Mathematics

This documentary explores the difficult ethical issues arising from advances in biotechnology that make it possible to identify genetic defects during pregnancy. What moral responsibilities are imposed on consumers by genetic diagnoses and other advances in bioechnology? Featuring interviews with seven couples who were offered prenatal testing, along with healthcare and genetic specialists and people with disabilities, this video raises questions about personal responsibility and the changing roles of science, medicine, and technology. What was once a lottery system has become a known quantity. Central to the film's exploration is the recognition of changing perspectives on disabilities which specifically calls into question the option of selective abortion.

While five-factor models of personality are widespread, there is still not universal agreement on this as a structural framework. Part of the reason for the lingering debate is its dependence on factor analysis. In particular, derivation or refutation of the model via other statistical means is a worthwhile project. In this paper we use the methodology of spectral clustering to articulate the structure in the dataset of responses of 20,993 subjects on a 300-item item version of the IPIP NEO personality questionnaire, and we compare our results to those obtained from a factor analytic solution. We found support for five-and six-cluster solutions. The five-cluster solution was similar to a conventional five-factor solution, but the six-cluster and six-factor solutions differed significantly, and only the six-cluster solution was readily interpretable: it gave a model similar to the HEXACO model. We suggest that spectral clustering provides a robust alternative view of personality data.

We establish the following theorem of Bernstein type for the first Heisenberg group H 1 : Let S be a C 2 connected H-minimal surface which is a graph over some plane P , then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.

We show that C 1 hypersurfaces in the Heisenberg group are countably Nrectifiable. As a corollary, this shows that all C 1 H graphs over the xy-plane are countable N -rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni [4] for such surfaces.

Background: Multi-gene interactions likely play an important role in the development of complex phenotypes, and relationships between interacting genes pose a challenging statistical problem in microarray analysis, since the genes involved in these interactions may not exhibit marginal differential expression. As a result, it is necessary to develop tools that can identify sets of interacting genes that discriminate phenotypes without requiring that the classification boundary between phenotypes be convex. Results: We describe an extension and application of a new unsupervised statistical learning technique, known as the Partition Decoupling Method (PDM), to gene expression microarray data. This method may be used to classify samples based on multi-gene expression patterns and to identify pathways associated with phenotype, without relying upon the differential expression of individual genes. The PDM uses iterated spectral clustering and scrubbing steps, revealing at each iteration progressively finer structure in the geometry of the data. Because spectral clustering has the ability to discern clusters that are not linearly separable, it is able to articulate relationships between samples that would be missed by distance-and tree-based classifiers. After projecting the data onto the cluster centroids and computing the residuals ("scrubbing"), one can repeat the spectral clustering, revealing clusters that were not discernible in the first layer. These iterations, each of which provide a partition of the data that is decoupled from the others, are carried forward until the structure in the residuals is indistinguishable from noise, preventing over-fitting. We describe the PDM in detail and apply it to three publicly available cancer gene expression data sets. By applying the PDM on a pathway-by-pathway basis and identifying those pathways that permit unsupervised clustering of samples that match known sample characteristics, we show how the PDM may be used to find sets of mechanistically-related genes that may play a role in disease. An R package to carry out the PDM is available for download. Conclusions: We show that the PDM is a useful tool for the analysis of gene expression data from complex diseases, where phenotypes are not linearly separable and multi-gene effects are likely to play a role. Our results demonstrate that the PDM is able to distinguish cell types and treatments with higher accuracy than is obtained through other approaches, and that the Pathway-PDM application is a valuable technique for identifying diseaseassociated pathways.

We present a new method for articulating scale-dependent topological descriptions of the network structure inherent in many complex systems. The technique is based on "Partition Decoupled Null Models," a new class of null models that incorporate the interaction of clustered partitions into a random model and generalize the Gaussian ensemble. As an application we analyze a correlation matrix derived from four years of close prices of equities in the NYSE and NASDAQ. In this example we expose (1) a natural structure composed of two interacting partitions of the market that both agrees with and generalizes standard notions of scale (eg., sector and industry) and (2) structure in the first partition that is a topological manifestation of a well-known pattern of capital flow called "sector rotation." Our approach gives rise to a natural form of multiresolution analysis of the underlying time series that naturally decomposes the basic data in terms of the effects of the different scales at which it clusters. The equities market is a prototypical complex system and we expect that our approach will be of use in understanding a broad class of complex systems in which correlation structures are resident.

We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli{Garofalo{ Nhieu or Lu{Manfredi{Stroolini) in terms of the positive semideniteness of the horizontal second fundamental form of their graph.

In this paper, we prove results concerning the large scale geometry of connected, simply connected nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT 0 metric space or an Alexandrov metric space with curvature bounded below. The main technical aspect of this work is the proof of a limited metric differentiability of Lipschitz maps between connected graded nilpotent Lie groups equipped with left invariant Carnot-Carathéodory metrics and complete metric spaces.

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C 1 H-minimal graphs whose components of the unit horizontal Gauss map are in W 1,1 are ruled surfaces with C 2 seed curves. In a different direction, we investigate ways in which patches of C 1 H-minimal graphs can be glued together to form continuous piecewise C 1 Hminimal surfaces.

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M , we say E is Nrectifiable if it is the Lipschitz image of a positive measure subset of N . First, we discuss the implications of N -rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N -approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones isomorphic to N almost everywhere in a set E, that E is N -rectifiable. Third, we investigate the rectifiability properties of level sets of C 1 N functions, f : N → R, where N is a Carnot group. We show that for almost every t ∈ R and almost every noncharacteristic x ∈ f −1 (t), there exists a subgroup Tx of H and r > 0 so that f −1 (t) ∩ BH (x, r) is Txapproximable at x and an approximate tangent cone isomorphic to Tx at x.

We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information.