inflammatory and metabolic markers between women who deliver preterm with and without preeclampsia.Study designA sample of 129 women (43 with preeclampsia) with preterm delivery was obtained from an existing population-based birth cohort. Banked second trimester serum samples were assayed for 267 inflammatory and metabolic markers. Backwards-stepwise logistic regression models were used to calculate odds ratios.ResultsHigher 5-α-pregnan-3β,20α-diol disulfate, and lower 1-linoleoylglycerophosphoethanolamine and octadecanedioate, predicted increased odds of preeclampsia.ConclusionsAmong women with preterm births, those who developed preeclampsia differed with respect metabolic markers. These findings point to potential etiologic underpinnings for preeclampsia as a precursor to preterm birth
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
Bayesian perspectives on mathematical practice
Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning