The STyLo seminar is a weekly internal meeting for members of the Systems, Types, and Logic group. It provides a space to present ongoing research, share preliminary results, and discuss papers relevant to the group’s themes such as type systems, categorical semantics, and formal methods for system verification.

When: Mondays at 11:00 (starting from week 38) Where: Location announced via the mailing list

20 October 2025 – 2-forms and Symplectic Manifolds

Given by Henning

We discuss examples of 2-forms and introduce symplectic manifolds.

13 October 2025 – Differential forms on Smooth Manifolds

Given by Henning

We construct a diffeology from an atlas and use that to define k-forms on smooth manifolds. We also look at 0-, 1- and n-forms.

06 October 2025 – Differential forms on Euclidean Spaces

Given by Henning

We introduce multi-linear maps and k-forms on open sets of Euclidean spaces

29 September 2025 – Smooth Maps

Given by Henning

We continue with the introduction to the smooth manifold theory. This time we do smooth maps.

22 September 2025 – Extra conditions on smooth manifolds

Given by Henning

We continue with the introduction to the smooth manifold theory and go over the long line and the line with two origins to show why Hausdorff and second countable matters.

15 September 2025 – Smooth Manifolds

Given by Henning

Today, I start to go through the equivalence between geometry and entropy and I will begin with a general introduction to smooth manifolds.

It is well-known that symplectic manifolds yield a measure, the Liouville measure, against one can define probability distributions and thus entropy. Surprisingly, this geometric structure, the symplectic form, can be reconstructed purely from the entropy. This comes from recent work by Gabriele Carcassi and collaborators, which I aim to go through in detail in the coming weeks. This will include the basics of symplectic manifolds and I aim to develop some category theoretical understanding around this result as well.

Reading

John M. Lee, Introduction to Smooth Manifolds, vol. 218. in Graduate Texts in Mathematics, vol. 218. New York, NY: Springer, 2012. doi.
P. Iglesias-Zemmour, Diffeology, 2nd edn. Beijing World Publishing Corporation, 2022. doi: 10.1090/surv/185. Download
G. Carcassi and C. A. Aidala, ‘Hamiltonian mechanics is conservation of information entropy’, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, vol. 71, pp. 60–71, Aug. 2020, doi.

Academic Year 2024 - 2025

Our group seminar takes place in the autumn semester of 2024 on Fridays at 10am, starting on 27 September.

For the first weeks of this semester, we will study abstract probability theory (measure theory, going via probability theory and probabilistic processes, all the way to stochastic differential equations) and build in parallel a category theoretic understanding of probability theory. For questions or if you wish to participate, please contact Berend van Starkenburg.

Literature

A. Klenke, Probability Theory: A Comprehensive Course. in Universitext. Cham: Springer International Publishing, 2020. doi: 10.1007/978-3-030-56402-5.
O. Kallenberg, Foundations of Modern Probability, vol. 99. in Probability Theory and Stochastic Modelling, vol. 99. Cham: Springer International Publishing, 2021. doi: 10.1007/978-3-030-61871-1.
M. Giry, ‘A categorical approach to probability theory’, in Categorical Aspects of Topology and Analysis, B. Banaschewski, Ed., Berlin, Heidelberg: Springer Berlin Heidelberg, 1982, pp. 68–85. doi: 10.1007/BFb0092872
B. Jacobs, Structured Probabilistic Reasoning. Draft, 2023. Available: online
C. Heunen, O. Kammar, S. Staton, and H. Yang, ‘A convenient category for higher-order probability theory’, in Proc. of LICS’17, IEEE Computer Society, 2017, pp. 1–12. doi: 10.1109/LICS.2017.8005137.