The hypertoric intersection cohomology ring

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.Comment: Significant revisions in Section 5, with several corrected proof

Degenerate flag varieties: moment graphs and Schr\"oder numbers

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schr\"oder number and the Poincar\'e polynomial is given by a natural statistics counting the number of diagonal steps in a Schr\"oder path. As an application we obtain a new combinatorial description of the large and small Schr\"oder numbers and their q-analogues.Comment: 25 page

Persistent Intersection Homology for the Analysis of Discrete Data

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional manifold. In such cases, persistent homology can be successfully employed to extract features, remove noise, and compare data sets. The underlying problems in some application domains, however, turn out to represent multiple manifolds with different dimensions. Algebraic topology typically analyzes such problems using intersection homology, an extension of homology that is capable of handling configurations with singularities. In this paper, we describe how the persistent variant of intersection homology can be used to assist data analysis in visualization. We point out potential pitfalls in approximating data sets with singularities and give strategies for resolving them.Comment: Topology-based Methods in Visualization 201

Notes on factorization algebras, factorization homology and applications

These notes are an expanded version of two series of lectures given at the winter school in mathematical physics at les Houches and at the Vietnamese Institute for Mathematical Sciences. They are an introduction to factorization algebras, factorization homology and some of their applications, notably for studying $E_n$-algebras. We give an account of homology theory for manifolds (and spaces), which give invariant of manifolds but also invariant of $E_n$-algebras. We particularly emphasize the point of view of factorization algebras (a structure originating from quantum field theory) which plays, with respect to homology theory for manifolds, the role of sheaves with respect to singular cohomology. We mention some applications to the study of mapping spaces and study several examples, including some over stratified spaces.Comment: 122 pages. A few examples adde

Assessment of age-related changes in pediatric gastrointestinal solubility

PurposeCompound solubility serves as a surrogate indicator of oral biopharmaceutical performance. Between infancy and adulthood, marked compositional changes in gastrointestinal (GI) fluids occur. This study serves to assess how developmental changes in GI fluid composition affects compound solubility.MethodsSolubility assessments were conducted in vitro using biorelevant media reflective of age-specific pediatric cohorts (i.e., neonates and infants). Previously published adult media (i.e., FaSSGF, FeSSGF, FaSSIF.v2, and FeSSIF.v2) were employed as references for pediatric media development. Investigations assessing age-specific changes in GI fluid parameters (i.e., pepsin, bile acids, pH, osmolality, etc.) were collected from the literature and served to define the composition of neonatal and infant media. Solubility assessments at 37°C were conducted for seven BCS Class II compounds within the developed pediatric and reference adult media.ResultsFor six of the seven compounds investigated, solubility fell outside an 80–125% range from adult values in at least one of the developed pediatric media. This result indicates a potential for age-related alterations in oral drug performance, especially for compounds whose absorption is delimited by solubility (i.e., BCS Class II).ConclusionDevelopmental changes in GI fluid composition can result in relevant discrepancies in luminal compound solubility between children and adults.<br/