Hilbert Domains, Conics, and Rigidity

A class of compact projective manifolds can be viewed as a convex set in projective space modulo a discrete group of isometries. This thesis explores the circumstances under which this convex set is a symmetric convex cone. The irreducible symmetric convex cones are analogous to symmetric spaces in Riemannian geometry and consist of hyperbolic space and positive definite Hermitian matrices. Having a properly embedded conic in the boundary of the convex set is equivalent to the existence of a subspace isometric to the hyperbolic plane. When enough of these conics exist, I will show that the convex set is a symmetric convex cone. This demonstrates how the shape of the boundary of the convex set determines its isometry class. Further, if enough twice differentiable curves are found in the boundary of the convex set, I will show that it must be hyperbolic space. This result also has applications to affine spheres.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163243/1/spinella_1.pd

Shattering Thresholds for Random Systems of Sets, Words, and Permutations

This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (specializing to t=3), all length 3 permutation patterns in specified positions. We uncover a very sharp zero-one probability threshold for the emergence of such shattering; Talagrand's isoperimetric inequality in product spaces is used as a key tool.Comment: 25 page

A recurrent 15q13.3 microdeletion syndrome associated with mental retardation and seizures.

We report a recurrent microdeletion syndrome causing mental retardation, epilepsy and variable facial and digital dysmorphisms. We describe nine affected individuals, including six probands: two with de novo deletions, two who inherited the deletion from an affected parent and two with unknown inheritance. The proximal breakpoint of the largest deletion is contiguous with breakpoint 3 (BP3) of the Prader-Willi and Angelman syndrome region, extending 3.95 Mb distally to BP5. A smaller 1.5-Mb deletion has a proximal breakpoint within the larger deletion (BP4) and shares the same distal BP5. This recurrent 1.5-Mb deletion contains six genes, including a candidate gene for epilepsy (CHRNA7) that is probably responsible for the observed seizure phenotype. The BP4-BP5 region undergoes frequent inversion, suggesting a possible link between this inversion polymorphism and recurrent deletion. The frequency of these microdeletions in mental retardation cases is approximately 0.3% (6/2,082 tested), a prevalence comparable to that of Williams, Angelman and Prader-Willi syndromes