Réunion d'hiver SMC 2016
Niagara Falls, 2 - 5 décembre 2016
This means, in particular, that that triangulated category is exact equivalent
to the derived category of modules of finite length over a (usually
noncommutative) Artin algebra. If the singularity is reduced, then
that Artin algebra is of finite global dimension and one can use this
to construct any graded torsionfree module over the curve singularity through
finitely many extensions of finitely many modules and their (co-)syzygies.
I will provide linear quotients orderings for powers of edge ideals of some graphs and present related results. In the second part of my talk, I will talk about the problem of describing edge ideals whose powers eventually have linear resolutions. In particular, I will discuss powers of edge ideals of some gap-free graphs.
The main question is whether one can uniquely determine a graph from its unlabeled vertex-deck. In 1964, Harary conjectured that any two graphs with at least four edges and the same edge-deck are isomorphic.
Long before that, in 1932 Whitney proved that if the line graphs of two simple graphs $G$ and $H$ are isomorphic, then $G$ and $H$ are also isomorphic except for the cases $K_3$ and $K_{1,3}$. Using this result, Hemminger proved that the edge reconstruction conjecture for graphs is equivalent to the vertex reconstruction conjecture for line graphs.
Trying to extend Whitney's theorem to hypergraphs, Berge introduced two hypergraphs $\mathcal{E}_p$ and $\mathcal{O}_p$ and proved that if two hypergraphs have isomorphic $(p-1)$-edge-decks then they are isomorphic only if they do not contain an $\mathcal{E}_p - \mathcal{O}_p$ pair. In 1987 Gardner proved the other direction under extra hypotheses.
In this talk we will discuss the ideal theory of hypergraphs with
isomorphic $(p-1)$-edge-decks, using the results mentioned above.
The Hilbert-Samuel multiplicity of an m-primary ideal is a well-known example of an interesting numerical invariant in Commutative Algebra. For rings of positive characteristic, there are many invariants that are analogous to the Hilbert-Samuel multiplicity, but instead are defined via sequences of ideals related by the Frobenius map. In general, such sequences exhibit non-polynomial growth, and existence of limits is a difficult question.
In this talk, we will discuss a proof of the existence of many limits in this context. Our method realizes these positive characteristic multiplicities as volumes of regions in euclidean space. In addition to existence, this association provides new insights into these numerical limits.
\begin{thebibliography}{9}
\bibitem {deg2} Abbasi, G.Q., Ahmad, S., Anwar, I. and Baig, W.A., 2012. f-Ideals of degree 2. Algebra Colloquium Vol. 19, No. spec01, pp. 921-926.
\bibitem{degd} Anwar, I., Mahmood, H., Binyamin, M.A. and Zafar, M.K., 2014. On the characterization of f-Ideals. Communications in Algebra, 42(9), pp.3736-3741.
\bibitem{tswu1} Guo, J., Wu, T. and Liu, Q., 2013. Perfect sets and $ f $-Ideals. arXiv preprint arXiv:1312.0324.
\bibitem{tswu2} Guo, J., Wu, T.,2015. On the $(n,d)^{th}$ $f$-ideals, J. Korean Math. Soc. 52(4), pp.685-697.
\bibitem{fgraph} Mahmood, H., Anwar, I. and Zafar, M.K., 2014. A construction of Cohen-Macaulay f-graphs. Journal of Algebra and Its Applications, 13(06), p.1450012.
\bibitem{fsimp} Mahmood, H., Anwar, I., Binyamin, M.A. and Yasmeen, S., 2016. On the connectedness of f-simplicial complexes. Journal of Algebra and Its Applications, p.1750017.
\end{thebibliography}
This is joint work with Federico Galetto (McMaster) and Tony Geramita (Queen's).