http://www.dagstuhl.de/15242

07. – 12. Juni 2015, Dagstuhl Seminar 15242

Complexity of Symbolic and Numerical Problems

Organisatoren

Peter Bürgisser (TU Berlin, DE)
Felipe Cucker (City University – Hong Kong, HK)
Marek Karpinski (Universität Bonn, DE)
Nicolai Vorobjov (University of Bath, GB)

Auskunft zu diesem Dagstuhl Seminar erteilt

Dagstuhl Service Team

Dokumente

Dagstuhl Report, Volume 5, Issue 6 Dagstuhl Report
Motivationstext
Teilnehmerliste
Gemeinsame Dokumente
Programm des Dagstuhl Seminars [pdf]

Summary

The seminar was dedicated to Prof. Dima Grigoriev on the occasion of his 60th birthday. Its aim was to discuss modern trends in computational real algebraic geometry, in particular, areas related to solving real algebraic and analytic equations and inequalities. Very recent new developments in the analysis of these questions from the point of view of tropical mathematics were also presented.

Historically there were two strands in the computational approach to polynomial systems' solving. One is the tradition of numerical analysis, a classical achievement of which is the Newton's method. Various other approximation algorithms were developed since then, some based on the idea of a homotopy. Numerical analysis did not bother to introduce formal models of computations (and hence computational complexity considerations) but developed refined methods of estimations of convergency rates. Another tradition emerged from algebra, particularly in classical works of Cayley, Sylvester and Macaulay. Algebraic results concerning real solutions go further back to the Descartes' rule and Sturm sequences. An important contribution to the subject from logic was Tarski's constructive quantifier elimination procedures for algebraically closed and real closed fields. The computations considered in this tradition are exact, under modern terminology - "symbolic". They naturally fit into standard models of computation (Turing Machines, straight-line programs, computation trees) thus lending themselves to complexity analysis.

Until 1990s these two strands developed largely independently. One of the important unifying ideas became the concept of a real numbers (or BSS) machine suggested by Blum, Shub and Smale which can be considered as a model of computation for the numerical analysis. This idea led to Smale's 9th and 17th problems, which became an inspiration for many researchers in the field.

The seminar considered a wide set of questions related to the current state of the symbolic and numeric approaches to algorithmic problems of real algebraic and analytic geometry, also from the novel perspective of tropical and max/plus mathematics.

License
  Creative Commons BY 3.0 Unported license
  Peter Bürgisser, Felipe Cucker, Marek Karpinski, and Nicolai Vorobjov

Classification

  • Data Structures / Algorithms / Complexity

Keywords

  • Symbolic computation
  • Algorithms in real algebraic geometry
  • Complexity lower bounds
  • Geometry of numerical algorithms

Buchausstellung

Bücher der Teilnehmer 

Buchausstellung im Erdgeschoss der Bibliothek

(nur in der Veranstaltungswoche).

Dokumentation

In der Reihe Dagstuhl Reports werden alle Dagstuhl-Seminare und Dagstuhl-Perspektiven-Workshops dokumentiert. Die Organisatoren stellen zusammen mit dem Collector des Seminars einen Bericht zusammen, der die Beiträge der Autoren zusammenfasst und um eine Zusammenfassung ergänzt.

 

Download Übersichtsflyer (PDF).

Publikationen

Es besteht weiterhin die Möglichkeit, eine umfassende Kollektion begutachteter Arbeiten in der Reihe Dagstuhl Follow-Ups zu publizieren.

Dagstuhl's Impact

Bitte informieren Sie uns, wenn eine Veröffentlichung ausgehend von
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