On Embedding Singular Poisson Spaces

This dissertation investigates the problem of locally embedding singular Poisson spaces. Specifically, it seeks to understand when a singular symplectic quotient V/G of a symplectic vector space V by a group G \subseteq Sp_2n(R) is realizable as a Poisson subspace of some Poisson manifold (R^n,{.,.}). The local embedding problem is recast in the language of schemes and reinterpreted as a problem of extending the Poisson bracket to infinitesimal neighborhoods of an embedded singular space. Such extensions of a Poisson bracket near a singular point p of V/G are then related to the cohomology and representation theory of the cotangent Lie algebra at p. Using this framework, it is shown that the real 4-dimensional quotient V/\ZZ_n (n odd) is not realizable as a Poisson subspace of any (R^{2n+6},{.,.}), even though the underlying variety algebraically embeds into R^{2n+6}. The proof of this nonembedding result hinges on a refinement of the Levi decomposition for Poisson manifolds to partially linearize any extension with respect to the Levi decomposition of the cotangent Lie algebra of V/G at the origin. Moreover, in the case n=3, this nonembedding result is complemented by a concrete realization of V/\ZZ_3 as a Poisson subspace of R^78

Genome-wide Analyses Identify KIF5A as a Novel ALS Gene

To identify novel genes associated with ALS, we undertook two lines of investigation. We carried out a genome-wide association study comparing 20,806 ALS cases and 59,804 controls. Independently, we performed a rare variant burden analysis comparing 1,138 index familial ALS cases and 19,494 controls. Through both approaches, we identified kinesin family member 5A (KIF5A) as a novel gene associated with ALS. Interestingly, mutations predominantly in the N-terminal motor domain of KIF5A are causative for two neurodegenerative diseases: hereditary spastic paraplegia (SPG10) and Charcot-Marie-Tooth type 2 (CMT2). In contrast, ALS-associated mutations are primarily located at the C-terminal cargo-binding tail domain and patients harboring loss-of-function mutations displayed an extended survival relative to typical ALS cases. Taken together, these results broaden the phenotype spectrum resulting from mutations in KIF5A and strengthen the role of cytoskeletal defects in the pathogenesis of ALS.Peer reviewe