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Let $X$ be a complex abelian variety. We prove an analogue of both the (cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on $X$ and the Betti moduli space of characters of the fundamental group of $X$. The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product., Comment: v2: minor corrections and updates, 23 pages, comments very welcome!

A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. Proving the fullness of such a collection is generally a nontrvial problem, usually solved on a case-by-case basis, with the aid of a deep understanding of the underlying geometry. Likewise, when an exceptional collection is not full, it is not straightforward to determine whether its "residual" category, i.e., its right orthogonal, is the derived category of a variety. We show how one can use the existence of Bridgeland stability condition these residual categories (when they exist) to address these problems. We examine a simple case in detail: the quadric threefold $Q_3$ in $\mathbb{P}^{4}$. We also give an indication how a variety of other classical results could be justified or re-discovered via this technique., e.g., the commutativity of the Kuznetsov component of the Fano threefold $Y_4$., Comment: 20 pages

Bridgeland stability manifolds of Calabi-Yau categories are of noticeable interest both in mathematics and in physics. By looking at some of the known example, a pattern clearly emerges and gives a fairly precise description of how they look like. In particular, they all seem to have missing loci, which tend to correspond to degenerate stability conditions vanishing on spherical objects. Describing such missing strata is also interesting from a mirror-symmetric perspective, as they conjecturally parametrize interesting types of degenerations of complex structures. All the naive attempts at constructing modular partial compactifications show how elusive and subtle the problem in fact is: ideally, the missing strata would correspond to stability manifolds of quotient triangulated categories, but establishing such correspondence on geometric level and viewing stability conditions on quotients of the original triangulated category as suitable degenerations of stability conditions is not straightforward. In this paper, we will present a method to construct such partial compactifications if some additional hypoteses are satisfied, by realizing our space of interest as a suitable metric completion of the stability manifold., Comment: 23 pages

The Origins Space Telescope (Origins) traces our cosmic history, from the formation of the first galaxies and the rise of metals to the development of habitable worlds and present-day life. Origins does this through exquisite sensitivity to infrared radiation from ions, atoms, molecules, dust, water vapor and ice, and observations of extra-solar planetary atmospheres, protoplanetary disks, and large-area extragalactic fields. Origins operates in the wavelength range 2.8 to 588 microns and is 1000 times more sensitive than its predecessors due to its large, cold (4.5 K) telescope and advanced instruments. Origins was one of four large missions studied by the community with support from NASA and industry in preparation for the 2020 Decadal Survey in Astrophysics. This is the final study report., Comment: 376 pages

Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the resulting weighted metric graph. In general, it is not known how to find the abstract tropicalization of a curve defined by polynomial equations, since an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction in practice. We solve this problem in the case of hyperelliptic curves by studying admissible covers. We also describe how to take a weighted metric graph and compute its period matrix, which gives its tropical Jacobian and tropical theta divisor. Lastly, we describe the present status of reversing this process, namely how to compute a curve which has a given matrix as its period matrix., Comment: 31 pages, 20 figures

In this article we explicitly compute equations of an Enriques surface via the involution on a K3 surface. We also discuss its tropicalization and compute the tropical homology, thus recovering a special case of the result of \cite{IKMZ}, and establish a connection between the dimension of the tropical homology groups and the Hodge numbers of the corresponding algebraic Enriques surface., Comment: 24 pages, comments most welcome!

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the Hilbert scheme. We then use these techniques to compute the cone of ample divisors on the Hilbert scheme for several surfaces where the cone was previously unknown. Our examples include families of surfaces of general type and del Pezzo surfaces of degree 1. The methods rely on Bridgeland stability and the Positivity Lemma of Bayer and Macri., Comment: Corrected an error in section 5: the action of the Weyl group on the extremal rays of Nef(X) is not transitive, and instead has two orbits (see Proposition 5.2). All the results from the section still hold after small modifications to the proofs

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph of a suitable arithmetically Cohen-Macaulay projective curve of regularity $\le 3$, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences of our work are: (1) Any graph is the Hochster-Huneke graph of a complete equidimensional local ring. (This answers a question by Sather-Wagstaff and Spiroff.) (2) The regularity of a curve is not larger than the sum of the regularities of its primary components., Comment: Added Remark 1.1 and Example 4.3; improved exposition

In the prequel to this paper, two versions of Le Potier's strange duality conjecture for sheaves over abelian surfaces were studied. A third version is considered here. In the current setup, the isomorphism involves moduli spaces of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai transform on one side, and moduli spaces where both determinants vary, on the other side. We first establish the isomorphism in rank one using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree 1 via Fourier-Mukai techniques. By degeneration to product geometries, the duality is obtained generically for a large number of numerical types. Finally, it is shown in great generality that the Verlinde sheaves encoding the variation of the spaces of theta functions are locally free over moduli.

The Origins Space Telescope will trace the history of our origins from the time dust and heavy elements permanently altered the cosmic landscape to present-day life. How did galaxies evolve from the earliest galactic systems to those found in the universe today? How do habitable planets form? How common are life-bearing worlds? To answer these alluring questions, Origins will operate at mid- and far-infrared wavelengths and offer powerful spectroscopic instruments and sensitivity three orders of magnitude better than that of Herschel, the largest telescope flown in space to date. After a 3 ½ year study, the Origins Science and Technology Definition Team will recommend to the Decadal Survey a concept for Origins with a 5.9-m diameter telescope cryo cooled to 4.5 K and equipped with three scientific instruments. A mid-infrared instrument (MISC-T) will measure the spectra of transiting exoplanets in the 2.8 – 20 μm wavelength range and offer unprecedented sensitivity, enabling definitive biosignature detections. The Far-IR Imager Polarimeter (FIP) will be able to survey thousands of square degrees with broadband imaging at 50 and 250 μm. The Origins Survey Spectrometer (OSS) will cover wavelengths from 25 – 588 μm, make wide-area and deep spectroscopic surveys with spectral resolving power R ~ 300, and pointed observations at R ~ 40,000 and 300,000 with selectable instrument modes. Origins was designed to minimize complexity. The telescope has a Spitzer-like architecture and requires very few deployments after launch. The cryo-thermal system design leverages JWST technology and experience. A combination of current-state-of-the-art cryocoolers and next-generation detector technology will enable Origins’ natural background limited sensitivity.

"How Much Can You Bench?" appears in the "Mathematical Lens" section of "Mathematics Teacher." "Mathematical Lens" uses photographs as a springboard for mathematical inquiry and appears in every issue of "Mathematics Teacher." This month the mathematics behind the photograph includes finding areas of regular polygons, right triangle trigonometry, the Pythagorean theorem, special right triangles, and similarity and scale factors.

This work presents an overview of the This work presents an overview of the Segmented Aperture Interferometric Nulling Testbed (SAINT), a project that will pair an actively-controlled macro-scale segmented mirror with the Visible Nulling Coronagraph (VNC). SAINT will incorporate the VNCs demonstrated wavefront sensing and control system to refine and quantify the end-to-end system performance for high-contrast starlight suppression. This pathfinder system will be used as a tool to study and refine approaches to mitigating instabilities and complex diffraction expected from future large segmented aperture telescopes., a project that will pair an actively-controlled macro-scale segmented mirror with the Visible Nulling Coronagraph (VNC). SAINT will incorporate the VNCs demonstrated wavefront sensing and control system to refine and quantify the end-to-end system performance for high-contrast starlight suppression. This pathfinder system will be used as a tool to study and refine approaches to mitigating instabilities and complex diffraction expected from future large segmented aperture telescopes.

