37,775 research outputs found

    Recent progress on truncated Toeplitz operators

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    This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason's seminal paper in 2007.Comment: 46 page

    Regularity of Edge Ideals and Their Powers

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    We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of  reg I(G)\text{ reg } I(G) and the asymptotic linear function  reg I(G)q\text{ reg } I(G)^q, for q≥1,q \geq 1, in terms of combinatorial data of the given graph G.G.Comment: 31 pages, 15 figure

    New mathematical structures in renormalizable quantum field theories

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    Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.Comment: 26p,4figs., Invited Contribution to Annals of Physics, minor typos correcte

    Failure of Brown representability in derived categories

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    Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and contravariant homological functors C --> Ab which are not restrictions of representables.Comment: 22 pages, to appear in Topology. http://jdc.math.uwo.ca Lots of minor revisions. This version should closely match the published versio

    Regularity of squarefree monomial ideals

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    We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.
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