1,998 research outputs found
Some applications of the ultrapower theorem to the theory of compacta
The ultrapower theorem of Keisler-Shelah allows such model-theoretic notions
as elementary equivalence, elementary embedding and existential embedding to be
couched in the language of categories (limits, morphism diagrams). This in turn
allows analogs of these (and related) notions to be transported into unusual
settings, chiefly those of Banach spaces and of compacta. Our interest here is
the enrichment of the theory of compacta, especially the theory of continua,
brought about by the immigration of model-theoretic ideas and techniques
Modular Koszul duality
We prove an analogue of Koszul duality for category of a
reductive group in positive characteristic larger than 1 plus the
number of roots of . However there are no Koszul rings, and we do not prove
an analogue of the Kazhdan--Lusztig conjectures in this context. The main
technical result is the formality of the dg-algebra of extensions of parity
sheaves on the flag variety if the characteristic of the coefficients is at
least the number of roots of plus 2.Comment: 62 pages; image displays best in pd
Quillen homology for operads via Gr\"obner bases
The main goal of this paper is to present a way to compute Quillen homology
of operads. The key idea is to use the notion of a shuffle operad we introduced
earlier; this allows to compute, for a symmetric operad, the homology classes
and the shape of the differential in its minimal model, although does not give
an insight on the symmetric groups action on the homology. Our approach goes in
several steps. First, we regard our symmetric operad as a shuffle operad, which
allows to compute its Gr\"obner basis. Next, we define a combinatorial
resolution for the "monomial replacement" of each shuffle operad (provided by
the Gr\"obner bases theory). Finally, we explain how to "deform" the
differential to handle every operad with a Gr\"obner basis, and find explicit
representatives of Quillen homology classes for a large class of operads. We
also present various applications, including a new proof of Hoffbeck's PBW
criterion, a proof of Koszulness for a class of operads coming from commutative
algebras, and a homology computation for the operads of Batalin-Vilkovisky
algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint
arXiv:0912.4895. Final version, to appear in Documenta Mat
Some Applications of the Ultrapower Theorem to the Theory of Compacta
The ultrapower theorem of Keisler and Shelah allows such model-theoretic notions as elementary equivalence, elementary embedding and existential embedding to be couched in the language of categories (limits, morphism diagrams). This in turn allows analogs of these (and related) notions to be transported into unusual settings, chiefly those of Banach spaces and of compacta. Our interest here is the enrichment of the theory of compacta, especially the theory of continua, brought about by the importation of model-theoretic ideas and techniques
Brill-Noether theory of squarefree modules supported on a graph
We investigate the analogy between squarefree Cohen-Macaulay modules
supported on a graph and line bundles on a curve. We prove a Riemann-Roch
theorem, we study the Jacobian and gonality of a graph, and we prove Clifford's
theorem.Comment: Major revision, new author added, paper restructured, results
correcte
Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
We investigate the set a) of positive, trace preserving maps acting on
density matrices of size N, and a sequence of its nested subsets: the sets of
maps which are b) decomposable, c) completely positive, d) extended by identity
impose positive partial transpose and e) are superpositive. Working with the
Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds
for the volumes of all five sets. A sample consequence is the fact that, as N
increases, a generic positive map becomes not decomposable and, a fortiori, not
completely positive.
Due to the Jamiolkowski isomorphism, the results obtained for quantum maps
are closely connected to similar relations between the volume of the set of
quantum states and the volumes of its subsets (such as states with positive
partial transpose or separable states) or supersets. Our approach depends on
systematic use of duality to derive quantitative estimates, and on various
tools of classical convexity, high-dimensional probability and geometry of
Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision
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