### 07.06.15 - 12.06.15, Seminar 15242

# Complexity of Symbolic and Numerical Problems

### Diese Seminarbeschreibung wurde vor dem Seminar auf unseren Webseiten veröffentlicht und bei der Einladung zum Seminar verwendet.

## Motivation

The seminar is dedicated to **Prof Dima Grigoriev** on the occasion of his 60^{th} 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.