Paolo Perrone

- Occupation: Mathematician and Computer Scientist
- Position: Research Associate
- Institution: University of Oxford
- Group leader: Sam Staton
- Some coworkers and coauthors: Carmen Constantin, Tobias Fritz, Tomáš Gonda, Sharwin Rezagholi, Eigil Rischel, Brandon Shapiro, Dario Stein, Walter Tholen

## Research Interests

- Category Theory
- Probability and Information Theory
- Complexity and Networks
- Theoretical Computer Science
- Geometry and Topology

## Curriculum Vitae

### Short CV:

- Born in Italy in 1989
- Bachelor degree in Physics, University of Milan-Bicocca, Italy, in 2011
- Master's degree in Physics, University of Milan, Italy, in 2013
- PhD in Mathematics, Max Planck Institute for Mathematics in the Sciences and University of Leipzig, Germany, in 2018
- Currently researcher at the University of Oxford, England

### Full academic CV:

*For a DORA/R4RI-style narrative CV, please contact me directly.*

## Writings

### Books:

- P. Perrone,
*Starting Category Theory*, World Scientific, 2024. Link here.

### Published articles:

- P. Perrone,
*Markov Categories and Entropy*. IEEE Transactions on Information Theory 70(3), 2024. Available here. - C. Constantin, T. Fritz, P. Perrone and B. Shapiro,
*Weak cartesian properties of simplicial sets*. Journal of Homotopy and Related Structures, 2023. Available here. - T. Fritz, T. Gonda, N. Gauguin Houghton-Larsen, P. Perrone and D. Stein,
*Dilations and information flow axioms in categorical probability*. Mathematical Structures in Computer Science, 2023. Available here. - T. Fritz, F. Gadducci, P. Perrone and D. Trotta,
*Weakly Markov categories and weakly affine monads*, Proceedings of CALCO, LIPIcs 10, 2023. Available here. - T. Fritz, T. Gonda, P. Perrone and E. Rischel,
*Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability*, Theoretical Computer Science 961, 2023. Available here. - C. Constantin, T. Fritz, P. Perrone and B. Shapiro,
*Partial Evaluations and the Compositional Structure of the Bar Construction*, Theory and Applications of Categories 39, 2023. Available here. - S. Moss and P. Perrone,
*A category-theoretic proof of the ergodic decomposition theorem*, Ergodic Theory and Dynamical Systems, 2023. Available here. - S. Moss and P. Perrone,
*Probability monads with submonads of deterministic states*. Proceedings of LICS, 2022. Extended version available here. - P. Perrone and W. Tholen,
*Kan extensions are partial colimits*, Applied Categorical Structures 30, 2022. Available here. - T. Fritz, P. Perrone, and S. Rezagholi,
*Probability, valuations, hyperspace: Three monads on Top and the support as a morphism*, Mathematical Structures in Computer Science 31(8), 2021. Available here. - T. Fritz, T. Gonda and P. Perrone,
*De Finetti's Theorem in Categorical Probability*, Journal of Stochastic Analysis 2(4), 2021. Available here. - T. Fritz and P. Perrone,
*Stochastic Order on Metric Spaces and the Ordered Kantorovich Monad*, Advances in Mathematics 366, 2020. Available here. - T. Fritz and P. Perrone,
*Monads, partial evaluations, and rewriting*. Proceedings of MFPS 36, ENTCS, 2020. Available here. - T. Fritz and P. Perrone,
*A Probability Monad as the Colimit of Spaces of Finite Samples*, Theory and Applications of Categories 34, 2019. Available here. - T. Fritz and P. Perrone,
*Bimonoidal Structure of Probability Monads*. Proceedings of MFPS 34, ENTCS, 2018. Available here. - P. Perrone and N. Ay,
*Hierarchical Quantification of Synergy in Channels*, Front. Robot. AI, 2016. Available here.

### Submitted articles:

- P. Perrone,
*How to Represent Non-Representable Functors*, 2024. Available here. - P. Perrone and R. Van Belle,
*Convergence of martingales via enriched dagger categories*, 2024. Available here. - N. Ensarguet and P. Perrone,
*Categorical probability spaces, ergodic decompositions, and transitions to equilibrium*, 2023. Available here. - T. Fritz, T. Gonda, A. Lorenzin, P. Perrone and D. Stein,
*Absolute continuity, supports and idempotent splitting in categorical probability*, 2023. Available here. - P. Perrone,
*Lifting couplings in Wasserstein spaces*, 2021. Available here.

