G-complete reducibility in non-connected groups

In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G^0 is G^0 -cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.Comment: 14 page

Cocharacter-closure and spherical buildings

Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.Comment: 16 pages; v 2 17 pages, exposition improved; to appear in the Robert Steinberg Memorial Issue of the Pacific Journal of Mathematic

Cocharacter-Closure and the Rational Hilbert-Mumford Theorem

For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure

Cocharacter-closure and spherical buildings

Let k be a field, let G be a reductive k-group and V an affine k-variety on which G acts. In this note we continue our study of the notion of cocharacter-closed G(k)-orbits in V . In earlier work we used a rationality condition on the point stabilizer of a G-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding G(k)-orbit in V . In the present paper we employ building-theoretic techniques to derive analogous results

G-complete reducibility in non-connected groups

The authors acknowledge the financial support of the DFG-priority programme SPP1388 “Representation Theory” and Marsden Grants UOC0501, UOC1009 and UOA1021. Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. The second author acknowledges additional support from ERC Advanced Grant No. 291512. The authors are grateful to the referee for helpful suggestions, including a strengthening of Proposition 3.4.Peer reviewedPostprin

Cocharacter-closure and the rational Hilbert-Mumford Theorem

For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V, we prove a rational version of the celebrated Hilbert–Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure

Planet Formation Imager (PFI): Science vision and key requirements

The Planet Formation Imager (PFI) project aims to provide a strong scientific vision for ground-based optical astronomy beyond the upcoming generation of Extremely Large Telescopes. We make the case that a breakthrough in angular resolution imaging capabilities is required in order to unravel the processes involved in planet formation. PFI will be optimised to provide a complete census of the protoplanet population at all stellocentric radii and over the age range from 0.1 to ∼100 Myr. Within this age period, planetary systems undergo dramatic changes and the final architecture of planetary systems is determined. Our goal is to study the planetary birth on the natural spatial scale where the material is assembled, which is the Hill Sphere of the forming planet, and to characterise the protoplanetary cores by measuring their masses and physical properties. Our science working group has investigated the observational characteristics of these young protoplanets as well as the migration mechanisms that might alter the system architecture. We simulated the imprints that the planets leave in the disk and study how PFI could revolutionise areas ranging from exoplanet to extragalactic science. In this contribution we outline the key science drivers of PFI and discuss the requirements that will guide the technology choices, the site selection, and potential science/technology tradeoffs

An Aged Canid with Behavioral Deficits Exhibits Blood and Cerebrospinal Fluid Amyloid Beta Oligomers

Many of the molecular and pathological features associated with human Alzheimer disease (AD) are mirrored in the naturally occurring age-associated neuropathology in the canine species. In aged dogs with declining learned behaviour and memory the severity of cognitive dysfunction parallels the progressive build up and location of Aβ in the brain. The main aim of this work was to study the biological behaviour of soluble oligomers isolated from an aged dog with cognitive dysfunction through investigating their interaction with a human cell line and synthetic Aβ peptides. We report that soluble oligomers were specifically detected in the dog’s blood and cerebrospinal fluid via anti-oligomer- and anti-Aβ specific binders. Importantly, our results reveal the potent neurotoxic effects of the dog’s cerebrospinal fluid on cell viability and the seeding efficiency of the cerebrospinal fluid-borne soluble oligomers on the thermodynamic activity and the aggregation kinetics of synthetic human Aβ. The value of further characterising the naturally occurring Alzheimer-like neuropathology in dogs using genetic and molecular tools is discussed

Planet Formation Imager (PFI): science vision and key requirements

The Planet Formation Imager (PFI) project aims to provide a strong scientific vision for ground-based optical astronomy beyond the upcoming generation of Extremely Large Telescopes. We make the case that a breakthrough in angular resolution imaging capabilities is required in order to unravel the processes involved in planet formation. PFI will be optimised to provide a complete census of the protoplanet population at all stellocentric radii and over the age range from 0.1 to ~100 Myr. Within this age period, planetary systems undergo dramatic changes and the final architecture of planetary systems is determined. Our goal is to study the planetary birth on the natural spatial scale where the material is assembled, which is the "Hill Sphere" of the forming planet, and to characterise the protoplanetary cores by measuring their masses and physical properties. Our science working group has investigated the observational characteristics of these young protoplanets as well as the migration mechanisms that might alter the system architecture. We simulated the imprints that the planets leave in the disk and study how PFI could revolutionise areas ranging from exoplanet to extragalactic science. In this contribution we outline the key science drivers of PFI and discuss the requirements that will guide the technology choices, the site selection, and potential science/technology tradeoffs.S.K. acknowledges support from an STFC Rutherford Fellowship (ST/J004030/1) and Philip Leverhulme Prize (PLP-2013-110). Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration