Opportunity Road: The Promise and Challenge of America's Forgotten Youth

There are millions of youth ages 16 to 24 who are out of school and out of work. They cost the nation billions of dollars every year and over their lifetimes in lost productivity and increased social services. They also represent an opportunity for the nation to tap the talents of millions of potential leaders and productive workers at a time when America's skills gap is significant. The central message of this report is that while these youth face significant life challenges, most start out with big dreams and remain confident or hopeful that they can achieve their goals; most accept responsibility for their futures; and most are looking to reconnect to school, work and service. They point the way to how they can effectively reconnect to education, productive work and civic life. On behalf of Civic Enterprises and the America's Promise Alliance, Peter D. Hart Research Associates undertook a national cross-section of opportunity youth in 23 diverse locations across the United States in August 2011 to learn about common elements in their personal histories and their lives today, and to explore opportunities to reconnect them to work and school. At the time of the survey, respondents were ages 16 to 24, neither enrolled in school nor planning to enroll in the coming year, were not working, and had not completed a college degree. In addition, they were not disabled such as to prevent long-term employment, were not incarcerated, and were not a stay-at-home parent with a working spouse. What the authors found was both heartbreaking and uplifting, frustrating and hopeful. Despite many growing up in trying circumstances of little economic means and weak family and social supports, the youth they surveyed were optimistic about their futures. More than half believed they would graduate college when they were growing up and their hopes remain high that they will achieve the American Dream with a strong family life of their own and a good job one day. For this reason, the authors believe they are truly "opportunity youth"--both for their belief in themselves that must be nurtured and for the opportunity they hold for America

Derived automorphism groups of K3 surfaces of Picard rank 1

We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank 1. We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences, and is contractible

Riemann-Hilbert problems from Donaldson-Thomas theory

We study a class of Riemann-Hilbert problems arising naturally in Donaldson-Thomas theory. In certain special cases we show that these problems have unique solutions which can be written explicitly as products of gamma functions. We briefly explain connections with Gromov-Witten theory and exact WKB analysis

Donaldson-Thomas invariants and wall-crossing formulas

Notes from the report at the Fields institute in Toronto. We introduce the Donaldson-Thomas invariants and describe the wall-crossing formulas for numerical Donaldson-Thomas invariants.Comment: 18 pages. To appear in the Fields Institute Monograph Serie

Equivalences between GIT quotients of Landau-Ginzburg B-models

We define the category of B-branes in a (not necessarily affine) Landau-Ginzburg B-model, incorporating the notion of R-charge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of Landau-Ginzburg B-models that arise as different GIT quotients of a vector space by a one-dimensional torus, and show that for each such pair the two categories of B-branes are quasi-equivalent. In fact we produce a whole set of quasi-equivalences indexed by the integers, and show that the resulting auto-equivalences are all spherical twists.Comment: v3: Added two references. Final version, to appear in Comm. Math. Phy

Quadratic differentials as stability conditions

We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.Comment: 123 pages; 38 figures. Version 2: hypotheses in the main results mildly weakened, to reflect improved results of Labardini-Fragoso and coauthors. Version 3: minor changes to incorporate referees' suggestions. This version to appear in Publ. Math. de l'IHE

Fourier–Mukai transforms for quotient varieties

We study Fourier–Mukai transforms for smooth projective varieties whose canonical bundles have finite order. Our results lead to new transforms for Enriques and bielliptic surfaces

Strings, Junctions and Stability

Identification of string junction states of pure SU(2) Seiberg-Witten theory as B-branes wrapped on a Calabi-Yau manifold in the geometric engineering limit is discussed. The wrapped branes are known to correspond to objects in the bounded derived category of coherent sheaves on the projective line \cp{1} in this limit. We identify the pronged strings with triangles in the underlying triangulated category using Pi-stability. The spiral strings in the weak coupling region are interpreted as certain projective resolutions of the invertible sheaves. We discuss transitions between the spiral strings and junctions using the grade introduced for Pi-stability through the central charges of the corresponding objects.Comment: 15 pages, LaTeX; references added. typos correcte

Moduli Spaces for D-branes at the Tip of a Cone

For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is isomorphic to the anticanonical cone over the surface.Comment: 8 page

Vertex Operators, Grassmannians, and Hilbert Schemes

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.Comment: 20 pages, 0 figure