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Dagstuhl Seminar 15242

Complexity of Symbolic and Numerical Problems

( Jun 07 – Jun 12, 2015 )

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Please use the following short url to reference this page: https://www.dagstuhl.de/15242

Organizers

Contact


Schedule

Motivation

The seminar is dedicated to Prof Dima Grigoriev on the occasion of his 60th birthday. It is going to discuss modern trends in computational real algebraic geometry, in particular in problems related to solving real algebraic and analytic equations and inequalities.

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. A somewhat trivialized digest of their work, most notably elimination theory can be found in older editions of van der Waerden's Moderne Algebra. On the other hand, 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 will consider the current state of the symbolic and numeric approaches to algorithmic problems of real algebraic and analytic geometry. There will be an emphasis on the hybrid symbolic-numerical methods, but also on the recent achievements in both separate directions.

The topics addressed by the seminar will include:

  • algorithmic real algebraic and o-minimal geometry;
  • upper bounds on the homology, lower complexity bounds, and monotonicity;
  • algebraic circuits and lower bounds;
  • Smale's 17th problem;
  • effective resolution of singularities;
  • quadratic maps;
  • tropical complexity;
  • group invariants cryptography.

The main objective of the seminar is to inform and familiarize researchers, particularly young ones with the background in both traditions, with the most significant recent achievements in complexity of symbolic and numerical computations. Each of the topics is still in the process of development, and in the need of new ideas. We expect some new collaborations to form during and after the seminar, and that the results and methods presented in Dagstuhl will be disseminated to a wider community through the participants.


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.

Copyright Peter Bürgisser, Felipe Cucker, Marek Karpinski, and Nicolai Vorobjov

Participants
  • Dennis Amelunxen (City University - Hong Kong, HK) [dblp]
  • Saugata Basu (Purdue University - West Lafayette, US) [dblp]
  • Peter Bürgisser (TU Berlin, DE) [dblp]
  • Michel Coste (University of Rennes, FR) [dblp]
  • Felipe Cucker (City University - Hong Kong, HK) [dblp]
  • James H. Davenport (University of Bath, GB) [dblp]
  • Dimitry Grigoriev (Lille I University, FR) [dblp]
  • Edward A. Hirsch (Steklov Institute - St. Petersburg, RU) [dblp]
  • Erich Kaltofen (North Carolina State University - Raleigh, US) [dblp]
  • Marek Karpinski (Universität Bonn, DE) [dblp]
  • Pascal Koiran (ENS - Lyon, FR) [dblp]
  • Martin Lotz (Manchester University, GB) [dblp]
  • Klaus Meer (BTU Cottbus, DE) [dblp]
  • Friedhelm Meyer auf der Heide (Universität Paderborn, DE) [dblp]
  • Dmitrii V. Pasechnik (University of Oxford, GB) [dblp]
  • Vladimir Podolskii (Steklov Institute - Moscow, RU) [dblp]
  • Ilia Ponomarenko (Steklov Institute - St. Petersburg, RU) [dblp]
  • Natacha Portier (ENS - Lyon, FR) [dblp]
  • Marie-Françoise Roy (University of Rennes, FR) [dblp]
  • Mohab Safey El Din (UPMC - Paris, FR) [dblp]
  • Éric Schost (University of Western Ontario - London, CA) [dblp]
  • Vladimir Shpilrain (City University of New York, US) [dblp]
  • Anatol Slissenko (Université Paris-Est Créteil, FR) [dblp]
  • Thorsten Theobald (Goethe-Universität Frankfurt am Main, DE) [dblp]
  • Nicolai Vorobjov (University of Bath, GB) [dblp]
  • Andreas Weber (Universität Bonn, DE) [dblp]

Classification
  • data structures / algorithms / complexity

Keywords
  • Symbolic computation
  • algorithms in real algebraic geometry
  • complexity lower bounds
  • geometry of numerical algorithms