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'm a mathematical pioneer with broad interests in foundational questions, often involving physics or computer science.
I am currently looking for a long-term research-focused position (CV, research statement, Google Scholar).
Papers and preprints
See my personal arXiv listing.
Notes and slides from talks
In reverse chronological order:
- Optimizing over iid distributions and the Beat the Average game.
Upper and lower bounds for the maximal probability of a strict inequality between iid random variables.
- What are quantum measurable spaces?
Motivation for why we should have a good definition of quantum/noncommutative measurable space and some ideas for what it could look like.
- 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
