ICMS 2020 - Braunschweig - The Classification Problem in Geometry

Anton Betten

Session Chair: Anton Betten (Colorado State University, USA)
Classification problems in discrete geometry including geometry over finite fields have long been an active field for people interested in computations. Very often, the problems can be expressed in terms of orbits of (mostly) finite groups acting on discrete or finite sets. Posets play a big role because they allow to generate the objects of interest in small steps. Classification algorithms must take advantage of the symmetry, in order to be sufficiently fast to attack interesting problems. Large numbers of CPU cycles are spent on some problems. Parallel computing is often used to get more CPU-cycles in shorter time. Computer Algebra systems play an important role because they facilitate group theoretic algorithms. On the other hand, computational primitives (clique finding, exact cover etc.) provide the backbone in many of these hard problems. Problems in discrete geometry often can be translated into a combinatorial setting and standard software tools can be applied. Isomorph rejection then turns into graph isomorphism, a notoriously difficult problem in theoretical computer science. Canonical forms are used frequently, but they are often difficult to compute. Tools for graph isomorphism have been developed, and are widely used. There are also approaches where the isomorphism problem is attacked in the original setting, using the group action on the partially ordered set. These approaches avoid backtracking altogether, but they are more memory intense. Some of the software packages which are often used include Gap, Magma, Nauty, Dancing links, Partition backtrack etc. Recently, Orbiter is another system that can be added to the list. Many people have written special purpose software packages that are not well-known but deserve to be highlighted.


Program:
  1. Classification results for hyperovals of generalized quadrangles, Bart De Bruyn, Ghent University, Department of Mathematics: Algebra and Geometry Abstract
  2. The program Generation in the software package QextNewEdition, Iliya Bouyukliev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria Abstract
  3. Classification of linear codes by extending their residuals, Stefka Bouyuklieva and Iliya Bouyukliev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria. Abstract
  4. Isomorphism of parallelisms of projective spaces, Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria Abstract
  5. Classifying Spacial Triangulations of Polyhedra with Symmetry, Anton Betten, Tarun Mukthineni, Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, U.S.A.
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On 25 Feb 2020, 09:35.