33,004 research outputs found
Quiver Varieties, Category O for Rational Cherednik Algebras, and Hecke Algebras
We relate the representations of the rational Cherednik algebras associated with the complex reflection group Āµā ā Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Z-algebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the c-function) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the a-function) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZ-functor.) This is related to a conjecture of BonnafĆ© and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties
Robustness to fundamental uncertainty in AGI alignment
The AGI alignment problem has a bimodal distribution of outcomes with most outcomes clustering around the poles of total success and existential, catastrophic failure. Consequently, attempts to solve AGI alignment should, all else equal, prefer false negatives (ignoring research programs that would have been successful) to false positives (pursuing research programs that will unexpectedly fail). Thus, we propose adopting a policy of responding to points of metaphysical and practical uncertainty associated with the alignment problem by limiting and choosing necessary assumptions to reduce the risk false positives. Herein we explore in detail some of the relevant points of uncertainty that AGI alignment research hinges on and consider how to reduce false positives in response to them
Global evolution of the magnetic field in a thin disc and its consequences for protoplanetary systems
The strength and structure of the large-scale magnetic field in
protoplanetary discs are still unknown, although they could have important
consequences for the dynamics and evolution of the disc. Using a mean-field
approach in which we model the effects of turbulence through enhanced diffusion
coefficients, we study the time-evolution of the large-scale poloidal magnetic
field in a global model of a thin accretion disc, with particular attention to
protoplanetary discs. With the transport coefficients usually assumed, the
magnetic field strength does not significantly increase radially inwards,
leading to a relatively weak magnetic field in the inner part of the disc. We
show that with more realistic transport coefficients that take into account the
vertical structure of the disc and the back-reaction of the magnetic field on
the flow as obtained by Guilet & Ogilvie (2012), the magnetic field can
significantly increase radially inwards. The magnetic-field profile adjusts to
reach an equilibrium value of the plasma parameter (the ratio of
midplane thermal pressure to magnetic pressure) in the inner part of the disc.
This value of depends strongly on the aspect ratio of the disc and on
the turbulent magnetic Prandtl number, and lies in the range for
protoplanetary discs. Such a magnetic field is expected to affect significantly
the dynamics of protoplanetary discs by increasing the strength of MHD
turbulence and launching an outflow. We discuss the implications of our results
for the evolution of protoplanetary discs and for the formation of powerful
jets as observed in T-Tauri star systems.Comment: 19 pages, 12 figures, accepted for publication in MNRA
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