# Eigil Fjeldgren Rischel

My name is Eigil Fjeldgren Rischel. I’m studying for a PhD in Computer and Information Sciences at the Mathematically Structured Programming group at the University of Strathclyde. My advisor is Radu Mardare.

## Research Interests

The unifying buzzword for all the stuff I’m interested in is ‘Category Theory’. I’m particularly interested in the applications of category theory(in the sense of applied mathematics). Currently I’m studying applications to statistics and probability. I used to be really into algebraic topology, but I don’t do much work in that space anymore. But if you want to chat about it, feel free to send me an email.

Some questions:

- What sort of “correspondence” or “transformations” between causal models allows one to “transfer information” between them? I investigated this in my MSc thesis, titled The Category Theory of Causal Models. See also the work of my advisor for that thesis, Sebastian Weichwald.
- In practical situations, we can almost never find laws that hold
*exactly*. Instead we have equations that hold up to a very small degree of error, and (ideally) and understanding of this error. How can we adapt the methods of “abstract mathematics”, e.g. category theory and algebra, to work in this situation? - Probability theory describes how to reason in situations where we
*lack information*. Often we have access to information, but not enough*computing power*to make use of it. For example, the \(10^{10^{10^{10}}}\) th prime number is “known information” for me, in the sense that I can calculate it from things I already know. Yet it is just as inaccessible to me as the outcome of a fair coin. Can we find a good compositional framework for reasoning about situations with bounded computing power, which is as useful as probability theory? - In the context of machine learning, people often reason informally with game-theoretic arguments.
For example, in a GAN, we might say that the generator is “incentivized” to output a large variety of examples (that all look like the training set), because otherwise the discriminator can simply downgrade the small number of outputs from the generator while still maintaining a high score on most of the training set. This is implicitly an argument about
*Nash equilibria*- can we make this type of reasoning tight? - Probability theory based on measure theory involves a number of techincal difficulties that seem largely disconnected from the actual practice of statistics. Can we find an approach to the foundations of probability with a more usable “interface”?

## Contact

The preferred avenue is email at ayegill (at) gmail (dot) com.

I’m interested in hearing from *anyone*.
I really want to emphasize this - so far, I’ve never been unhappy to receive a mail from someone who wanted to talk to me.
In particular, if you want to chat about any of my research interests, I want to talk to you.

## Online presence

- I have some code on github, but I’m not super active there. Not a good way to follow my work.
- I’m fairly active on twitter, and can probably be reached via DM there (but would prefer email).
- I try to maintain my goodreads account.

## Some things I’ve written

- The Zero-One Laws of Kolmogorov and Hewitt-Savage in Categorical Probability. A statement and proof of a classical theorem of probability theory in an abstract (i.e categorical) setting. With Tobias Fritz. Now published by Compositionality
- Localizations in higher algebra and algebraic K-theory. My undergraduate thesis. Essentially an account of the results in this paper. Describes how to construct monoidal structures on certain infinity-categories using universal properties. Supervised by Rune Haugseng
- Bounded ultraspaces. A note on some ideas of Lurie concerning higher categorical logic.
- On 2020-06-06, I gave a talk for the Categorical Probability and Statistics conference. There’s a video online here, and my slides are here
- The Category Theory of Causal Models. My MSc thesis in mathematics, supervised by Sebastian Weichwald
- Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability. With Tobias Fritz, Paolo Perrone and Tomas Gonda
- On 2020-11-19, I gave a talk for the MIT Categories Seminar. The youtube video is here. The slides are here

You can also take a look at my blog