12,569 research outputs found
Mirror Symmetry, N=1 Superpotentials and Tensionless Strings on Calabi-Yau Four-Folds
We study aspects of Calabi-Yau four-folds as compactification manifolds of
F-theory, using mirror symmetry of toric hypersurfaces. Correlation functions
of the topological field theory are determined directly in terms of a natural
ring structure of divisors and the period integrals, and subsequently used to
extract invariants of moduli spaces of rational curves subject to certain
conditions. We then turn to the discussion of physical properties of the
space-time theories, for a number of examples which are dual to
heterotic N=1 theories. Non-critical strings of various kinds, with low tension
for special values of the moduli, lead to interesting physical effects. We give
a complete classification of those divisors in toric manifolds that contribute
to the non-perturbative four-dimensional superpotential; the physical
singularities associated to it are related to the apppearance of tensionless
strings. In some cases non-perturbative effects generate an everywhere non-zero
quantum tension leading to a combination of a conventional field theory with
light strings hiding at a low energy scale related to supersymmetry breaking.Comment: 68 pages, harvmac.tex, references added, minor corrections, Comments
on phase transitions in sect. 5. added, two figures adde
Secondary homotopy groups
Secondary homotopy groups supplement the structure of classical homotopy
groups. They yield a track functor on the track category of pointed spaces
compatible with fiber sequences, suspensions and loop spaces. They also yield
algebraic models of homotopy types with homotopy groups concentrated in two
consecutive dimensions.Comment: We added further commets and references to make the paper more easily
readabl
Elimination of cusps in dimension 4 and its applications
We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M×S1, where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory
Fat-shattering dimension of -fold maxima
We provide improved estimates on the fat-shattering dimension of the -fold
maximum of real-valued function classes. The latter consists of all ways of
choosing functions, one from each of the classes, and computing their
pointwise maximum. The bound is stated in terms of the fat-shattering
dimensions of the component classes. For linear and affine function classes, we
provide a considerably sharper upper bound and a matching lower bound,
achieving, in particular, an optimal dependence on . Along the way, we point
out and correct a number of erroneous claims in the literature
Event-Based Modeling with High-Dimensional Imaging Biomarkers for Estimating Spatial Progression of Dementia
Event-based models (EBM) are a class of disease progression models that can
be used to estimate temporal ordering of neuropathological changes from
cross-sectional data. Current EBMs only handle scalar biomarkers, such as
regional volumes, as inputs. However, regional aggregates are a crude summary
of the underlying high-resolution images, potentially limiting the accuracy of
EBM. Therefore, we propose a novel method that exploits high-dimensional
voxel-wise imaging biomarkers: n-dimensional discriminative EBM (nDEBM). nDEBM
is based on an insight that mixture modeling, which is a key element of
conventional EBMs, can be replaced by a more scalable semi-supervised support
vector machine (SVM) approach. This SVM is used to estimate the degree of
abnormality of each region which is then used to obtain subject-specific
disease progression patterns. These patterns are in turn used for estimating
the mean ordering by fitting a generalized Mallows model. In order to validate
the biomarker ordering obtained using nDEBM, we also present a framework for
Simulation of Imaging Biomarkers' Temporal Evolution (SImBioTE) that mimics
neurodegeneration in brain regions. SImBioTE trains variational auto-encoders
(VAE) in different brain regions independently to simulate images at varying
stages of disease progression. We also validate nDEBM clinically using data
from the Alzheimer's Disease Neuroimaging Initiative (ADNI). In both
experiments, nDEBM using high-dimensional features gave better performance than
state-of-the-art EBM methods using regional volume biomarkers. This suggests
that nDEBM is a promising approach for disease progression modeling.Comment: IPMI 201
Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
We give examples over arbitrary fields of rings of invariants that are not
finitely generated. The group involved can be as small as three copies of the
additive group, as in Mukai's examples over the complex numbers. The failure of
finite generation comes from certain elliptic fibrations or abelian surface
fibrations having positive Mordell-Weil rank.
Our work suggests a generalization of the Morrison-Kawamata cone conjecture
from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in
dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic
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