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:

Articles

Published:

Submitted:

  • C. Constantin, T. Fritz, P. Perrone and B. Shapiro, Weak cartesian properties of simplicial sets, 2021. Available here.
  • T. Fritz, T. Gonda and P. Perrone, De Finetti's Theorem in Categorical Probability, 2021. Available here.
  • P. Perrone and W. Tholen, Kan extensions are partial colimits, 2021. Available here.
  • T. Fritz, T. Gonda, P. Perrone and E. Rischel, Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability, 2020. Available here.
  • C. Constantin, T. Fritz, P. Perrone and B. Shapiro, Partial Evaluations and the Compositional Structure of the Bar Construction, 2020. Available here.
  • T. Fritz, P. Perrone, and S. Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, 2019. Available here.

Lecture notes, conference papers and other material:

  • P. Perrone, Notes on Category Theory with examples from basic mathematics, 2019. Lecture notes. Available here.
  • P. Perrone, Categorical Probability and Stochastic Dominance in Metric Spaces, 2018. PhD thesis, University of Leipzig. Full text here, abstract 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:

  • Course, Applied Category Theory, at the University of Oxford, Department of Mathematics. Trinity term 2021. Course page here.
  • Tutorials, Groups and Group Actions, at the University of Oxford, Mansfield College. Trinity term 2021.

Past lectures and seminars:

  • 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.
  • 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.
  • Student seminar on Geometric Group Theory at the University of Leipzig. Summer semester 2017.
  • Student seminar on Characteristic Classes at the University of Leipzig. Summer semester 2015.
For further material about current and past courses please contact me directly.

Events

Upcoming and current events:

  • Kan extensions are partial colimits. Category Theory 2021, University of Genoa (Italy), 30 August-4 September 2021. Possibly held online. Main page here.

Past events

2021 (all events held online):

  • De Finetti's theorem in categorical probability. Applied Category Theory 2021, University of Cambridge (UK), July 2021. 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. Main page here.
  • Weighted limits in metric geometry. Metric geometry seminar. Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany), May 2021.
  • Markov categories: towards a syntax for probability. Logics and Semantics for Dummies seminar. University of Cambridge (UK), May 2021. Link here.
  • The law of large numbers in categorical probability. Topos Colloquium. Topos Institute, Berkeley, CA (USA), May 2021. Speaker: Tobias Fritz (joint work). Link 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. 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 (USA), March 2021. Speaker: Tobias Fritz (joint work). 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 (USA), March 2021. 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. 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.
  • Kan extensions are partial colimits. Invited talk, Masaryk University Algebra Seminar, Masaryk University, Brno (Czech Republic), February 2021. Main page here, video here.
  • An introduction to monads. Invited lecture, Applied Compositional Thinking for Engineers (ACT4E), ETH Zürich (Switzerland), January 2021. Main page here, video here.
  • Markov categories: randomness and information flow. Oxford Quantum Group Workshop 2021, University of Oxford (UK), January 2021.

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
    (Currently in home office.)

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