Fundamental domains and generators for lattice Veech groups

The moduli space QMg of non-zero genus g quadratic differentials has a natural action of G=GL+2(R) / ⟨±(1001) ⟩. The Veech group PSL(X,q) is the stabilizer of (X,q)∈QMg in G. We describe a new algorithm for finding elements of PSL(X,q) which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H / PSL(X,q) is greater than zero

A saturated map of common genetic variants associated with human height

Common single-nucleotide polymorphisms (SNPs) are predicted to collectively explain 40–50% of phenotypic variation in human height, but identifying the specific variants and associated regions requires huge sample sizes1. Here, using data from a genome-wide association study of 5.4 million individuals of diverse ancestries, we show that 12,111 independent SNPs that are significantly associated with height account for nearly all of the common SNP-based heritability. These SNPs are clustered within 7,209 non-overlapping genomic segments with a mean size of around 90 kb, covering about 21% of the genome. The density of independent associations varies across the genome and the regions of increased density are enriched for biologically relevant genes. In out-of-sample estimation and prediction, the 12,111 SNPs (or all SNPs in the HapMap 3 panel2) account for 40% (45%) of phenotypic variance in populations of European ancestry but only around 10–20% (14–24%) in populations of other ancestries. Effect sizes, associated regions and gene prioritization are similar across ancestries, indicating that reduced prediction accuracy is likely to be explained by linkage disequilibrium and differences in allele frequency within associated regions. Finally, we show that the relevant biological pathways are detectable with smaller sample sizes than are needed to implicate causal genes and variants. Overall, this study provides a comprehensive map of specific genomic regions that contain the vast majority of common height-associated variants. Although this map is saturated for populations of European ancestry, further research is needed to achieve equivalent saturation in other ancestries.publishedVersionPeer reviewe

Odbifold points and Jacobians with complex multiplication on Teichmüller curves in genus two

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 83-85).For each integer D >/= 5 with D =/- 0 or 1 mod 4, the Weierstrass curve WD is an algebraic curve and a finite volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces. The Weierstrass curves are the main examples of Teichmüller curves in genus two. The primary goal of this thesis is to determine the number and type of orbifold points on each component of WD. Our enumeration of the orbifold points, together with [Ba] and [Mc3], completes the determination of the homeomorphism type of WD and gives a formula for the genus of its components. We use our formula to give bounds on the genus of WD and determine the Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point on WD.by Ronen E. Mukamel.Ph.D

Large mosaic copy number variations confer autism risk

Although germline de novo copy number variants are known causes of autism spectrum disorder (ASD), the contribution of mosaic (early-developmental) copy number variants (mCNVs) has not been explored. Here, we assessed the contribution of mCNVs to ASD by ascertaining mCNVs in genotype array intensity data from 12,077 ASD probands and 5,500 unaffected siblings. We detected 46 mCNVs in probands and 19 mCNVs in siblings affecting 2.8–73.8% of cells. Probands carried a significant burden of large (>4 Mb) mCNVs, which were detected in 25 probands but only 1 sibling (OR=11.4, 95% CI=1.5–84.2, P=7.4×10(−4)). Event size positively correlated with severity of ASD symptoms (P=0.016). Surprisingly, we did not observe mosaic analogues of the short de novo CNVs recurrently observed in ASD (e.g. 16p11.2). We further experimentally validated two mCNVs in post-mortem brain tissue from 59 additional probands. These results indicate that mosaic CNVs contribute a previously unexplained component of ASD risk