### Lecture notes, conference papers and other material:

- P. Perrone,
*Categorical Information Geometry*, 2023. Geometric Science of Information (GSI 2023), Lecture Notes in Computer Science 14071. Available here. - P. Perrone,
*Notes on Category Theory with examples from basic mathematics*, 2019. Lecture notes. Available here. Extended to a book here. - P. Perrone,
*Categorical Probability and Stochastic Dominance in Metric Spaces*, 2018. PhD thesis, University of Leipzig. Full text here, here. - T. Fritz and P. Perrone,
*A Criterion for Kan Extensions of Lax Monoidal Functors*, 2018. Available here. - P. Perrone and N. Ay,
*Iterative Scaling Algorithm for Channels*, 2016. Available here. - P. Perrone and N. Ay,
*Decomposition of Markov Kernels*. Proceedings of WUPES 10, 2015. Available here. - P. Perrone,
*Dual Connections and Holonomy*, 2015. Available here.

## Teaching

### Current:

- Project, ACT Adjoint School, together with Tomáš Gonda.

### Past lectures and seminars:

- Tutorials,
*Integration*, at the University of Oxford, Mansfield College. Hilary term 2023. - Tutorials,
*Integral Transforms*, at the University of Oxford, Mansfield College. Hilary term 2023. - Tutorials,
*Probability I*, at the University of Oxford, Mansfield College. Michaelmas term 2022. - Tutorials,
*Probability II*, at the University of Oxford, Mansfield College. Michaelmas term 2022. - Tutorials,
*Quantum Computing and Modelling*, Cambridge Center for International Research. Summer 2022. - Online course,
*Category Theory*(in Italian), together with the ItaCa project. Main page here, videos here. - Tutorials,
*Groups and Group Actions*, at the University of Oxford, Mansfield College. Hilary term 2021-22. - Tutorials,
*Linear Algebra II*, at the University of Oxford, Mansfield College. Hilary term 2021-22. - Tutorials,
*Integration*, at the University of Oxford, Mansfield College. Hilary term 2021-22. - Tutorials,
*Integral Transforms*, at the University of Oxford, Mansfield College. Hilary term 2021-22. - Course,
*Bayesian Statistical Probabilistic Programming*, at the University of Oxford, Department of Computer Science. Michaelmas term 2021. Together with Hugo Paquet and Sam Staton. - Tutorials,
*Introduction to Complex Numbers*, at the University of Oxford, Mansfield College. Michaelmas term 2021. - Tutorials,
*Linear Algebra I*, at the University of Oxford, Mansfield College. Michaelmas term 2021. - Course,
*Applied Category Theory*, at the University of Oxford, Department of Mathematics. Trinity term 2021. - Tutorials,
*Groups and Group Actions*, at the University of Oxford, Mansfield College. Trinity term 2021. - Applied Category Theory Adjoint School 2020, based at MIT (moved online), together with Carmen Constantin and Eliana Lorch. More information here.
- Reading seminar,
*Categories and gauge theory*, at MIT. - Course,
*Applied Linear Algebra*, at the York University of Toronto. Fall semester 2019.*For notes and other teaching material see the Moodle page (students only)*. - Course,
*Category theory and applications*, at the Max Planck Institute of Leipzig. Summer semester 2019. Lecture notes here (which were extended to this book). - Teaching assistant at the Applied Category Theory Adjoint School, based at the University of Oxford. March-July 2019.
*For notes and other teaching material see the shared folder (students only)*. - Reading seminar on Applied Category Theory at the Max Planck Institute of Leipzig. Summer semester 2018.
- "LikBez" seminar at the Max Planck Institute of Leipzig. Years 2017 and 2018.

*For further material about current and past courses please contact me directly*.

## Events

### Upcoming and current events:

*Mathematics and modelling are the keys we need to safely unlock transformative AI*. Invited event, ARIA, Birmingham (UK), 1-4 October 2024. More information here.*The Formal Theory of Random Variables*. Distinguished Lecture, Peripathetic Seminar on Sheaves and Logic (PSSL), University of Leiden (Netherlands), 15-17 November 2024. Main page here.*A Categorical Theory of Conditional Expectations*. Invited talk, AMS Special Session, Joint Mathematics Meetings (JMM), Seattle, WA (USA), 8-11 January 2025. Main page here.- Currently coordinating the OASIS Seminar at the University of Oxford.

