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Welcome to the Algebra Group at the University of Innsbruck!

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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.


 News

  • New paper 30 September 2022 
    New paper on Polynomial Equations over Algebras by Maximilian Illmer and Tim Netzer.

  • Atlas der guten Lehre 27 September 2022
    The module Lineare Algebra und Analytische Geometrie 1 has been included into Atlas der guten Lehre. Thanks for this honor!

  • New paper 22 September 2022 
    New paper on Magic squares: Latin, Semiclassical and Quantum by Gemma De las Cuevas, Tim Netzer and Inga Valentiner-Branth.

  • New paper 19 August 2022 
    New paper on A hierarchy of semidefinite programs for generalised Einstein-Podolsky-Rosen scenarios by Matty Hoban, Tom Drescher and Ana Belén Sainz.

  • New paper 13 July 2022 
    New paper on The D-Separation Criterion in Categorical Probability by Tobias Fritz and Andreas Klingler.

  • New paper 24 June 2022 
    New paper on Non-Abelian ε-Homology by Tobias Fritz.

  • Successful FWF Project  May 2022
    The Project Proposal Probability and Statistics with Markov Categories by Tobias Fritz has been approved by the Austrian Science Fund (FWF). The group looks forward to working on this exciting topic during the following years.

  • New paper 24 May 2022 
    New paper on Asymptotic and Catalytic Containment of Representations of 𝖲𝖴(n) by Tobias Fritz. Read it here.

  • Article in The Science Breaker  May 2022
    We have written a short article in The Science Breaker, explaining our paper on Quantum Magic Squares.

  • Successful PhD defense by Paria Abbasi  25 February 2022
    Paria Abbasi has successfully defended her PhD thesis, titled "Approximation Techniques for Positive Matrices". In her thesis she studied the convex cone of completely positive semidefinite matrices and the corresponding notion of rank. We congratulate her to this great success!
  • Joint group retreat with the Mathematical Quantum Physics group at Maria Waldrast  16 February - 18 February 2022 (picture)

  • New papers 14 December 2021 
    Two new papers on Vergleichsstellensätze by Tobias Fritz. Read them here and here.

  • Some media coverage November 2020
    Our recent paper on Quantum Magic Squares has attracted some interest in the media. See herehere and here.

  • We are happy to announce that Tobias Fritz has joined our department  1 October 2020
    Tobias has accepted a tenure track position in Computational Algebraic Geometry. This winter he will teach the Algebra 1 course. You can find him in room 722b in our institute. Welcome!

  • We started recording and sharing lectures on Linear Algebra (in German) for the upcoming Winter Term  25 September 2020
    Watch them here.

  • DOC-Scholarship for Martin Berger  3 June 2020

    Martin Berger has been awarded a DOC-Scholarship by the Austrian Academy of Sciences. He will be funded for two years to carry out his research on foundations of free algebraic statistics, starting in Fall 2020. We congratulate him on this great success and look forward to working with him in the coming years!

  • 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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