A long-standing conjecture of Stanley states that every Cohen-Macaulay
simplicial complex is partitionable. We disprove the conjecture by constructing
an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our
construction also disproves the conjecture that the Stanley depth of a monomial
ideal is always at least its depth.Comment: Final version. 13 pages, 2 figure
The paper surveys several results on the topology of the space of arcs of an
algebraic variety and the Nash problem on the arc structure of singularities.Comment: 29 pages; v3 corrects some typos. To appear in the Proceedings of the
2015 Summer Institute on Algebraic Geometr
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
This paper represents a step in our program towards the proof of the
Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the
Pierce-Birkhoff conjecture for a ring Aisequivalenttoastatementaboutanarbitrarypairofpoints\alpha,\beta\in\sper\ Aandtheirseparatingideal;werefertothisstatementastheLocalPierce−Birkhoffconjectureat\alpha,\beta.Inthispaper,foreachpair(\alpha,\beta)withht()=\dim A,wedefineanaturalnumber,calledcomplexityof(\alpha,\beta).Complexity0correspondstothecasewhenoneofthepoints\alpha,\betaismonomial;thiscasewasalreadysettledinalldimensionsinaprecedingpaper.Hereweintroduceanewconjecture,calledtheStrongConnectednessconjecture,andprovethatthestrongconnectednessconjectureindimensionn−1impliestheconnectednessconjectureindimensionninthecasewhenht()islessthann−1.WeprovetheStrongConnectednessconjectureindimension2,whichgivestheConnectednessandthePierce−−Birkhoffconjecturesinanydimensioninthecasewhenht()lessthan2.Finally,weprovetheConnectedness(andhencealsothePierce−−Birkhoff)conjectureinthecasewhendimensionofAisequaltoht()=3,thepair(\alpha,\beta)isofcomplexity1andA$ is excellent with residue field the field of real numbers