A Geometric Perspective on Sparse Filtrations

We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any convex metric. We also give an algorithm for finding the simplices in such a filtration and prove that the vertex removal can be implemented as a sequence of elementary edge collapses

Visualizing Sparse Filtrations

Over the last few years, there have been several approaches to building sparser complexes that still give good approximations to the persistent homology. In this video, we have illustrated a geometric perspective on sparse filtrations that leads to simpler proofs, more general theorems, and a more visual explanation. We hope that as these techniques become easier to understand, they will also become easier to use

The Long-Baseline Neutrino Experiment: Exploring Fundamental Symmetries of the Universe

The preponderance of matter over antimatter in the early Universe, the dynamics of the supernova bursts that produced the heavy elements necessary for life and whether protons eventually decay --- these mysteries at the forefront of particle physics and astrophysics are key to understanding the early evolution of our Universe, its current state and its eventual fate. The Long-Baseline Neutrino Experiment (LBNE) represents an extensively developed plan for a world-class experiment dedicated to addressing these questions. LBNE is conceived around three central components: (1) a new, high-intensity neutrino source generated from a megawatt-class proton accelerator at Fermi National Accelerator Laboratory, (2) a near neutrino detector just downstream of the source, and (3) a massive liquid argon time-projection chamber deployed as a far detector deep underground at the Sanford Underground Research Facility. This facility, located at the site of the former Homestake Mine in Lead, South Dakota, is approximately 1,300 km from the neutrino source at Fermilab -- a distance (baseline) that delivers optimal sensitivity to neutrino charge-parity symmetry violation and mass ordering effects. This ambitious yet cost-effective design incorporates scalability and flexibility and can accommodate a variety of upgrades and contributions. With its exceptional combination of experimental configuration, technical capabilities, and potential for transformative discoveries, LBNE promises to be a vital facility for the field of particle physics worldwide, providing physicists from around the globe with opportunities to collaborate in a twenty to thirty year program of exciting science. In this document we provide a comprehensive overview of LBNE's scientific objectives, its place in the landscape of neutrino physics worldwide, the technologies it will incorporate and the capabilities it will possess.Comment: Major update of previous version. This is the reference document for LBNE science program and current status. Chapters 1, 3, and 9 provide a comprehensive overview of LBNE's scientific objectives, its place in the landscape of neutrino physics worldwide, the technologies it will incorporate and the capabilities it will possess. 288 pages, 116 figure

Acoustic Oddball during NREM Sleep: A Combined EEG/fMRI Study

Background: A condition vital for the consolidation and maintenance of sleep is generally reduced responsiveness to external stimuli. Despite this, the sleeper maintains a level of stimulus processing that allows to respond to potentially dangerous environmental signals. The mechanisms that subserve these contradictory functions are only incompletely understood. Methodology/Principal Findings: Using combined EEG/fMRI we investigated the neural substrate of sleep protection by applying an acoustic oddball paradigm during light NREM sleep. Further, we studied the role of evoked K-complexes (KCs), an electroencephalographic hallmark of NREM sleep with a still unknown role for sleep protection. Our main results were: (1) Other than in wakefulness, rare tones did not induce a blood oxygenation level dependent (BOLD) signal increase in the auditory pathway but a strong negative BOLD response in motor areas and the amygdala. (2) Stratification of rare tones by the presence of evoked KCs detected activation of the auditory cortex, hippocampus, superior and middle frontal gyri and posterior cingulate only for rare tones followed by a KC. (3) The typical high frontocentral EEG deflections of KCs were not paralleled by a BOLD equivalent. Conclusions/Significance: We observed that rare tones lead to transient disengagement of motor and amygdala responses during light NREM sleep. We interpret this as a sleep protective mechanism to delimit motor responses and to reduce the sensitivity of the amygdala towards further incoming stimuli. Evoked KCs are suggested to originate from a brain state wit

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

Methods in Homology Inference

In this dissertation we introduce novel techniques to infer the shape of a geometric space from local data. The major contribution of this research is that the techniques aim to minimize the hypotheses of the sample and the space compared to existing results in the field. We first prove an equivalence between the two standard geometric sampling methods: adaptive and uniform. From this we create a collection of topological interleavings derived from this equivalence, which we use to compute the homology of an arbitrary Euclidean compact set only assuming a bound on the weak feature size of the domain and a sample of an arbitrary reference set for sampling. Next we provide an algorithm to check for k-coverage of sensor network from a collection of finite sensors using computable structures built upon from sensors\u27 local data. This algorithm generalizes the original Topological Coverage Criterion by De Silva and Ghrist to domains with non-smooth boundaries among other relaxed assumptions. We then prove that if one has coverage of the sensor network, there exists subsampling parameters such that one can infer the homology of the entire domain. We culminate the dissertation by proving an approximation of the Persistent Nerve Theorem assuming a relaxation of the standard topological cover assumption. Namely, given such a cover exists, there is a tight bottleneck distance between the persistence diagrams of the nerve filtration and the space filtration, generalizing the Persistent Nerve Theorem. This results provides information about the shape of a subdivisions of triangulations with no global geometric data provided, as well as broadens the applications of nerves to cases there are not nice sampling guarantees or there exists approximation errors during triangulation