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Welcome to the Algebra Group at the University of Innsbruck!  

Our research focuses on Real Algebra and Geometry, with connections to Convexity, Optimization, Functional Analysis and Theoretical Quantum Physics.

Real Algebraic Geometry studies semialgebraic sets over the reals or general real closed fields. In algebraic terms, one studies nonnegative polynomials on such sets, and describes them as (generalized) sums of squares. This approach goes back to Hilbert’s 17th Problem. It helps understanding semialgebraic sets and establishes a close connection to optimization, mostly via semidefinite programming. Convex hulls of semialgebraic sets can also be understood via their nonnegative polynomials, and this helps classifying sets for convex optimization. There are also connections to Functional Analysis, via moment problems and non-commutative polynomial inequalities.

If you want to know more, or have questions concerning these topics, feel free to contact us.


  • The Department of Mathematics announces a tenure track position for Computational Algebraic Geometry.

  • Alexander Müller-Hermes visits our group, seminar talk on June 12 12 June - 14 June 2019

  • Our group visits the conference Arctic Applied Algebra 1 April - 5 April 2019

  • Andreas Thom visits our group 18 February - 22 February 2019

  • Seminar talk of Marek Kaluba 31 January 2019

  • DOC-Scholarship for Paria Abbasi 12 December 2018

    Paria Abbasi has been awarded a DOC-Scholarship by the Austrian Academy of Sciences. She will be funded for three years to carry out her research project on completely positive semidefinite matrices, starting in Spring 2019. We congratulate her on this great success and look forward to working with her in the coming years.

  • Seminar talk of Igor Klep 15 November 2018

  • Successful PhD defense by Tom Drescher 24 September 2018

    Tom Drescher's PhD thesis, titled "Free Convex Semi-Algebraic Geometry - The Limits of Quantifier Elimination, Projection Properties, and Operator Systems", was written under the joint supervision of Tim Netzer and Andreas Thom. It examines non-commutative semialgebraic sets from a viewpoint of model theory and functional analysis. Today he completed his doctoral degree with a successful defense. We congratulate him on this great success!

  • Seminar talk of Laurens Wittchow 7 June 2018

  • Seminar talk of Milon Brunner 6 June 2018

  • Successful FWF Project 1 January 2017

    Our FWF Project Free Semialgebraic Geometry and Convexity has been approved by the Austrian Science Fund (FWF) under project number P 29496-N35. We look forward to working on this exciting topic during the following three years.

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