Browsing by Author "Harbourne, Brian"
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- ItemArrangements of lines(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, TomaszWe discuss certain open problems in the context of arrangements of lines in the plane.
- ItemGeproci Sets: a New Perspective in Algebraic Geometry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Chiantini, Luca; Harbourne, BrianGeproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple X-ray images, we aim at a classification of sets with certain algebraic properties under multiple projections.
- ItemMini-Workshop: Arrangements of Subvarieties, and their Applications in Algebraic Geometry(Zürich : EMS Publ. House, 2016) Di Rocco, Sandra; Harbourne, Brian; Szemberg, TomaszWhile arrangements of hyperplanes have been studied in algebra, combinatorics and geometry for a long time, recent discoveries suggest that they (and more generally arrangements of nonlinear subvarieties) play an even more fundamental role in major problems in algebraic geometry than has yet been understood. The workshop brought into contact experts from commutative algebra and algebraic geometry working on these problems – it provided opportunities to get updated on the latest developments through talks of the participants, but also reserved time for working groups in which participants brainstormed ideas and insights in the context of high-intensity discussions aimed at initiating immediate progress on proposed problems, thereby setting the stage for on-going collaborations after the workshop.
- ItemMini-Workshop: Asymptotic Invariants of Homogeneous Ideals(Zürich : EMS Publ. House, 2018) Cooper, Susan; Harbourne, Brian; Szpond, JustynaRecent decades have witnessed a shift in interest from isolated objects to families of objects and their limit behavior, both in algebraic geometry and in commutative algebra. A series of various invariants have been introduced in order to measure and capture asymptotic properties of various algebraic objects motivated by geometrical ideas. The major goals of this workshop were to refine these asymptotic ideas, to articulate unifying themes, and to identify the most promising new directions for study in the near future. We expect the ideas discussed and originated during this workshop to be poised to have a broad impact beyond the areas of algebraic geometry and commutative algebra.
- ItemMini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems(Zürich : EMS Publ. House, 2015) Carlini, Enrico; Guardo, Elena; Harbourne, BrianIt is a fundamental challenge for many problems of significant current interest in algebraic geometry and commutative algebra to understand symbolic powers $I^{(m)}$ of homogeneous ideals $I$ in polynomial rings, particularly ideals of linear varieties. Such problems include computing Waring ranks of polynomials, determining the occurrence of equality $I^{(m)} = I^m$ (or, more generally, of containments $I^{(m)} \subseteq I^r$), computing Waldschmidt constants (i.e., determining the limit of the ratios of the least degree of an element in $I^{(m)}$ to the least degree of an element of $I^m$), and studying major conjectures such as Nagata’s Conjecture and the uniform SHGH Conjecture (which respectively specify the Waldschmidt constant of ideals of generic points in the plane and the Hilbert functions of their symbolic powers).
- ItemMini-Workshop: Linear Series on Algebraic Varieties(Zürich : EMS Publ. House, 2010) Di Rocco, Sandra; Harbourne, Brian; Szemberg, TomaszLinear series have long played a central role in algebraic geometry. In recent years, starting with seminal papers by Demailly and Ein-Lazarsfeld, local properties of linear series – in particular local positivity, as measured by Seshadri constants – have come into focus. Interestingly, in their multi-point version they are closely related to the famous Nagata conjecture on plane curves. While a number of important basic results are available by now, there are still a large number of open questions and even completely open lines of research.
- ItemVery general monomial valuations of P2 and a Nagata type conjecture(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Dumnicki, Marcin; Harbourne, Brian; Küronya, Alex; Roé, Joaquim; Sze,berg, Tomasz[no abstract available]