# Tobias Fritz

Office: 722b

Department of Mathematics

University of Innsbruck, Austria

For secure communication, please use PGP together with my public key.

## Introduction

Welcome to my page. I currently work on a range of topics in pure mathematics, many with connections to applied mathematics and mathematical physics. I am also an editor for Compositionality, a journal on applied category theory.
## Papers and preprints

All my papers are accessible through my personal arXiv listing.
I have also contributed to Polymath14 and supervised Paolo Perrone's thesis.

## Notes and slides from talks

In reverse chronological order:
- 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.
- Asymptotic and Catalytic Resource Orderings: Beyond Majorization.

The relation between resource theories and real algebra, and a new result in real algebra giving a characterization of asymptotic and catalytic resource orderings.
- 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

## Teaching

## Event Calendar

### 2020

### 2021

### 2022