Tobias Fritz
Office: 722c
Department of Mathematics
University of Innsbruck, Austria
For secure communication, please use PGP together with my public key.
Introduction
Welcome to my page. I work on a range of topics involving category theory, probability and mathematical physics.
I am currently looking for an attractive research position.
Papers and preprints
All my papers are accessible through my personal arXiv listing.
See also Google Scholar.
Notes and slides from talks
In reverse chronological order:
- An approach to homological algebra up to ε.
A proposal for doing homological algebra with differentials that do not square to zero, but rather to an operator of small norm, and a result on nonabelian homological algebra with arrow categories.
- Resource theories in the asymptotic and catalytic regime.
An overview of known results on asymptotic and catalytic majorization, and a theorem which generalizes them to a large class of resource theories.
- Categorical Probability and the de Finetti theorem.
An introduction to categorical probability with Markov categories, including the de Finetti theorem and brief overview of its proof.
- Comparison of statistical experiments beyond the discrete case.
An overview of our results on the comparison of statistical experiments on standard Borel spaces, and a general introduction to Markov categories (with many bonus slides).
- Markov Categories: Probability and Statistics as a Theory of Information Flow.
I argue that Markov categories provide a general theory of information flow, that this is synonymous with probability theory maximally generalized, and that there is a vast unexplored landscape of Markov categories.
- A synthetic introduction to probability and statistics.
An attempt at explaining some elementary aspects of probability and statistics in terms of category theory rather than measure theory.
- Categorical probability: results and challenges.
A big-picture overview of two categorical approaches to probability and statistics.
- Towards synthetic Euclidean QFT.
An emerging approach to statistical field theory (or Euclidean QFT) based on intuitionistic logic and topos theory.
- Real algebra, random walks, and information theory.
Positive polynomials, a new Positivstellensatz, and its application to random walks and majorization.
- Measurement functors.
Understanding a quantum systems through keeping track of all ways of performing classical measurements on it.
- The energy-entropy diagram as a fundamental tool of thermodynamics.
A simple method for treating aspects of thermodynamics arbitrarily far away from equilibrium.
- A simple formalism for resource efficiency in thermodynamics.
A simple method for treating aspects of thermodynamics arbitrarily far away from equilibrium.
- The inflation technique for causal inference.
A general method for approaching causal inference for Bayesian networks in the presence of latent variables.
- Quantum logic and computability of noncommutative sums of squares.
Satisfiability in quantum logic and approaches via sums of squares.
- The Kitaev model and aspects of semisimple Hopf algebras via the graphical calculus.
An idiosyncratic perspective on the quantum double models.
- Some thoughts on inferring system structure.
Beginnings of a theory of how much can be inferred about the internal structure of a system by observing its behaviour from the outside.
- Quantum logic is undecidable.
The complexity of quantum logic, the hypergraph approach to quantum contextuality, and the undecidability as a consequence of Slofstra's theorem.
- The inflation technique for causal inference with latent variables.
A general method for approaching causal inference for Bayesian networks in the presence of latent variables.
- A mathematical toolbox for resource theories.
A formalism for analyzing resource efficiency with wide applicability.
- Almost C*-algebras.
A conjectural reaxiomatization of C*-algebras in terms of generalized functional calculus and Noether's theorem.
- Characterizations of Shannon and Rényi entropy.
Broad overview of characterizations of Shannon and Rényi entropies.
- Characterizing Entropy.
Categorical characterizations of Shannon entropy and relative entropy (Kullback-Leibler divergence). Slides by John Baez with minor modifications.
- Equality.
High-level perspective on the nature of equality with an emphasis on homotopy type theory and univalent foundations.
- A Combinatorial Approach to Nonlocality and Contextuality.
A hypergraph formalism for quantum nonlocality and contextuality with implications for graph theory.
- Bell's Theorem on arbitrary causal structures.
Generalizing quantum nonlocality to causal structures other than the one of the Bell scenario.
- Resources.
A mathematical toolbox for analyzing resource efficiency (early stages).
- Turning Weyl's tile argument into a no-go theorem.
If space is discrete and periodic, then it cannot be isotropic even on large scales.
- Quantum correlations and group C*-algebras.
Maximal group C*-algebras and their relation to quantum Bell inequalities and to Tsirelson's problem.
- Witnessing Infinite-dimensional State Spaces.
Using the Baumslag-Solitar groups to witness that a given quantum system is infinite-dimensional.
- Horn Formulas as Linear Inequalities and Nonlocality Paradoxes.
How Hardy's paradox and its variants can be understood as linear inequalities, and how to reason about such linear inequalities systematically.
- Tsirelson's problem and Kirchberg's conjecture.
How maximal group C*-algebras relate to quantum Bell inequalities, and how this connects Tsirelson's problem with Kirchberg's QWEP conjecture and the Connes embedding problem.
- Cuntz' proof of Bott periodicity for C*-algebras.
A detailed exposition of Cuntz's clever proof of Bott periodicity in C*-algebra K-theory using the Toeplitz C*-algebra.
- Quantum correlations and group C*-algebras.
Using maximal group C*-algebras in the context of quantum nonlocality.
- Curious properties of iterated measurements.
Surprising phenomena that appear when repeating binary projective measurements on a quantum system.
- Abstract Convexity: Results and Speculations.
Convex spaces, the convex space of polytopes, a Hahn—Banach theorem, hints at a theory of convex homotopy, and metric aspects of convex spaces.
- Abstract Convexity.
Introduction to convex spaces and their intrinsic geometry.
- The Geometry of the Standard Model.
Overview of the different kinds of fields making up the standard model.
- The Geometry of Black Holes.
General relativity, the Schwarzschild solution and the Kruskal extension.
In German:
Essays
Event Calendar
2026
2025
2024
2023
2022
2021
2020
