32,764 research outputs found
Toward classifying unstable theories
The paper deals with two issues: the existence of universal models of a
theory T and related properties when cardinal arithmetic does not give this
existence offhand. In the first section we prove that simple theories (e.g.,
theories without the tree property, a class properly containing the stable
theories) behaves ``better'' than theories with the strict order property, by
criterion from [Sh:457]. In the second section we introduce properties SOP_n
such that the strict order property implies SOP_{n+1}, which implies SOP_n,
which in turn implies the tree property. Now SOP_4 already implies
non-existence of universal models in cases where earlier the strict order
property was needed, and SOP_3 implies maximality in the Keisler order, again
improving an earlier result which had used the strict order property
General Properties of Multiscalar RG Flows in
Fixed points of scalar field theories with quartic interactions in
dimensions are considered in full generality. For such
theories it is known that there exists a scalar function of the couplings
through which the leading-order beta-function can be expressed as a gradient.
It is here proved that the fixed-point value of is bounded from below by a
simple expression linear in the dimension of the vector order parameter, .
Saturation of the bound requires a marginal deformation, and is shown to arise
when fixed points with the same global symmetry coincide in coupling space.
Several general results about scalar CFTs are discussed, and a review of known
fixed points is given.Comment: 29 pages, 4 figures; see section 3 for a prize problem. v2: small
correction in appendix, typos fixed. v3: minor additions. v4: some
next-to-leading order results added, typos fixe
Saturating the random graph with an independent family of small range
Motivated by Keisler's order, a far-reaching program of understanding basic
model-theoretic structure through the lens of regular ultrapowers, we prove
that for a class of regular filters on , , the
fact that P(I)/\de has little freedom (as measured by the fact that any
maximal antichain is of size , or even countable) does not prevent
extending to an ultrafilter on which saturates ultrapowers of the
random graph. "Saturates" means that M^I/\de_1 is -saturated
whenever M is a model of the theory of the random graph. This was known to be
true for stable theories, and false for non-simple and non-low theories. This
result and the techniques introduced in the proof have catalyzed the authors'
subsequent work on Keisler's order for simple unstable theories. The
introduction, which includes a part written for model theorists and a part
written for set theorists, discusses our current program and related results.Comment: 14 page
Braids, mapping class groups, and categorical delooping
Dehn twists around simple closed curves in oriented surfaces satisfy the
braid relations. This gives rise to a group theoretic from the braid group to
the mapping class group. We prove here that this map is trivial in stable
homology with any trivial coefficients. In particular this proves an old
conjecture of J. Harer. The main tool is categorical delooping. In an appendix
we discuss geometrically defined homomorphisms from the braid to the mapping
class group.Comment: 19 pages, 9 figures, late
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