Joint analysis of functional genomic data and genome-wide association studies of 18 human traits

Annotations of gene structures and regulatory elements can inform genome-wide association studies (GWAS). However, choosing the relevant annotations for interpreting an association study of a given trait remains challenging. We describe a statistical model that uses association statistics computed across the genome to identify classes of genomic element that are enriched or depleted for loci that influence a trait. The model naturally incorporates multiple types of annotations. We applied the model to GWAS of 18 human traits, including red blood cell traits, platelet traits, glucose levels, lipid levels, height, BMI, and Crohn's disease. For each trait, we evaluated the relevance of 450 different genomic annotations, including protein-coding genes, enhancers, and DNase-I hypersensitive sites in over a hundred tissues and cell lines. We show that the fraction of phenotype-associated SNPs that influence protein sequence ranges from around 2% (for platelet volume) up to around 20% (for LDL cholesterol); that repressed chromatin is significantly depleted for SNPs associated with several traits; and that cell type-specific DNase-I hypersensitive sites are enriched for SNPs associated with several traits (for example, the spleen in platelet volume). Finally, by re-weighting each GWAS using information from functional genomics, we increase the number of loci with high-confidence associations by around 5%.Comment: Fixed typos, included minor clarification

Werner's Measure on Self-Avoiding Loops and Welding

Werner's conformally invariant family of measures on self-avoiding loops on Riemann surfaces is determined by a single measure $\mu_0$ on self-avoiding loops in ${\mathbb C} \setminus\{0\}$ which surround $0$. Our first major objective is to show that the measure $\mu_0$ is infinitesimally invariant with respect to conformal vector fields (essentially the Virasoro algebra of conformal field theory). This makes essential use of classical variational formulas of Duren and Schiffer, which we recast in representation theoretic terms for efficient computation. We secondly show how these formulas can be used to calculate (in principle, and sometimes explicitly) quantities (such as moments for coefficients of univalent functions) associated to the conformal welding for a self-avoiding loop. This gives an alternate proof of the uniqueness of Werner's measure. We also attempt to use these variational formulas to derive a differential equation for the (Laplace transform of) the "diagonal distribution" for the conformal welding associated to a loop; this generalizes in a suggestive way to a deformation of Werner's measure conjectured to exist by Kontsevich and Suhov (a basic inspiration for this paper)

Homogeneous Poisson structures on symmetric spaces

We calculate, in a relatively explicit way, the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. A corollary is that the Hamiltonian system arising in the noncompact case is isomorphic to the generic Hamiltonian system arising in the compact case. In the group case these systems are also isomorphic to those arising from the Bruhat Poisson structure on the flag space, and hence, by results of Lu, can be completely factored.Comment: 28 pages, substantially revised exposition, corrected proof of Thm 2.1