109,162 research outputs found
Cut distance identifying graphon parameters over weak* limits
The theory of graphons comes with the so-called cut norm and the derived cut
distance. The cut norm is finer than the weak* topology. Dole\v{z}al and
Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin.
Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a
cut distance accumulation graphon can be pinpointed in the set of weak*
accumulation points as a minimizer of the entropy. Motivated by this, we study
graphon parameters with the property that their minimizers or maximizers
identify cut distance accumulation points over the set of weak* accumulation
points. We call such parameters cut distance identifying. Of particular
importance are cut distance identifying parameters coming from subgraph
densities, t(H,*). This concept is closely related to the emerging field of
graph norms, and the notions of the step Sidorenko property and the step
forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step
Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory
Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if
and only if it is step Sidorenko, and that if a graph is norming then it is
step forcing. Further, we study convexity properties of cut distance
identifying graphon parameters, and find a way to identify cut distance limits
using spectra of graphons. We also show that continuous cut distance
identifying graphon parameters have the "pumping property", and thus can be
used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming
graphs, and a new section 3.2 on index pumping in the regularity lemm
On the Crepant Resolution Conjecture in the Local Case
In this paper we analyze four examples of birational transformations between
local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial
resolution, and a flop. We study the effect of these transformations on
genus-zero Gromov-Witten invariants, proving the
Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in
each case. Our results suggest that this form of the Crepant Resolution
Conjecture may also hold for more general crepant birational transformations.
They also suggest that Ruan's original Crepant Resolution Conjecture should be
modified, by including appropriate "quantum corrections", and that there is no
straightforward generalization of either Ruan's original Conjecture or the
Cohomological Crepant Resolution Conjecture to the case of crepant partial
resolutions. Our methods are based on mirror symmetry for toric orbifolds.Comment: 27 pages. This is a substantially revised and shortened version of my
preprint "Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples";
all results contained here are also proved there. To appear in Communications
in Mathematical Physic
From wormhole to time machine: Comments on Hawking's Chronology Protection Conjecture
The recent interest in ``time machines'' has been largely fueled by the
apparent ease with which such systems may be formed in general relativity,
given relatively benign initial conditions such as the existence of traversable
wormholes or of infinite cosmic strings. This rather disturbing state of
affairs has led Hawking to formulate his Chronology Protection Conjecture,
whereby the formation of ``time machines'' is forbidden. This paper will use
several simple examples to argue that the universe appears to exhibit a
``defense in depth'' strategy in this regard. For appropriate parameter regimes
Casimir effects, wormhole disruption effects, and gravitational back reaction
effects all contribute to the fight against time travel. Particular attention
is paid to the role of the quantum gravity cutoff. For the class of model
problems considered it is shown that the gravitational back reaction becomes
large before the Planck scale quantum gravity cutoff is reached, thus
supporting Hawking's conjecture.Comment: 43 pages,ReV_TeX,major revision
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