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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 and Optimization, Operator Algebra, Category Theory 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. There are also connections to Operator Algebra and Theoretical Quantum Physics, via non-commutative semialgebraic sets and non-commutative notions of convexity. We also study Category Theory and its applications, for example to Probability.
If you want to know more, or have questions concerning these topics, feel free to contact us.
News
- New paper 22 April 2025
Categories of abstract and noncommutative measurable spaces by Tobias Fritz and Antonio Lorenzin. - New paper 28 March 2025
Empirical Measures and Strong Laws of Large Numbers in Categorical Probability by Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone and Areeb Shah Mohammed. - New paper 25 March 2025
Constructive Quantifier Elimination with a Focus on Matrix Rings by Max Illmer and Tim Netzer. - New paper 20 December 2024
Optimizing over iid distributions and the Beat the Average game by Tobias Fritz at. al. - New paper 21 November 2024
The Aldous-Hoover Theorem in Categorical Probability by Tobias Fritz et. al. - New group member October 2024
We warmly welcome our new group member Maximilian Illmer! - Successful PhD defense by Andreas Klingler 10 September 2024
Andreas Klingler has successfully defended his PhD thesis, titled "Positive and Invariant Tensor Decompositions: Approximations and Computational Complexity" (joint supervision with Gemma De les Coves). In his thesis he studied how tensors can be decomposed into easier parts, and how complex this is from a computational point of view. We congratulate Andreas to this great success! - New paper 19 August 2024
A Note on the Carathéodory Number of the Joint Numerical Range, by Beatrice Maier and Tim Netzer. - New paper 15 August 2024
Self-Dual Cone Systems and Tensor Products, by Tim Netzer. - Successful PhD defense by Mirte van der Eyden 20 June 2024
Mirte van der Eyden has successfully defended her PhD thesis, titled "Where Convex Cones Meet Tensor Products: A Dimension Free Perspective" (joint supervision with Gemma De les Coves). In her thesis she studied questions from quantum information theory in the context of operator systems and complexity theory. We congratulate Mirte to this great success! - New paper 24 April 2024
Positive Moments Forever: Undecidable and Decidable Cases, by Gemma De les Coves, Joshua Graf, Andreas Klingler and Tim Netzer. - New paper 21 March 2024
Self-distributive structures in physics by Tobias Fritz. - New paper 11 March 2024
Differential geometry and general relativity with algebraifolds by Tobias Fritz. - New paper 4 March 2024
Homotopy Methods for Convex Optimization by Andreas Klingler and Tim Netzer. - New paper 29 January 2024
Hidden Markov Models and the Bayes Filter in Categorical Probability by Tobias Fritz et. al. - New paper 2 January 2024
Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs by Lukas Braun et. al. - New paper 21 December 2023
Beyond Operator Systems by Gemma De les Coves, Mirte van der Eyden and Tim Netzer. - New paper 15 December 2023
Involutive Markov categories and the quantum de Finetti theorem by Tobias Fritz and Antonio Lorenzin. - New group members 1 December 2023
We warmly welcome our two new group members, Lukas Braun and Markus Dannemüller! - Atlas der guten Lehre 18 September 2023
The module Lineare Algebra 1 has been included into Atlas der guten Lehre. Thanks for this honor! - New paper 7 August 2023
Absolute Continuity, Supports and Idempotent Splitting in Categorical Probability by Tobias Fritz et. al. - New paper 27 April 2023
Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations by Andreas Klingler, Tim Netzer and Gemma De les Coves. - Successful FWF Project March 2023
The Project Proposal Semialgebraic Operator Algebra With Applications by Tim Netzer has been approved by the Austrian Science Fund (FWF). We look forward to working on this exciting topic during the following years. - New paper 27 January 2023
Classifying Linear Matrix Inequalities via Abstract Operator Systems by Martin Berger, Tom Drescher and Tim Netzer. - New paper 19 January 2023
Asymptotic and Catalytic Matrix Majorization by Tobias Fritz et. al. - Successful PhD defense by Martin Berger 25 November 2022
Martin Berger has successfully defended his PhD thesis, titled "On the Free Carathéodory Number and Operator Systems over the Cone of Positive Semidefinite Matrices". In his thesis he studied different concepts from quantum information theory in the context of free semialgebraic geometry and operator algebra. We congratulate Martin to this great success! - New paper 8 November 2022
Dilations and Information Flow Axioms in Categorical Probability by Tobias Fritz et. al. - New paper 30 September 2022
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
Magic squares: Latin, Semiclassical and Quantum by Gemma De las Cuevas, Tim Netzer and Inga Valentiner-Branth. - New paper 19 August 2022
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
The D-Separation Criterion in Categorical Probability by Tobias Fritz and Andreas Klingler. - New paper 24 June 2022
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
Asymptotic and Catalytic Containment of Representations of 𝖲𝖴(n) by Tobias Fritz. - 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.