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Engineering Systems and Free Semi-Algebraic Geometry

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Emerging Applications of Algebraic Geometry

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 149))

Abstract

This article sketches a few of the developments in the recently emerging area of real algebraic geometry (in short RAG) in a free* algebra, in particular on “noncommutative inequalities”. Also we sketch the engineering problems which both motivated them and are expected to provide directions for future developments. The free* algebra is forced on us when we want to manipulate expressions where the unknowns enter naturally as matrices. Conditions requiring positive definite matrices force one to noncommutative inequalities. The theory developed to treat such situations has two main parts, one parallels classical semialgebraic geometry with sums of squares representations (Positivstellensatze) and the other has a new flavor focusing on how noncommutative convexity (similarly, a variety with positive curvature) is very constrained, so few actually exist.

To Scott Joplin and his eternal RAGs

The authors received partial support respectively from the Ford Motor Co., NSF award DMS-0700758 and the Ford Motor Co., NSF award DMS-0457504 and NSF award DMS-0701094.

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Author information

Authors and Affiliations

  1. Mechanics and Aerospace Engineering Department, UC at San Diego, La Jolla, CA, 92093, USA

    Mauricio C. De Oliveira

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    J. William Helton

  3. Department of Mathematics, University of Florida, Gainesville, FL, 32611-8105, USA

    Scott A. Mccullough

  4. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Mihai Putinar

Authors
  1. Mauricio C. De Oliveira
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  2. J. William Helton
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  3. Scott A. Mccullough
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  4. Mihai Putinar
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Corresponding author

Correspondence to Mauricio C. De Oliveira .

Editor information

Editors and Affiliations

  1. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

  2. Department of Mathematics Harvard University, Cambridge, 02138, U.S.A.

    Seth Sullivant

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De Oliveira, M.C., Helton, J.W., Mccullough, S.A., Putinar, M. (2009). Engineering Systems and Free Semi-Algebraic Geometry. In: Putinar, M., Sullivant, S. (eds) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09686-5_2

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