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"Boundaries Maps."
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The global north-south atlas : mapping global change
\"This innovative atlas deconstructs the contemporary image of the North-South divide and advocates the need for the international community to redraw the global map, as the contemporary world map with the 1980 Brandt Line drawing a stark divide between developed and underdeveloped countries no longer serves its purpose in the twenty-first century. Throughout the book a range of colourful maps and charts graphically demonstrate the ways in which the world has changed over the last two thousand years. The atlas firstly analyses the genesis, nature and validity of the Brandt Line, before going on to discuss its validity through centuries, especially in 1980 and after, and finally demonstrating the many definitions and philosophies of development that exist or may exist, which make it difficult to define a single notion of a Global North and South. The book concludes by proposing a new schemes of division between developed and developing countries. This book will serve as a perfect textbook for students studying global divisions within geography, politics, economics, international relations, and development departments, as well as being a useful guide for researchers, and for those working in NGOs and government institutions\"-- Provided by publisher.
Book
NATURAL MAPS FOR MEASURABLE COCYCLES OF COMPACT HYPERBOLIC MANIFOLDS
Let
$\\operatorname {\\mathrm {{\\rm G}}}(n)$
be equal to either
$\\operatorname {\\mathrm {{\\rm PO}}}(n,1),\\operatorname {\\mathrm {{\\rm PU}}}(n,1)$
or
$\\operatorname {\\mathrm {\\textrm {PSp}}}(n,1)$
and let
$\\Gamma \\leq \\operatorname {\\mathrm {{\\rm G}}}(n)$
be a uniform lattice. Denote by
$\\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}}$
the hyperbolic space associated to
$\\operatorname {\\mathrm {{\\rm G}}}(n)$
, where
$\\operatorname {\\mathrm {{\\rm K}}}$
is a division algebra over the reals of dimension d. Assume
$d(n-1) \\geq 2$
. In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability
$\\Gamma $
-space
$(X,\\mu _X)$
, we assume that a measurable cocycle
$\\sigma :\\Gamma \\times X \\rightarrow \\operatorname {\\mathrm {{\\rm G}}}(m)$
admits an essentially unique boundary map
$\\phi :\\partial _\\infty \\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}} \\times X \\rightarrow \\partial _\\infty \\operatorname {\\mathrm {\\mathbb {H}^m_{{\\rm K}}}$
whose slices
$\\phi _x:\\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}} \\rightarrow \\operatorname {\\mathrm {\\mathbb {H}^m_{{\\rm K}}}$
are atomless for almost every
$x \\in X$
. Then there exists a
$\\sigma $
-equivariant measurable map
$F: \\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}} \\times X \\rightarrow \\operatorname {\\mathrm {\\mathbb {H}^m_{{\\rm K}}}$
whose slices
$F_x:\\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}} \\rightarrow \\operatorname {\\mathrm {\\mathbb {H}^m_{{\\rm K}}}$
are differentiable for almost every
$x \\in X$
and such that
$\\operatorname {\\mathrm {\\textrm {Jac}}}_a F_x \\leq 1$
for every
$a \\in \\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}}$
and almost every
$x \\in X$
. This allows us to define the natural volume
$\\operatorname {\\mathrm {\\textrm {NV}}}(\\sigma )$
of the cocycle
$\\sigma $
. This number satisfies the inequality
$\\operatorname {\\mathrm {\\textrm {NV}}}(\\sigma ) \\leq \\operatorname {\\mathrm {\\textrm {Vol}}}(\\Gamma \\backslash \\operatorname {\\mathrm {\\mathbb {H}^n_{{\\rm K}}})$
. Additionally, the equality holds if and only if
$\\sigma $
is cohomologous to the cocycle induced by the standard lattice embedding
$i:\\Gamma \\rightarrow \\operatorname {\\mathrm {{\\rm G}}}(n) \\leq \\operatorname {\\mathrm {{\\rm G}}}(m)$
, modulo possibly a compact subgroup of
$\\operatorname {\\mathrm {{\\rm G}}}(m)$
when
$m>n$
. Given a continuous map
$f:M \\rightarrow N$
between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.
Journal Article
The bulk Hilbert space of double scaled SYK
A
bstract
The emergence of the bulk Hilbert space is a mysterious concept in holography. In [
1
], the SYK model was solved in the double scaling limit by summing chord diagrams. Here, we explicitly construct the bulk Hilbert space of double scaled SYK by slicing open these chord diagrams; this Hilbert space resembles that of a lattice field theory where the length of the lattice is dynamical and determined by the chord number. Under a calculable bulk-to-boundary map, states of fixed chord number map to particular entangled 2-sided states with a corresponding size. This bulk reconstruction is well-defined even when quantum gravity effects are important. Acting on the double scaled Hilbert space is a Type II
1
algebra of observables, which includes the Hamiltonian and matter operators. In the appropriate quantum Schwarzian limit, we also identify the JT gravitational algebra including the physical SL(2
,
ℝ) symmetry generators, and obtain explicit representations of the algebra using chord diagram techniques.