### Past events

#### 2024:

*When limits are limits: topological enrichment with an application to probability*. Category Theory 2024, Universidad de Santiago de Compostela (Spain), June 2024. Main page here, related video here.*Convergence of martingales via enriched dagger categories*. Keynote talk, Applied Category Theory 2024, University of Oxford (UK), June 2024. Speaker: Ruben Van Belle (joint work). Main page here, video here.*Markov categories, Bayesian inversions, and statistical experiments*. Tweag, Paris (France), June 2024. Video here.*Markov categories: towards user-friendly probability theory*. Invited talk, Finance and Stochastics seminar, Imperial College, London (UK), May 2024. Link here.*Conditional expectation for category theorists*. PUDDLE Seminar, Department of Computer Science, University of Oxford (UK), May 2024.*Monoidal preorders and enrichment*. Guest lecture at the Applied Category Theory course, Department of Mathematics, University of Oxford (UK), May 2024.*Martingales and their convergence, categorically*. Invited talk, Four decades of probabilistic semantics, Bellairs Research Institute (Mc Gill University), Holetown (Barbados), April 2024.*Higher abstraction for probability and statistics*. Logic, Uncertainty, Computation and Information (LUCI) Lab, Università degli Studi di Milano (Italy), March 2024.*Held online*. Main page here.*Safeguarded AI: Constructing safety by design*. Invited event, ARIA, Birmingham (UK), March 2024. More information here.

#### 2023:

*Conditional expectations as split idempotents*. ItaCa 2023, Università degli Studi di Torino (Italy), December 2023. Main page here.*Mathematics and modelling are the keys we need to safely unlock transformative AI*. Invited event, ARIA, Birmingham (UK), December 2023. More information here.*Probability theory with Markov categories*. Invited talk, University College London (UK), October 2023.*Information theory with Markov categories*. Invited talk, Geometry of Information Theory, Universidad Carlos III, Madrid (Spain), September 2023. Main page here.*Categorical Information Geometry*. 6th International Conference on Geometric Science of Information (GSI), Saint-Malo (France), August-September 2023. Main page here.*Markov categories and entropy*. Applied Category Theory 2023, University of Maryland, College Park, MD (USA), July-August 2023. Main page here, video here.*Absolute continuity, supports and idempotent splitting in Markov categories*. Applied Category Theory 2023, University of Maryland, College Park, MD (USA), July-August 2023. Speaker: Tomáš Gonda (joint work). Main page here.*Markov categories, a tutorial*. Tutorial session, Applied Category Theory 2023, University of Maryland, College Park, MD (USA), July-August 2023. Main page here, video here.*Universal properties in probability theory*. Keynote talk, Category Theory 2023, Université Catholique de Louvain, Louvain-la-Neuve (Belgium), July 2023. Main page here, video here.*Weakly Markov categories and weakly affine monads*. CALCO 2023, University of Indiana, Bloomington, IN (USA), June 2023. Main page here, slides here.*Compositional Robotics: Mathematics and Tools*, ICRA 2023 Workshop. ExCeL London (UK), June 2023. Main page here.*Probability, Symmetry, and Entropy with Markov Categories*. Invited talk, Computational Statistics and Machine Learning Seminar (OxCSML), Department of Statistics, University of Oxford (UK), May 2023.*Categorical Probability and Information Theory*. Invited talk, Q-Math Seminar, Universidad Carlos III, Madrid (Spain), May 2023. Video here (in Spanish).*Probabilistic programming: its challenges and opportunities*. Invited event, Alan Turing Institute, London (UK), April 2023.*Research week on Markov categories*. Invited event, Universität Innsbruck (Austria), January 2023.

