Logical starting point:

A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions \(\phi^t\) that for any element of t ∈ T, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.

I'm going to work through some of this here. Feel free anyone to add any comments that come to mind (the notes can be interleaved).

]]>- Tai-Danae Bradley, At the interface of algebra and statistics, PhD thesis, 2020.

]]>This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.

]]>Hey guys, if anybody is interested, I host two Sunday 2pm reading groups on an alternating weekly basis. We've started meeting on Zoom now so it's accessible to everybody.

In the first one, we're working through Steve Roman's

An Introduction to the Language of Category Theory. It's for people who want to fill holes in their basic Category Theory knowledge. We're just starting in on Chapter 2 so it's still possible to catch up easily.In the second group, we're reading Coecke and Kissinger's

Picturing Quantum Processes. We've been working through Kissinger's problem sets from when he taught a class based on the book. We're just starting on assignment 2 so it's still accessible to newcomers as well.If you're interested, I post the events on http://www.meetup.com/category_theory/.

Would you like to exchange ideas related to subtlety theory and information theory? I am curious how you look at these subjects. Perhaps we have overlapping interests. Could would explore them here?

]]>**Syllabus.**

MIT 2020 Programming with Categories course:

David Dalrymple's class summaries

Please add comments with your recommended entries for study!

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