Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections

We prove two conjectures from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the expected number of critical points of random holomorphic sections of a positive line bundle. We show that, on average, the critical points of minimal Morse index are the most plentiful for holomorphic sections of {\mathcal O}(N) \to \CP^m and, in an asymptotic sense, for those of line bundles over general K\"ahler manifolds. We calculate the expected number of these critical points for the respective cases and use these to obtain growth rates and asymptotic bounds for the total expected number of critical points in these cases. This line of research was motivated by landscape problems in string theory and spin glasses.Comment: 14 pages, corrected typo

Single-cell analysis of the 3D topologies of genomic loci using genome architecture mapping

Although each cell within an organism contains a nearly identical genome sequence, the three-dimensional (3D) packing of the genome varies among individual cells, influencing cell-type-specific gene expression. Genome Architecture Mapping (GAM) is the first genome-wide experimental method for capturing 3D proximities between any number of genomic loci without ligation. GAM overcomes several limitations of 3C-based methods by sequencing DNA from a large collection of thin sections sliced from individual nuclei. The GAM technique measures locus co-segregation, extracts radial positions, infers chromatin compaction, requires small numbers of cells, does not depend on ligation, and provides rich single-cell information. However, previous analyses of GAM data focused exclusively on population averages, neglecting the variation in 3D topology among individual cells. We present the first single-cell analysis of GAM data, demonstrating that the slices from individual cells reveal intercellular heterogeneity in chromosome conformation. By simultaneously clustering both slices and genomic loci, we identify topological variation among single cells, including differential compaction of cell cycle genes. We also develop a geometric model of the nucleus, allowing prediction of the 3D positions of each slice. Using GAM data from mouse embryonic stem cells, we make new discoveries about the structure of the major mammalian histone gene locus, which is incorporated into the Histone Locus Body (HLB), including structural fluctuations and putative causal molecular mechanisms. Our methods are packaged as SluiceBox, a toolkit for mining GAM data. Our approach represents a new method of investigating variation in 3D genome topology among individual cells across space and time

Inverse-kinematics one-neutron pickup with fast rare-isotope beams

New measurements and reaction model calculations are reported for single neutron pickup reactions onto a fast \nuc{22}{Mg} secondary beam at 84 MeV per nucleon. Measurements were made on both carbon and beryllium targets, having very different structures, allowing a first investigation of the likely nature of the pickup reaction mechanism. The measurements involve thick reaction targets and $\gamma$-ray spectroscopy of the projectile-like reaction residue for final-state resolution, that permit experiments with low incident beam rates compared to traditional low-energy transfer reactions. From measured longitudinal momentum distributions we show that the \nuc{12}{C} (\nuc{22}{Mg},\nuc{23}{Mg}+\gamma)X reaction largely proceeds as a direct two-body reaction, the neutron transfer producing bound \nuc{11}{C} target residues. The corresponding reaction on the \nuc{9}{Be} target seems to largely leave the \nuc{8}{Be} residual nucleus unbound at excitation energies high in the continuum. We discuss the possible use of such fast-beam one-neutron pickup reactions to track single-particle strength in exotic nuclei, and also their expected sensitivity to neutron high-$\ell$ (intruder) states which are often direct indicators of shell evolution and the disappearance of magic numbers in the exotic regime.Comment: 8 pages, 5 figure

Octupole strength in the neutron-rich calcium isotopes

Low-lying excited states of the neutron-rich calcium isotopes $^{48-52}$Ca have been studied via $\gamma$-ray spectroscopy following inverse-kinematics proton scattering on a liquid hydrogen target using the GRETINA $\gamma$-ray tracking array. The energies and strengths of the octupole states in these isotopes are remarkably constant, indicating that these states are dominated by proton excitations.Comment: 15 pages, 3 figure

Intermediate-energy inverse-kinematics one-proton pickup reactions on neutron-deficient f p-shell nuclei

Background: Thick-target-induced nucleon-adding transfer reactions onto energetic rare-isotope beams are an emerging spectroscopic tool. Their sensitivity to single-particle structure complements one-nucleon removal reaction capabilities in the quest to reveal the evolution of nuclear shell structure in very exotic nuclei. Purpose: Our purpose is to add intermediate-energy, carbon-target-induced one-proton pickup reactions to the arsenal of γ-ray-tagged direct reactions applicable in the regime of low beam intensities and to apply these for the first time to fp-shell nuclei. Methods: Inclusive and partial cross sections were measured for the 12C(48Cr,49Mn+γ)X and 12C(50Fe,51Co+γ)X proton pickup reactions at 56.7 and 61.2 MeV/nucleon, respectively, using coincident particle-γ spectroscopy at the National Superconducting Cyclotron Laboratory. The results are compared to reaction theory calculations using fp-shell-model nuclear structure input. For comparison with our previous work, the same reactions were measured on 9Be targets. Results: The measured partial cross sections confirm the specific population pattern predicted by theory, with pickup into high-ℓ orbitals being strongly favored, driven by linear and angular momentum matching. Conclusion: Carbon-target-induced pickup reactions are well suited, in the regime of modest beam intensity, to study the evolution of nuclear structure, with specific sensitivities that are well described by theory

Spectroscopy of 54^{54}Ti and the systematic behavior of low energy octupole states in Ca and Ti isotopes

Excited states of the $N=32$ nucleus $^{54}$Ti have been studied, via both inverse-kinematics proton scattering and one-neutron knockout from $^{55}$Ti by a liquid hydrogen target, using the GRETINA $\gamma$-ray tracking array. Inelastic proton-scattering cross sections and deformation lengths have been determined. A low-lying octupole state has been tentatively identified in $^{54}$Ti for the first time. A comparison of $(p,p')$ results on low-energy octupole states in the neutron-rich Ca and Ti isotopes with the results of Random Phase Approximation calculations demonstrates that the observed systematic behavior of these states is unexpected.Comment: 7 pages, 8 figure