Let $\ell$ be a prime and $q = p^{\nu}$ where $p$ is a prime different from $\ell$. We show that the $\ell$-completion of the $n$th stable homotopy group of spheres is a summand of the $\ell$-completion of the $(n, 0)$ motivic stable homotopy group of spheres over the finite field with $q$ elements $F_q$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems $\pi_{n, 0}(F_q)^{\wedge}_2$ for $0\leq n\leq 18$. Additionally, we compute $\pi_{19, 0}(F_q)^{\wedge}_2$ and $\pi_{20, 0}(F_q)^{\wedge}_2$ when $q \equiv 1 \bmod 4$ assuming Morel's connectivity theorem for $F_q$ holds.Comment: 4 figures, 2 tables. Published version now availabl

Abstract. We show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and (ii) determine displaceability for most of the moment fibers of a symplectic ellipsoid.

We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map η over fields of cohomological dimension at most 2 with characteristic different from 2 (this includes the p-adic fields Qp and the finite fields Fq of odd characteristic) and the field of rational numbers; the ring structure is also determined.

Background: Repeat expansion (RE) disorders affect ~1 in 3000 individuals and are clinically heterogeneous diseases caused by expansions of short tandem DNA repeats. Genetic testing is often locus-specific, resulting in under diagnosis of atypical clinical presentations, especially in paediatric patients without a prior positive family history. Whole genome sequencing (WGS) is emerging as a first-line test for rare genetic disorders, but until recently REs were thought to be undetectable by this approach. Methods: WGS pipelines for RE disorder detection were deployed by the 100,000 Genomes Project and Illumina Clinical Services Laboratory. Performance was retrospectively assessed across the 13 most common neurological RE loci using 793 samples with prior orthogonal testing (182 with expanded alleles and 611 with alleles within normal size) and prospectively interrogated in 13,331 patients with suspected genetic neurological disorders. Findings: WGS RE detection showed minimum 97.3% sensitivity and 99.6% specificity across all 13 disease-associated loci. Applying the pipeline to patients from the 100,000 Genomes Project identified pathogenic repeat expansions which were confirmed in 69 patients, including seven paediatric patients with no reported family history of RE disorders, with a 0.09% false positive rate. Interpretation: We show here for the first time that WGS enables the detection of causative repeat expansions with high sensitivity and specificity, and that it can be used to resolve previously undiagnosed neurological disorders. This includes children with no prior suspicion of a RE disorder. These findings are leading to diagnostic implementation of this analytical pipeline in the NHS Genomic Medicine Centres in England.

Leveraging the heightened levels of polymorphism in NB-ARC-related protein encoding genes in higher plants, a bioinformatic pipeline was created to identify regions in this gene family from sequenced plant genomes that exhibit fragment length or single nucleotide differences in different accessions of the same species. Testing this approach with the aquatic plant Spirodela polyrhiza demonstrated its superior performance in comparison with currently available genotyping technologies based on PCR amplification. Rapid and economical genotyping tools that can reliably distinguish species and intraspecific variations in plants can be powerful tools for biogeographical and ecological studies. Clones of the cosmopolitan duckweed species, Spirodela polyrhiza, are difficult to distinguish morphologically due to their highly abbreviated architecture and inherently low levels of sequence variation. The use of plastidic markers and generic Amplification Fragment Length Polymorphism approaches have met with limited success in resolving clones of S. polyrhiza from diverse geographical locales. Using whole genome sequencing data from nine S. polyrhiza clones as a training set, we created an informatic pipeline to identify and rank polymorphic regions from nuclear-encoded NB-ARC-related genes to design markers for PCR, Sanger sequencing (barcoding), and fragment length analysis. With seven primer sets, we found 21 unique fingerprints from a set of 23 S. polyrhiza clones. However, three of these clones share the same fingerprint and are indistinguishable by these markers. These primer sets can also be used as interspecific barcoding tools to rapidly resolve S. polyrhiza from the closely related S. intermedia species without the need for DNA sequencing. Our work demonstrates a general approach of using hyper-polymorphic loci within genomes as a resource to produce facile tools that can have high resolving power for genotyping applications.

We use elementary linear algebra to explicitly calculate a basis for, and the dimension of, the space of degree-three covariants for a binary form of arbitrary degree. We also give an explicit basis for the subspace of covariants complementary to the space of degree-three reducible covariants. The study of invariant functions was one of the main influences on the development of modern algebra. Consider the following simple example. The group G = ‫ޚ‬ acts on ‫ޒ‬ by addition: g • x = g + x. We define a G-invariant function to be a real-valued function f (x) on ‫ޒ‬ such that f • g = f for all g ∈ G. In other words, f (x) = f (g + x) for all g ∈ ‫,ޚ‬ x ∈ ‫.ޒ‬ The invariant functions are precisely the real-valued functions with period one. Hence, geometric information, such as periodicity, can be recovered by studying functions with certain algebraic properties. In Section 1, we introduce the concepts of an invariant and covariant function for a binary form Q(x, y). The problem of determining the complete set of these functions was widely worked on during the late nineteenth century. Gordan [1868] proved that the ring of invariants (and the ring of covariants) for a degree-n binary form is finitely generated. A milestone in the history of modern algebra was Hilbert's nonconstructive proof [1890] of the following fundamental theorem. Theorem [Hilbert 1890]. The ring of invariants (and the ring of covariants) for a degree-n homogeneous polynomial in m variables is finitely generated. Hilbert's theorem says that all invariants (resp. covariants) for a homogeneous polynomial can be expressed as polynomials in a certain finite set of invariants (resp. covariants). Hilbert [1893] subsequently gave a constructive proof of this theorem. The minimal size of the generating set is only known for a few values of (m, n). When m = 2, this number has been determined for n ≤ 8 [Bedratyuk 2009; Bedratyuk and Bedratyuk 2008].

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