Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in Algebraic Combinatoric

Alcove Walks and GKM Theory for Affine Flags

We develop the GKM theory for the torus-equivariant cohomology of the affine flag variety using the combinatorics of alcove walks. Dual to the usual GKM setup, which depicts the orbits of the small torus action on a graph, alcove walks take place in tessellations of Euclidean space. Walks in affine rank two occur on triangulations of the plane, providing a more direct connection to splines used for approximating surfaces. Alcove walks in GKM theory also need not be minimal length, and can instead be randomly generated, giving rise to more flexible implementation. This work reinterprets and recovers classical results in GKM theory on the affine flag variety, generalizing them to both non-minimal and folded alcove walks, all motivated by applications to splines.Comment: 30 pages, 8 figures best viewed in color; final version to appear in Springer INdAM Series, "Approximation Theory and Numerical Analysis Meet Algebra, Geometry, Topology

An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.Comment: 27 pages, 9 figures best viewed in color; updated discussion of several reference

An equivariant rim hook rule for quantum cohomology of Grassmannians

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus

Somatic MED12 Nonsense Mutation Escapes mRNA Decay and Reveals a Motif Required for Nuclear Entry

MED12 is a key component of the transcription-regulating Mediator complex. Specific missense and in-frame insertion/deletion mutations in exons 1 and 2 have been identified in uterine leiomyomas, breast tumors, and chronic lymphocytic leukemia. Here, we characterize the first MED12 5 end nonsense mutation (c.97G > T, p.E33X) identified in acute lymphoblastic leukemia and show that it escapes nonsense-mediated mRNA decay (NMD) by using an alternative translation initiation site. The resulting N-terminally truncated protein is unable to enter the nucleus due to the lack of identified nuclear localization signal (NLS). The absence of NLS prevents the mutant MED12 protein to be recognized by importin- and subsequent loading into the nuclear pore complex. Due to this mislocalization, all interactions between the MED12 mutant and other Mediator components are lost. Our findings provide new mechanistic insights into the MED12 functions and indicate that somatic nonsense mutations in early exons may avoid NMD. (C) 2017 Wiley Periodicals, Inc.Peer reviewe

WNT2 activation through proximal germline deletion predisposes to small intestinal neuroendocrine tumors and intestinal adenocarcinomas

Many hereditary cancer syndromes are associated with an increased risk of small and large intestinal adenocarcinomas. However, conditions bearing a high risk to both adenocarcinomas and neuroendocrine tumors are yet to be described.We studied a family with 16 individuals in four generations affected by a wide spectrum of intestinal tumors, including hyperplastic polyps, adenomas, small intestinal neuroendocrine tumors, and colorectal and small intestinal adenocarcinomas.To assess the genetic susceptibility and understand the novel phenotype, we utilized multiple molecular methods, including whole genome sequencing, RNA sequencing, single cell sequencing, RNA in situ hybridization and organoid culture.We detected a heterozygous deletion at the cystic fibrosis locus (7q31.2) perfectly segregating with the intestinal tumor predisposition in the family. The deletion removes a topologically associating domain border between CFTR and WNT2, aberrantly activating WNT2 in the intestinal epithelium. These consequences suggest that the deletion predisposes to small intestinal neuroendocrine tumors and small and large intestinal adenocarcinomas, and reveals the broad tumorigenic effects of aberrant WNT activation in the human intestine.Peer reviewe

Cost-effectiveness analysis of pemetrexed versus docetaxel in the second-line treatment of non-small cell lung cancer in Spain: results for the non-squamous histology population

BackgroundThe objective of this study was to conduct a cost-effectiveness evaluation of pemetrexed compared to docetaxel in the treatment of advanced or metastatic non-small cell lung cancer (NSCLC) for patients with predominantly non-squamous histology in the Spanish healthcare setting.MethodsA Markov model was designed consisting of stable, responsive, progressive disease and death states. Patients could also experience adverse events as long as they received chemotherapy. Clinical inputs were based on an analysis of a phase III clinical trial that identified a statistically significant improvement in overall survival for non-squamous patients treated with pemetrexed compared with docetaxel. Costs were collected from the Spanish healthcare perspective.ResultsOutcomes of the model included total costs, total quality-adjusted life years (QALYs), total life years gained (LYG) and total progression-free survival (PFS). Mean survival was 1.03 years for the pemetrexed arm and 0.89 years in the docetaxel arm; QALYs were 0.52 compared to 0.42. Per-patient lifetime costs were € 34677 and € 32343, respectively. Incremental cost-effectiveness ratios were € 23967 per QALY gained and € 17225 per LYG.ConclusionsPemetrexed as a second-line treatment option for patients with a predominantly non-squamous histology in NSCLC is a cost-effective alternative to docetaxel according to the € 30000/QALY threshold commonly accepted in Spain