#### 2022:

*Dilations and information flow axioms in categorical probability*. 10th Symposium on Compositional Structures, University of Edinburgh (Scotland), December 2022. Speaker: Tomáš Gonda (joint work). Main page here.*GS-monoidal structures and thunkability in Kleisli categories*. Invited talk, Università di Pisa (Italy), December 2022.*Introduction to Markov categories*. Sheaf Talk Seminar, University of Oxford (UK), November 2022. Link here, related video here.*When measurable spaces don't have enough points*. Theoretical Computer Science Seminar, University of Birmingham (UK), November 2022. Main page here, related video here.*Lifting weights: enriched lenses between transport plans*. 9th Symposium on Compositional Structures, Como (Italy), September 2022. Main page here.*Probability monads with submonads of deterministic states*. Logic in Computer Science (LICS), Technion, Haifa (Israel), August 2022. Speaker: Sean Moss (joint work). Main page here.*When measurable spaces don't have enough points*. Invited talk, Seminário de Teoria das Categorias, University of Coimbra (Portugal), July 2022.*Held online.*Related video here.*A category-theoretic proof of the ergodic decomposition theorem*. Applied Category Theory 2022, University of Strathclyde, Glasgow (Scotland), July 2022. Main page here, video here.*Introduction to categorical probability*. Guest lecture at the Applied Category Theory course, University of Oxford (UK), June 2022.*Compositional Robotics: Mathematics and Tools ICRA 2022 Workshop*, Philadelphia, PA (USA), May 2022.*Hybrid event*. Main page here.

#### 2021:

*The rise of quantitative category theory*. Topos Colloquium. Topos Institute, Berkeley, CA (USA), November 2021.*Held online.*Main page here, video here.*Kan extensions are partial colimits*. Category Theory 2021, University of Genoa (Italy), August-September 2021. Main page here, video here.*De Finetti's theorem in categorical probability*. Applied Category Theory 2021, University of Cambridge (UK), July 2021.*Held online.*Speaker: Tobias Fritz (joint work). Main page here.*An invitation to categorical probability and statistics with Markov categories*. Invited talk, Categories and Companions. Macquarie University, Sydney (Australia), June 2021.*Held online.*Speaker: Tobias Fritz (joint work). Main page here.*Compositional structure of partial evaluations*. Categories and Companions. Macquarie University, Sydney (Australia), June 2021.*Held online.*Speaker: Brandon Shapiro (joint work). Main page here.*Compositional Robotics: Mathematics and Tools*, ICRA 2021 Workshop. ETH Zürich (Switzerland) May 2021.*Held online.*Main page here.*Weighted limits in metric geometry*. Metric geometry seminar. Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany), May 2021.*Held online.**Markov categories: towards a syntax for probability*. Logics and Semantics for Dummies seminar. University of Cambridge (UK), May 2021.*Held online.*Link here.*The law of large numbers in categorical probability*. Topos Colloquium. Topos Institute, Berkeley, CA (USA), May 2021.*Held online.*Speaker: Tobias Fritz (joint work). Main page here, video here.*The de Finetti theorem in categorical probability*. Seminar talk, Cohomology in algebra, geometry, physics and statistics, Institute of Mathematics, Czech Academy of Sciences (Czech Republic), April 2021.*Held online.*Speaker: Tobias Fritz (joint work).*Categorical probability, Markov categories and the De Finetti theorem*. Seminar talk, New York City Category Theory Seminar, CUNY, New York, NY (USA), March 2021. Speaker: Tobias Fritz (joint work).*Held online.*Main page here, video here.*Categorical probability, Markov categories and the De Finetti theorem*. Impromptu seminar talk, New York City Category Theory Seminar, CUNY, New York, NY (USA), March 2021.*Held online.*Main page here, video here.*Partial evaluations: the results so far*. Invited talk, Seminario de categorías UNAM, Universidad Nacional Autónoma de México, Mexico City (Mexico), February 2021.*Held online.*Main page here, video here.*Partial evaluations for monads and 2-monads*. Invited talk, Seminário de Teoria das Categorias, University of Coimbra (Portugal), February 2021.*Held online.**Kan extensions are partial colimits*. Invited talk, Masaryk University Algebra Seminar, Masaryk University, Brno (Czech Republic), February 2021.*Held online.*Main page here, video here.*An introduction to monads*. Invited lecture, Applied Compositional Thinking for Engineers (ACT4E), ETH Zürich (Switzerland), January 2021.*Held online.*Main page here, video here.*Markov categories: randomness and information flow*. Oxford Quantum Group Workshop 2021, University of Oxford (UK), January 2021.*Held online.*

