A Simple Geometric Representative for μ\mu of a Point

For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $\mu$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p)$ is reducible $\}$, with coefficient -1/4 and appropriate orientation, is our desired geometric representative.Comment: Updated 2018 to published version. 8 pages, AmS-TeX, no figure

Uncovering Bugs in Distributed Storage Systems during Testing (not in Production!)

Testing distributed systems is challenging due to multiple sources of nondeterminism. Conventional testing techniques, such as unit, integration and stress testing, are ineffective in preventing serious but subtle bugs from reaching production. Formal techniques, such as TLA+, can only verify high-level specifications of systems at the level of logic-based models, and fall short of checking the actual executable code. In this paper, we present a new methodology for testing distributed systems. Our approach applies advanced systematic testing techniques to thoroughly check that the executable code adheres to its high-level specifications, which significantly improves coverage of important system behaviors. Our methodology has been applied to three distributed storage systems in the Microsoft Azure cloud computing platform. In the process, numerous bugs were identified, reproduced, confirmed and fixed. These bugs required a subtle combination of concurrency and failures, making them extremely difficult to find with conventional testing techniques. An important advantage of our approach is that a bug is uncovered in a small setting and witnessed by a full system trace, which dramatically increases the productivity of debugging

Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry

Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y should act by families of Fourier-Mukai transforms over the complex moduli space of the mirror X. The conjecture generalizes a proposal of Kontsevich relating monodromy transformations and self-equivalences. Supporting evidence is given in the case of elliptic curves, lattice-polarized K3 surfaces and Calabi-Yau threefolds. A relation to the global Torelli problem is discussed.Comment: Approx. 20 pages LaTeX. One reference adde

Stability of Affine G-varieties and Irreducibility in Reductive Groups

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-R\"ohrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.Comment: 29 pages. To appear in Int. J. Math. Note: this version 4 is identical with version 2 (version 3 is empty

Khovanov homology is an unknot-detector

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.Comment: 124 pages, 13 figure

Optical Coronagraphic Spectroscopy of AU Mic: Evidence of Time Variable Colors?

We present coronagraphic long slit spectra of AU Mic's debris disk taken with the STIS instrument aboard the Hubble Space Telescope (HST). Our spectra are the first spatially resolved, scattered light spectra of the system's disk, which we detect at projected distances between approximately 10 and 45 AU. Our spectra cover a wavelength range between 5200 and 10200 angstroms. We find that the color of AU Mic's debris disk is bluest at small (12-35 AU) projected separations. These results both confirm and quantify the findings qualitatively noted by Krist et al. (2005), and are different than IR observations that suggested a uniform blue or gray color as a function of projected separation in this region of the disk. Unlike previous literature that reported the color of AU Mic's disk became increasingly more blue as a function of projected separation beyond approximately 30 AU, we find the disk's optical color between 35-45 AU to be uniformly blue on the southeast side of the disk and decreasingly blue on the northwest side. We note that this apparent change in disk color at larger projected separations coincides with several fast, outward moving "features" that are passing through this region of the southeast side of the disk. We speculate that these phenomenon might be related, and that the fast moving features could be changing the localized distribution of sub-micron sized grains as they pass by, thereby reducing the blue color of the disk in the process. We encourage follow-up optical spectroscopic observations of the AU Mic to both confirm this result, and search for further modifications of the disk color caused by additional fast moving features propagating through the disk.Comment: Accepted by AJ, 13 pages, 8 figures, 1 tabl

Implementing and evaluating candidate-based invariant generation

The discovery of inductive invariants lies at the heart of static program verification. Presently, many automatic solutions to inductive invariant generation are inflexible, only applicable to certain classes of programs, or unpredictable. An automatic technique that circumvents these deficiencies to some extent is candidate-based invariant generation , whereby a large number of candidate invariants are guessed and then proven to be inductive or rejected using a sound program analyser. This paper describes our efforts to apply candidate-based invariant generation in GPUVerify, a static checker of programs that run on GPUs. We study a set of 383 GPU programs that contain loops, drawn from a number of open source suites and vendor SDKs. Among this set, 253 benchmarks require provision of loop invariants for verification to succeed. We describe the methodology we used to incrementally improve the invariant generation capabilities of GPUVerify to handle these benchmarks, through candidate-based invariant generation , whereby potential program invariants are speculated using cheap static analysis and subsequently either refuted or proven. We also describe a set of experiments that we used to examine the effectiveness of our rules for candidate generation, assessing rules based on their generality (the extent to which they generate candidate invariants), hit rate (the extent to which the generated candidates hold), effectiveness (the extent to which provable candidates actually help in allowing verification to succeed), and influence (the extent to which the success of one generation rule depends on candidates generated by another rule). We believe that our methodology for devising and evaluation candidate generation rules may serve as a useful framework for other researchers interested in candidate-based invariant generation. The candidates produced by GPUVerify help to verify 231 of these 253 programs. An increase in precision, however, has created sluggishness in GPUVerify because more candidates are generated and hence more time is spent on computing those which are inductive invariants. To speed up this process, we have investigated four under-approximating program analyses that aim to reject false candidates quickly and a framework whereby these analyses can run in sequence or in parallel. Across two platforms, running Windows and Linux, our results show that the best combination of these techniques running sequentially speeds up invariant generation across our benchmarks by 1 . 17 × (Windows) and 1 . 01 × (Linux), with per-benchmark best speedups of 93 . 58 × (Windows) and 48 . 34 × (Linux), and worst slowdowns of 10 . 24 × (Windows) and 43 . 31 × (Linux). We find that parallelising the strategies marginally improves overall invariant generation speedups to 1 . 27 × (Windows) and 1 . 11 × (Linux), maintains good best-case speedups of 91 . 18 × (Windows) and 44 . 60 × (Linux), and, importantly, dramatically reduces worst-case slowdowns to 3 . 15 × (Windows) and 3 . 17 × (Linux)

Development of a new laser Doppler velocimeter for the Ames High Reynolds Channel No. 2

A new two-channel laser Doppler velocimeter developed for the Ames High Reynolds Channel No. 2 is described. Design features required for the satisfactory operation of the optical system in the channel environment are discussed. Fiber optics are used to transmit the megahertz Doppler signal to the photodetectors located outside the channel pressure vessel, and provision is made to isolate the optical system from pressure and thermal strain effects. Computer-controlled scanning mirrors are used to position the laser beams in the channel flow. Techniques used to seed the flow with 0.5-micron-diam polystyrene spheres avoiding deposition on the test-section windows and porous boundary-layer removal panels are described. Preliminary results are presented with a discussion of several of the factors affecting accuracy

On the Topology of Black Hole Event Horizons in Higher Dimensions

In four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two-sphere $S^2$. We consider the topology of event horizons in higher dimensions. First, we reconsider Hawking's theorem and show that the integrated Ricci scalar curvature with respect to the induced metric on the event horizon is positive also in higher dimensions. Using this and Thurston's geometric types classification of three-manifolds, we find that the only possible geometric types of event horizons in five dimensions are $S^3$ and $S^2 \times S^1$. In six dimensions we use the requirement that the horizon is cobordant to a four-sphere (topological censorship), Friedman's classification of topological four-manifolds and Donaldson's results on smooth four-manifolds, and show that simply connected event horizons are homeomorphic to $S^4$ or $S^2\times S^2$. We find allowed non-simply connected event horizons $S^3\times S^1$ and $S^2\times \Sigma_g$, and event horizons with finite non-abelian first homotopy group, whose universal cover is $S^4$. Finally, following Smale's results we discuss the classification in dimensions higher than six.Comment: 12 pages, minor edits 27/09/0