150 research outputs found
Problems on homology manifolds
A list of problems prepared for the proceedings of the Workshop on Exotic
Homology Manifolds, Oberwolfach June 29-July 5 2003.Comment: This is the version published by Geometry & Topology Monographs on 22
April 200
Realcompactifications of frames
Bibliography: leaves 53-57.The first notion of realcompactness in frames was introduced by Reynolds [1979], and it was shown by Madden and Vermeer [1986] that this coincides with the Lindelof property. My thesis advisor suggested that more general realcompactifications of a frame L could be constructed by considering regular sub σ-frames which join generate L. This was motivated by the fact that the Alexandroff bases, which are used to construct the Wallman realcompactifications of a space X, are, as shown by Gilmour, simply the regular sub σ-frames of the frame of open sets of X. The key definition of realcompactness needed here is due to Schlitt [1990] and it is his construction of the universal realcompactification that we modify in order to obtain the Wallman realcompactifications
Regularity theory for fully nonlinear integro-differential equations
We consider nonlinear integro-differential equations, like the ones that
arise from stochastic control problems with purely jump L\`evy processes. We
obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior
regularity for general fully nonlinear integro-differential
equations. Our estimates remain uniform as the degree of the equation
approaches two, so they can be seen as a natural extension of the regularity
theory for elliptic partial differential equations.Comment: Minor typos corrected, and some extra comments adde
Specialization orders on atom spectra of Grothendieck categories
We introduce systematic methods to construct Grothendieck categories from
colored quivers and develop a theory of the specialization orders on the atom
spectra of Grothendieck categories. We show that any partially ordered set is
realized as the atom spectrum of some Grothendieck category, which is an analog
of Hochster's result in commutative ring theory. We also show that there exists
a Grothendieck category which has empty atom spectrum but has nonempty
injective spectrum.Comment: 39 page
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