Journal Article
Cartographic humanism : the making of early modern Europe
What is \"Europe,\" and when did it come to be? In the Renaissance, the term \"Europe\" circulated widely. But as Katharina N. Piechocki argues in this compelling book, the continent itself was only in the making in the fifteenth and sixteenth centuries. Cartographic Humanism sheds new light on how humanists negotiated and defined Europe's boundaries at a momentous shift in the continent's formation: when a new imagining of Europe was driven by the rise of cartography. As Piechocki shows, this tool of geography, philosophy, and philology was used not only to represent but, more importantly, also to shape and promote an image of Europe quite unparalleled in previous centuries. Engaging with poets, historians, and mapmakers, Piechocki resists an easy categorization of the continent, scrutinizing Europe as an unexamined category that demands a much more careful and nuanced investigation than scholars of early modernity have hitherto undertaken. Unprecedented in its geographic scope, Cartographic Humanism is the first book to chart new itineraries across Europe as it brings France, Germany, Italy, Poland, and Portugal into a lively, interdisciplinary dialogue.
Book
A bulk manifestation of Krylov complexity
A
bstract
There are various definitions of the concept of complexity in Quantum Field Theory as well as for finite quantum systems. For several of them there are conjectured holographic bulk duals. In this work we establish an entry in the AdS/CFT dictionary for one such class of complexity, namely Krylov or K-complexity. For this purpose we work in the double-scaled SYK model which is dual in a certain limit to JT gravity, a theory of gravity in AdS
2
. In particular, states on the boundary have a clear geometrical definition in the bulk. We use this result to show that Krylov complexity of the infinite-temperature thermofield double state on the boundary of AdS
2
has a precise bulk description in JT gravity, namely the length of the two-sided wormhole. We do this by showing that the Krylov basis elements, which are eigenstates of the Krylov complexity operator, are mapped to length eigenstates in the bulk theory by subjecting K-complexity to the bulk-boundary map identifying the bulk/boundary Hilbert spaces. Our result makes extensive use of chord diagram techniques and identifies the Krylov basis of the boundary quantum system with fixed chord number states building the bulk gravitational Hilbert space.
Journal Article
Cosmology from random entanglement
A
bstract
We construct entangled microstates of a pair of holographic CFTs whose dual semiclassical description includes big bang-big crunch AdS cosmologies in spaces without boundaries. The cosmology is supported by inhomogeneous heavy matter and it partially purifies the bulk entanglement of two disconnected auxiliary AdS spacetimes. We show that the island formula for the fine grained entropy of one of the CFTs follows from a standard gravitational replica trick calculation. In generic settings, the cosmology is contained in the entanglement wedge of one of the two CFTs. We then investigate properties of the cosmology-to-boundary encoding map, and in particular, its non-isometric character. Restricting our attention to a specific class of states on the cosmology, we provide an explicit, and state-dependent, boundary representation of operators acting on the cosmology. Finally, under genericity assumptions, we argue for a non-isometric to approximately-isometric transition of the cosmology-to-boundary map for “simple” states on the cosmology as a function of the bulk entanglement, with tensor network toy models of our setup as a guide.
Journal Article
Rigidity of Kleinian groups via self-joinings
Let Γ
Journal Article
Beyond toy models: distilling tensor networks in full AdS/CFT
A
bstract
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-
N
CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
Journal Article
Unlocking Large-Scale Crop Field Delineation in Smallholder Farming Systems with Transfer Learning and Weak Supervision
Crop field boundaries aid in mapping crop types, predicting yields, and delivering field-scale analytics to farmers. Recent years have seen the successful application of deep learning to delineating field boundaries in industrial agricultural systems, but field boundary datasets remain missing in smallholder systems due to (1) small fields that require high resolution satellite imagery to delineate and (2) a lack of ground labels for model training and validation. In this work, we use newly-accessible high-resolution satellite imagery and combine transfer learning with weak supervision to address these challenges in India. Our best model uses 1.5 m resolution Airbus SPOT imagery as input, pre-trains a state-of-the-art neural network on France field boundaries, and fine-tunes on India labels to achieve a median Intersection over Union (mIoU) of 0.85 in India. When we decouple field delineation from cropland classification, a model trained in France and applied as-is to India Airbus SPOT imagery delineates fields with a mIoU of 0.74. If using 4.8 m resolution PlanetScope imagery instead, high average performance (mIoU > 0.8) is only achievable for fields larger than 1 hectare. Experiments also show that pre-training in France reduces the number of India field labels needed to achieve a given performance level by as much as 10× when datasets are small. These findings suggest our method is a scalable approach for delineating crop fields in regions of the world that currently lack field boundary datasets. We publicly release 10,000 Indian field boundary labels and our delineation model to facilitate the creation of field boundary maps and new methods by the community.
Journal Article