Browsing by Author "Szpond, Justyna"

Now showing 1 - 6 of 6
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    A few shades of interpolation
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Szpond, Justyna
    The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already-known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient Babylonians and Greeks, and will arrive at subjects of current research activity.
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    Lefschetz Properties in Algebra, Geometry and Combinatorics (hybrid meeting)
    (Zürich : EMS Publ. House, 2020) Migliore, Juan; Miró-Roig, Rosa; Szpond, Justyna
    The themes of the workshop are the Weak Lefschetz Property - WLP - and the Strong Lefschetz Property - SLP. The name of these properties, referring to Artinian algebras, is motivated by the Lefschetz theory for projective manifolds, initiated by S. Lefschetz, and well established by the late 1950's. In fact, Lefschetz properties of Artinian algebras are algebraic generalizations of the Hard Lefschetz property of the cohomology ring of a smooth projective complex variety. The investigation of the Lefschetz properties of Artinian algebras was started in the mid 1980's and nowadays is a very active area of research. Although there were limited developments on this topic in the 20th century, in the last years this topic has attracted increasing attention from mathematicians of different areas, such as commutative algebra, algebraic geometry, combinatorics, algebraic topology and representation theory. One of the main features of the WLP and the SLP is their ubiquity and the quite surprising and still not completely understood relations with other themes, including linear configurations, interpolation problems, vector bundle theory, plane partitions, splines, $d$-webs, differential geometry, coding theory, digital image processing, physics and the theory of statistical designs, etc. among others.
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    Mini-Workshop: Asymptotic Invariants of Homogeneous Ideals
    (Zürich : EMS Publ. House, 2018) Cooper, Susan; Harbourne, Brian; Szpond, Justyna
    Recent 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.
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    Mini-Workshop: Subvarieties in Projective Spaces and Their Projections
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2022) Bauer, Thomas; Favacchio, Giuseppe; Migliore, Juan; Szpond, Justyna
    The major goals of this workshop are to lay paths for a systematic study of geproci (and related, e.g., projecting to almost complete intersections or full intersections) sets of points in projective spaces, study algebraic properties of their ideals (e.g. in the spirit of the Cayley-Bacharach properties), and to identify the most promising new directions for study.
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    On the containment problem
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Szemberg, Tomasz; Szpond, Justyna
    Mathematicians routinely speak two languages: the language of geometry and the language of algebra. When translating between these languages, curves and lines become sets of polynomials called “ideals”. Often there are several possible translations. Then the mystery is how these possible translations relate to each other. We present how geometry itself gives insights into this question.
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    Unexpected Properties of the Klein Configuration of 60 Points in P³
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2020) Pokora, Piotr; Szemberg, Tomasz; Szpond, Justyna
    Felix Klein in course of his study of the regular and its symmetries encountered a highly symmetric configuration of 60 points in P³. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the 60 reflection planes in the group G₃₁ in the Shephard-Todd list. In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities 4; 2 and 2. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in P³ with the surprising property that their general projection to P² is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in P³ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.