18 research outputs found
Desingularization in Computational Applications and Experiments
After briefly recalling some computational aspects of blowing up and of
representation of resolution data common to a wide range of desingularization
algorithms (in the general case as well as in special cases like surfaces or
binomial varieties), we shall proceed to computational applications of
resolution of singularities in singularity theory and algebraic geometry, also
touching on relations to algebraic statistics and machine learning. Namely, we
explain how to compute the intersection form and dual graph of resolution for
surfaces, how to determine discrepancies, the log-canoncial threshold and the
topological Zeta-function on the basis of desingularization data. We shall also
briefly see how resolution data comes into play for Bernstein-Sato polynomials,
and we mention some settings in which desingularization algorithms can be used
for computational experiments. The latter is simply an invitation to the
readers to think themselves about experiments using existing software, whenever
it seems suitable for their own work.Comment: notes of a summer school talk; 16 pages; 1 figur
Einstieg in die Ingenieurmathematik aus der Berufspraxis - Unterstützung in Mathematik und fachadäquaten Lernstrategien
Das Projekt „Einstieg in die Ingenieurmathematik aus der Berufspraxis“
wurde im Wintersemester 2014/2015 an der Leibniz Universität Hannover
pilotiert und richtet sich an Studierende, die nach längerer Zeit der Berufspraxis
ihr Studium ohne bzw. mit länger zurückliegender Allgemeiner
Hochschulreife aufnehmen. Für diese Gruppe von Studierenden stellt die
Veranstaltung Mathematik für Ingenieure I in der Regel ein großes Hindernis
für den erfolgreichen Einstieg ins Studium dar
Towards Massively Parallel Computations in Algebraic Geometry
Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka’s celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores