Real Algebraic Geometry and Emerging Applications

Description

Research Proposal

Real algebraic geometry is concerned with specifically real questions in geometry and algebra. In the 19th century, classical algebraic  geometers had a well-developed sense for the real point of view. While during the 20th century the general interest in this field had been somewhat uneven, it was only towards the end of the century that this began to change. Within the past two decades, the general perception of real algebraic geometry has changed in a way that could almost be called dramatic. Today, real algebraic geometry represents a very active and innovative area of research, rapidly developing at many ends, and with close ties to manifold other fields in- and outside of mathematics.

Hilbert (in the 1880s and 1890s) was the first to study the relation between positive polynomials and sums of squares. He considered this subject as so important that he included a central open question in his famous list of mathematical problems, the 17th problem. Its groundbreaking solution by Artin in 1927 opened the door for a systematic development of real algebra. However, it was not before 1980s that real algebra and geometry startet to be advanced thoroughly and with considerable emphasis. The discovery of new algebraic and geometric methods startet to make problems accessible that formerly had been beyond hope.

From the 1990s on, the field gained additional momentum (still growing) through surprising interactions with other mathematical subjects, and also through the increasing general interest in applications. In the early 1990s, new results on positive polynomials started a flow of ideas between real algebraic geometry and the theory of moment problems from analysis, from which both fields have profited strongly to this date. The discovery of powerful new algorithms, combined with the enormously growing power of electronic computation in general, makes questions from real algebraic geometry a central issue in polynomial optimization since around the year 2000; conversely, the needs coming from problems in optimization shaped some of the recent directions that research in real algebraic geometry took. Relations to control theory, and to problems from theoretical physics and from engineering, lead to problems in noncommutative algebra related to notions of positivity. Largely, this area is still terra incognita, but analogues from the traditional commutative theory (themselves only proved in the last 10 to 15 years) have already successfully served as excellent guiding principles, leading to startling recent results. For more details on all this, we refer to the project descriptions below.

This reserach proposal consists of three parts (projects):

1. Positive polynomials, sums of squares and applications,

2. Linear matrix inequalities, semidefinite representations, hyperbolic polynomials and Lax-type questions,

3. Noncommutative real algebraic geometry and related problems.

Institutions
  • Department of Mathematics and Statistics
  • Zukunftskolleg
Funding sources
Name Finanzierungstyp Kategorie Project no.
Exzellenzinitiative third-party funds research funding program 791/09
Further information
Period: 01.12.2009 – 31.10.2012