From Koszul duality to Poincar\'e duality

We discuss the notion of Poincar\'e duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincar\'e duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.Comment: Dedicated to Raymond Stora. 27 page

Noncommutative resolutions of ADE fibered Calabi-Yau threefolds

In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by F. Cachazo, S. Katz and C. Vafa. The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by V. Ginzburg, which we call the "N=1 ADE quiver algebra"

The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants

We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU(3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.Comment: 46 pages which constitutes the published version, plus an Appendix detailing some long calculations. arXiv admin note: text overlap with arXiv:1110.454

Graded D-branes and skew-categories

I describe extended gradings of open topological field theories in two dimensions in terms of skew categories, proving a result which alows one to translate between the formalism of graded open 2d TFTs and equivariant cyclic categories. As an application of this formalism, I describe the open 2d TFT of graded D-branes in Landau-Ginzburg models in terms of an equivariant cyclic structure on the triangulated category of `graded matrix factorizations' introduced by Orlov. This leads to a specific conjecture for the Serre functor on the latter, which generalizes results known from the minimal and Calabi-Yau cases. I also give a description of the open 2d TFT of such models which manifestly displays the full grading induced by the vector-axial R-symmetry group.Comment: 37 page

De novo assembly of Euphorbia fischeriana root transcriptome identifies prostratin pathway related genes

Background Euphorbia fischeriana is an important medicinal plant found in Northeast China. The plant roots contain many medicinal compounds including 12-deoxyphorbol-13-acetate, commonly known as prostratin that is a phorbol ester from the tigliane diterpene series. Prostratin is a protein kinase C activator and is effective in the treatment of Human Immunodeficiency Virus (HIV) by acting as a latent HIV activator. Latent HIV is currently the biggest limitation for viral eradication. The aim of this study was to sequence, assemble and annotate the E. fischeriana transcriptome to better understand the potential biochemical pathways leading to the synthesis of prostratin and other related diterpene compounds. Results In this study we conducted a high throughput RNA-seq approach to sequence the root transcriptome of E. fischeriana. We assembled 18,180 transcripts, of these the majority encoded protein-coding genes and only 17 transcripts corresponded to known RNA genes. Interestingly, we identified 5,956 protein-coding transcripts with high similarity (>=75%) to Ricinus communis, a close relative to E. fischeriana. We also evaluated the conservation of E. fischeriana genes against EST datasets from the Euphorbeacea family, which included R. communis, Hevea brasiliensis and Euphorbia esula. We identified a core set of 1,145 gene clusters conserved in all four species and 1,487 E. fischeriana paralogous genes. Furthermore, we screened E. fischeriana transcripts against an in-house reference database for genes implicated in the biosynthesis of upstream precursors to prostratin. This identified 24 and 9 candidate transcripts involved in the terpenoid and diterpenoid biosyntehsis pathways, respectively. The majority of the candidate genes in these pathways presented relatively low expression levels except for 1-hydroxy-2-methyl-2-(E)-butenyl 4-diphosphate synthase (HDS) and isopentenyl diphosphate/dimethylallyl diphosphate synthase (IDS), which are required for multiple downstream pathways including synthesis of casbene, a proposed precursor to prostratin. Conclusion The resources generated in this study provide new insights into the upstream pathways to the synthesis of prostratin and will likely facilitate functional studies aiming to produce larger quantities of this compound for HIV research and/or treatment of patients

Genome-wide association study of male sexual orientation

Family and twin studies suggest that genes play a role in male sexual orientation. We conducted a genome-wide association study (GWAS) of male sexual orientation on a primarily European ancestry sample of 1,077 homosexual men and 1,231 heterosexual men using Affymetrix single nucleotide polymorphism (SNP) arrays. We identified several SNPs with p < 10 -5 , including regions of multiple supporting SNPs on chromosomes 13 (minimum p = 7.5 × 10 -7 ) and 14 (p = 4.7 × 10 -7 ). The genes nearest to these peaks have functions plausibly relevant to the development of sexual orientation. On chromosome 13, SLITRK6 is a neurodevelopmental gene mostly expressed in the diencephalon, which contains a region previously reported as differing in size in men by sexual orientation. On chromosome 14, TSHR genetic variants in intron 1 could conceivably help explain past findings relating familial atypical thyroid function and male homosexuality. Furthermore, skewed X chromosome inactivation has been found in the thyroid condition, Graves' disease, as well as in mothers of homosexual men. On pericentromeric chromosome 8 within our previously reported linkage peak, we found support (p = 4.1 × 10 -3 ) for a SNP association previously reported (rs77013977, p = 7.1 × 10 -8 ), with the combined analysis yielding p = 6.7 × 10 -9 , i.e., a genome-wide significant association