#### 2020 (all events held online):

*Colimits as algebraic operations*. Invited talk, ItaCaFest. Online seminar, Italy, September 2020. Link and abstract here, video here.*Compositional structure of partial evaluations*. MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), September 2020. Speaker: Brandon Shapiro (joint work). Main page here, video here.*Distribution functors, second-order stochastic dominance and the Blackwell-Sherman-Stein Theorem in Categorical Probability*. Keynote talk at the Applied Category Theory Conference, based at MIT, Cambridge, MA (USA), July 2020. Speaker: Tomáš Gonda (joint work). Main page here, video here.*Monads and comonads*. Tutorial talk, Applied Category Theory Conference, based at MIT, Cambridge, MA (USA), July 2020. Main page here, video here.*Applied Category Theory Adjoint School*, Massachusetts Institute of Technology, Cambridge, MA (USA), June-July 2020. Link here.*Monads, partial evaluations, and rewriting*. Principles of Programming and Verification seminar, Boston University, Boston, MA (USA), June 2020. Link here.*Kan extensions are partial colimits*. MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), June 2020.*Held online.*Main page here, video here.*Probability monads and stochastic dominance*, Categorical Probability and Statistics Workshop, University of Ottawa, ON (Canada), June 2020. Main page here, video here.*What is a probability monad?*Tutorial talk, Categorical Probability and Statistics Workshop, University of Ottawa, ON (Canada), June 2020. Main page here, video here.*Monads, partial evaluations, and rewriting*. Mathematical Foundations of Programming Semantics 36, Paris (France), June 2020. Main page here, video here.*The support as a morphism from probability to possibility*. TallCat Seminar. Tallinn Institute of Technology (TalTech), Tallinn (Estonia), May 2020. Speaker: Sharwin Rezagholi (joint work).*Held online.**Stochastic orders and Kantorovich duality*. Metric geometry seminar. Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany), April 2020.*Held online.**Composing partial evaluations*. MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), March 2020.*Held online.*Main page here, video here.

#### 2019:

*The compositional structure of partial evaluations*. Invited talk, University of Ottawa, ON (Canada), November 2019.*Partial evaluations, the bar construction, and second-order stochastic dominance*. Project at the Applied Category Theory Adjoint School, University of Oxford (UK), July 2019. Link here.*The support is a morphism of monads*. Applied Category Theory 2019, University of Oxford (UK), July 2019. Link here.*Categorical Probability: Results and Challenges*. 4th Symposium on Compositional Structures, Chapman University, CA (USA), May 2019. Speaker: Tobias Fritz (joint work). Link here.*The support is a morphism of monads*. 3rd Symposium on Compositional Structures, University of Oxford (UK), March 2019. Speaker: Sharwin Rezagholi (joint work). Link here.*Towards probability theory without measure theory*. Invited talk, MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), February 2019. Speaker: Tobias Fritz (joint work). Main page here, video here.

#### 2018:

*Monads, Partial Evaluations, and Martingales*. Invited talk, MIT Categories Seminar. Massachusetts Institute of Technology, Cambridge, MA (USA), October 2018. Link and abstract here.-
*Monads, Partial Evaluations, and Rewriting*. 1st Symposium on Compositional Structures, University of Birmingham (United Kingdom), September 2018. Link and slides here. *On the Operational Meaning of the Bar Construction*. Category Theory 2018, University of Azores, Ponta Delgada (Portugal), July 2018. Abstract and slides here.*Bimonoidal Structure of Probability Monads*. Mathematical Foundations of Programming Semantics 34, Halifax, NS (Canada), June 2018. Abstract and paper here.*A Kantorovich Monad for Ordered Spaces*. 26th Foundational Methods in Computer Science Workshop, Mount Allison University, Sackville, NB (Canada), May-June 2018. Abstract and slides here.

#### Earlier:

*The Wasserstein Monad in Categorical Probability*. Category Theory 2017, Vancouver, BC (Canada), July 2017. Abstract and slides here.*A Conceptual Viewpoint on Information Decomposition*. Invited talk, Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada), November 2016. Abstract and video here.*Decomposition of Markov Kernels*. 10th Workshop on Uncertainty Processing, Monínec (Czech Republic), September 2015. Link and abstract here.*Synergy as a Linear Operator*. Guided Self Organization 2014, Freiburg (Germany), December 2014. Link here.

## Contact

**Email:**paolo.perrone at cs.ox.ac.uk**Address:**

University of Oxford

Department of Computer Science

Wolfson Building, Parks Road

Oxford, OX1 3QD, U.K.**Office:**Wolfson Building 331