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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 and Theoretical Quantum Physics, mostly via non-commutative semialgebraic sets and non-commutative notions of convexity.

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


  • Due to the current situation, if you have any questions or concerns regarding your studies, please do not hesitate to contact us!
  • Successful PhD defense by Philipp Jukic  11 March 2020
    Philipp Jukic's PhD thesis, titled "Stability of Non-Commutative Quadratic Modules", was written under the supervision of Tim Netzer. It examines the important notion of stability of quadratic modules, for which only a commutative theory existed so far, in the context of non-commutative algebras. Today he completed his doctoral degree with a successful defense. We congratulate him on this great success!
  • Joint group retreat with the Mathematical Quantum Physics group at Maria Waldrast  20 February - 22 February 2020 (picture)

  • Martha Lewis gives a talk on Compositional Hyponymy with Positive Operators in the Mathematical Quantum Physics group 5 February 2020 (2S17)

  • Our group visits the workshop "Algebraic Geometry in Austria" Seewalchen am Attersee, 9 January - 11 January 2020

  • 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 (picture)

  • 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.

  • 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!

  • 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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