The presentation will be given at the 26th Annual Thermal Fluids Analysis Workshop (TFAWS 2015) hosted by the Goddard Spaceflight Center (GSFC) Thermal Engineering Branch (Code 545). A STOP analysis is a multidiscipline analysis, consisting of Structural, Thermal and Optical Performance Analyses, that is performed for all space flight instruments and satellites. This course will explain the different parts of performing this analysis. The student will learn how to effectively interact with each discipline in order to accurately obtain the system analysis results.

The GeoCAPE Wide Angle Spectrometer (WAS) Study was a revisit of the COEDI Study from 2012. The customer primary goals were to keep mass, volume and cost to a minimum while meeting the science objectives and maximizing flight opportunities by fitting on the largest number of GEO accommodations possible. Riding on a commercial GEO satellite minimizes total mission costs. For this study, it is desired to increase the coverage rate,km2min, while maintaining ground sample size, 375m, and spectral resolution, 0.4-0.5nm native resolution. To be able to do this, the IFOV was significantly increased, hence the wide angle moniker. The field of view for COEDI was +0.6 degrees or (2048) 375m ground pixels. The WAS Threshold (the IDL study baseline design) is +2.4 degrees IDL study baseline design) is +2.4 degrees.

The International X-ray Observatory requires mirror assemblies with unprecedented characteristics that cannot be provided by existing optical technologies. In the past several years, the project has supported a vigorous mirror technology development program. This program includes the fabrication of lightweight mirror segments by slumping commercially available thin glass sheets, the support and mounting of these thin mirror segments for accurate metrology, the mounting and attachment of these mirror segments for the purpose of X-ray tests, and development of methods for aligning and integrating these mirror segments into mirror assemblies. This paper describes our efforts and developments in these areas.

Area and mass requirements for the Constellation-X Spectroscopy X-Ray Telescopes restrict the thickness of the mirror segment to below a mm. Requirement of angular resolution of 15" over the soft x-ray band implies that allowable optic deformation is sub-micrometer for these thin segments. These requirements place stringent constraint on the mounting, alignment and affixing of these mirror segments in both the metrology and integration processes. We present analyses and optimization of the Constellation-X mirrors under relevant mechanical and thermal environments.

The Fourier-Kelvin Stellar Interferometer (FKSI) has been proposed to detect and characterize extra solar giant planets. The baseline configuration for FKSI is a two- aperture, structurally connected nulling interferometer, capable of providing null depth less than lo4 in the infrared. The objective of this paper is to summarize the process for setting the top level requirements and the jitter analysis performed on FKSI to date. The first part of the paper discusses the derivation of dynamic stability requirements, necessary for meeting the FKSI nulling demands. An integrated model including structures, optics, and control systems has been developed to support dynamic jitter analysis and requirements verification. The second part of the paper describes how the integrated model is used to investigate the effects of reaction wheel disturbances on pointing and optical path difference stabilities.

This report describes the development of the NASA Goddard Space Flight Center Virtual Science Fair, including its history and outgrowth from the traditional regional science fairs supported by NASA. The results of the 1999 Virtual Science Fair pilot program, the mechanics of running the 2000 Virtual Science Fair and its results, and comments and suggestions for future Virtual Science Fairs are provided. The appendices to the report include the original proposal for this project, the judging criteria, the user's guide and the judge's guide to the Virtual Science Fair Web site, the Fair publicity brochure and the Fair award designs, judges' and students' responses to survey questions about the Virtual Science Fair, and lists of student entries to both the 1999 and 2000 Fairs.

This report describes the development of the NASA Goddard Space Flight Center Virtual Science Fair, including its history and outgrowth from the traditional regional science fairs supported by NASA. The results of the 1999 Virtual Science Fair pilot program, the mechanics of running the 2000 Virtual Science Fair and its results, and comments and suggestions for future Virtual Science Fairs are provided. The appendices to the report contain supporting documentation, including the original proposal for this project, the judging criteria, the user's guide and the judge's guide to the Virtual Science Fair Web site, the Fair publicity brochure and the Fair award designs, judges' and students' responses to survey questions about the Virtual Science Fair, and lists of student entries to both the 1999 and 2000 Fairs.