57 research outputs found
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Wachpress Conjecture Restricted To Arrangements Of Three Conics
This thesis discusses Wachpress conjecture restricted to arrangements of three conics. Wachpress conjectured the existence of a set of barycentric coordinates, namely Wachpress coordinates, on all polycons. Barycentric coordinates are very useful in many different fields as they can be used to define a finite element approximation scheme with linear precision. This thesis focuses on the conjecture on the real projective plane. The polycons of lowest degree for which the conjecture has not been proven completely yet are those which arise from arrangements of three conics. We state the current knowledge on the veracity of the conjecture on the polycons of this family. Throughout the thesis we view real rational polycons as positive geometries which encode both differential and algebraic properties in their unique canonical form
Two Counting Problems in Geometric Triangulations and Pseudoline Arrangements
The purpose of this dissertation is to study two problems in combinatorial geometry in regard to obtaining better bounds on the number of geometric objects of interest: (i) monotone paths in geometric triangulations and (ii) pseudoline arrangements.
\medskip(i) A directed path in a graph is monotone in direction of if every edge in the path has a positive inner product with . A path is monotone if it is monotone in some direction. Monotone paths are studied in optimization problems, specially in classical simplex algorithm in linear programming. We prove that the (maximum) number of monotone paths in a geometric triangulation of points in the plane is . This improves an earlier upper bound of ; the current best lower bound is (Dumitrescu~\etal, 2016).
\medskip (ii) Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, for instance in the study of sorting networks. Let be the number of nonisomorphic arrangements of pseudolines and let . The problem of estimating was posed by Knuth in 1992. Knuth conjectured that and also derived the first upper and lower bounds: and . The upper bound underwent several improvements, (Felsner, 1997), and (Felsner and Valtr, 2011), for large . Here we show that for some constant . In particular, for large . This improves the previous best lower bound, , due to Felsner and Valtr (2011). Our arguments are elementary and geometric in nature. Further, our constructions are likely to spur new developments and improved lower bounds for related problems, such as in topological graph drawings.
\medskip Developing efficient algorithms and computer search were key to verifying the validity of both results
Unexpected Stein fillings, rational surface singularities, and plane curve arrangements
We compare Stein fillings and Milnor fibers for rational surface
singularities with reduced fundamental cycle. Deformation theory for this class
of singularities was studied by de Jong-van Straten in [dJvS98]; they
associated a germ of a singular plane curve to each singularity and described
Milnor fibers via deformations of this singular curve. We consider links of
surface singularities, equipped with their canonical contact structures, and
develop a symplectic analog of de Jong-van Straten's construction. Using planar
open books and Lefschetz fibrations, we describe all Stein fillings of the
links via certain arrangements of symplectic disks, related by a homotopy to
the plane curve germ of the singularity. As a consequence, we show that many
rational singularities in this class admit Stein fillings that are not strongly
diffeomorphic to any Milnor fibers. This contrasts with previously known cases,
such as simple and quotient surface singularities, where Milnor fibers are
known to give rise to all Stein fillings. On the other hand, we show that if
for a singularity with reduced fundamental cycle, the self-intersection of each
exceptional curve is at most -5 in the minimal resolution, then the link has a
unique Stein filling (given by a Milnor fiber).Comment: 70 pages, 29 figures, additions to v2 are Theorem 1.3 and its
corresponding discussion in 4.3, along with added references, v3 updated
Remark 4.
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