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"Berman, Robert"
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Projects and their consequences
Projects and Their Consequences presents fifteen key projects from leading architectural thinkers Reiser + Umemoto. Projects and Their Consequences traces thirty years of innovative, multidisciplinary investigations of form, structure, technique, and planning. Projects include large-scale studies of infrastructure for the East River Corridor and Hudson Yards areas in Manhattan and the Alishan Railway in Taiwan, as well as schemes for cultural institutions including the New Museum, Children's Museum of Pittsburgh, and University of Applied Arts Vienna. Also included are thought-provoking \"textual projects\": narrative works that blur the boundaries of art and architecture. Projects and Their Consequences balances incisive interviews and essays with more than 400 strikingly original drawings, collages, and paintings. Large-format and beautifully designed, it is a necessary volume for architects and those interested in the intersection of architecture, art, and culture.
BOOK
Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
Journal Article
The nanny diaries
Annie is a young girl from a working-class neighborhood who suddenly finds herself working as a nanny for a wealthy family in Manhattan's Upper East Side. Between catering to the every whim of her employers and their precocious son and falling in love with their gorgeous neighbor, Annie tries to figure out what she wants to do with her life.
DVD
The probabilistic vs the quantization approach to Kähler–Einstein geometry
In the probabilistic construction of Kähler–Einstein metrics on a complex projective algebraic manifold
X
—involving random point processes on
X
—a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that
X
admits a Kähler–Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler–Einstein geometry.
Journal Article
Sharp bounds on the height of K-semistable Fano varieties I, the toric case
Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\\mathcal {X}$ of relative dimension $n$ is maximal when $\\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\\leq 6$ (the extension to higher dimensions is conditioned on a conjectural ‘gap hypothesis’ for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\\mathcal {X}$. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.
Journal Article
The spherical ensemble and quasi-Monte-Carlo designs
The spherical ensemble is a well-known ensemble of
N
repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to
N
nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter
s
≤
2
.
The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble.
Journal Article
Ensemble Equivalence for Mean Field Models and Plurisubharmonicity
We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the point vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher-dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding “microscopic” concavity result at finite N, shown by leveraging an unexpected link to Kähler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions, strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application, it is shown that the critical inverse negative temperatures—in the macroscopic as well as the microscopic setting—coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way, we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.
Journal Article
Emergent Sasaki-Einstein geometry and AdS/CFT
A central problem in any quantum theory of gravity is to explain the emergence of the classical spacetime geometry in some limit of a more fundamental, microscopic description of nature. The gauge/gravity-correspondence provides a framework in which this problem can, in principle, be addressed. This is a holographic correspondence which relates a supergravity theory in five-dimensional Anti-deSitter space to a strongly coupled superconformal gauge theory on its 4-dimensional flat Minkowski boundary. In particular, the classical geometry should therefore emerge from some quantum state of the dual gauge theory. Here we confirm this by showing how the classical metric emerges from a canonical state in the dual gauge theory. In particular, we obtain approximations to the Sasaki-Einstein metric underlying the supergravity geometry, in terms of an explicit integral formula involving the canonical quantum state in question. In the special case of toric quiver gauge theories we show that our results can be computationally simplified through a process of tropicalization.
It is an outstanding question in quantum gravity how to describe the emergence of classical spacetime geometry from a quantum state. Here, the authors propose a construction in the context of the gauge/gravity correspondence, producing the classical geometry from a quantum state at the boundary of spacetime.
Journal Article
Behavioral Phenotyping of Juvenile Long-Evans and Sprague-Dawley Rats: Implications for Preclinical Models of Autism Spectrum Disorders
The laboratory rat is emerging as an attractive preclinical animal model of autism spectrum disorder (ASD), allowing investigators to explore genetic, environmental and pharmacological manipulations in a species exhibiting complex, reciprocal social behavior. The present study was carried out to compare two commonly used strains of laboratory rats, Sprague-Dawley (SD) and Long-Evans (LE), between the ages of postnatal day (PND) 26-56 using high-throughput behavioral phenotyping tools commonly used in mouse models of ASD that we have adapted for use in rats. We detected few differences between young SD and LE strains on standard assays of exploration, sensorimotor gating, anxiety, repetitive behaviors, and learning. Both SD and LE strains also demonstrated sociability in the 3-chamber social approach test as indexed by spending more time in the social chamber with a constrained age/strain/sex matched novel partner than in an identical chamber without a partner. Pronounced differences between the two strains were, however, detected when the rats were allowed to freely interact with a novel partner in the social dyad paradigm. The SD rats in this particular testing paradigm engaged in play more frequently and for longer durations than the LE rats at both juvenile and young adult developmental time points. Results from this study that are particularly relevant for developing preclinical ASD models in rats are threefold: (i) commonly utilized strains exhibit unique patterns of social interactions, including strain-specific play behaviors, (ii) the testing environment may profoundly influence the expression of strain-specific social behavior and (iii) simple, automated measures of sociability may not capture the complexities of rat social interactions.
Journal Article
Growth of balls of holomorphic sections and energy at equilibrium
Let
L
be a big line bundle on a compact complex manifold
X
. Given a non-pluripolar compact subset
K
of
X
and a continuous Hermitian metric
e
−
φ
on
L
, we define the energy at equilibrium of (
K
,
φ
) as the Monge-Ampère energy of the extremal psh weight associated to (
K
,
φ
). We prove the differentiability of the energy at equilibrium with respect to
φ
, and we show that this energy describes the asymptotic behaviour as
k
→∞ of the volume of the sup-norm unit ball induced by (
K
,
k
φ
) on the space of global holomorphic sections
H
0
(
X
,
kL
). As a consequence of these results, we recover and extend Rumely’s Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.
Journal Article