Research Interests

  • Category Theory
  • Probability and Information Theory
  • Complexity and Networks
  • Theoretical Computer Science
  • Geometry and Topology
Here is an elementary introduction to some of the work I do.

Curriculum Vitae

Short CV:

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:

Submitted articles:

  • 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.
See also my arXiv page.

Teaching

Current:

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:

  • 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.
  • 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.
  • Currently coordinating the OASIS Seminar at the University of Oxford.

Past events

2024:

  • Markov categories, Bayesian inversions, and statistical experiments. Tweag, Paris (France), June 2024.
  • 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

Links