Dear scientists, mathematicians, linguists, philosophers, and hackers:

We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 Adjoint School. It will begin February 18, 2019 and culminate in a meeting in Oxford, July 22–26, right after the associated conference. Applications are due January 30th; see below for details.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our community’s members is as varied as the systems being studied.

The goal of the ACT2019 School is to help grow this community by pairing ambitious young researchers together with established researchers in order to work on questions, problems, and conjectures in applied category theory.

### Who should apply

Anyone from anywhere who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

We will consider advanced undergraduates, PhD students, and post-docs. We ask that you commit to the full program as laid out below.

Instructions for how to apply can be found below the research topic descriptions.

### Senior research mentors and their topics

Below is a list of the senior researchers, each of whom describes a research project that their team will pursue, as well as the background reading that will be studied between now and July 2019.

#### Miriam Backens

Title: Simplifying quantum circuits using the ZX-calculus

Description: The ZX-calculus is a graphical calculus based on the category-theoretical formulation of quantum mechanics. A complete set of graphical rewrite rules is known for the ZX-calculus, but not for quantum circuits over any universal gate set. In this project, we aim to develop new strategies for using the ZX-calculus to simplify quantum circuits.

Background reading:

- Matthes Amy, Jianxin Chen, Neil Ross. A finite presentation of CNOT-Dihedral operators.
- Miriam Backens. The ZX-calculus is complete for stabiliser quantum mechanics.

#### Tobias Fritz

Title: Partial evaluations, the bar construction, and second-order stochastic dominance

Description: We all know that 2+2+1+1 evaluates to 6. A less familiar notion is that it can *partially evaluate* to 5+1. In this project, we aim to study the compositional structure of partial evaluation in terms of monads and the bar construction and see what this has to do with financial risk via second-order stochastic dominance.

Background reading:

- Tobias Fritz and Paolo Perrone. Monads, partial evaluations, and rewriting.
- Maria Manuel Clementino, Dirk Hofmann, George Janelidze. The monads of classical algebra are seldom weakly cartesian.
- Todd Trimble. On the bar construction.

#### Pieter Hofstra

Title: Complexity classes, computation, and Turing categories

Description: Turing categories form a categorical setting for studying computability without bias towards any particular model of computation. It is not currently clear, however, that Turing categories are useful to study practical aspects of computation such as complexity. This project revolves around the systematic study of step-based computation in the form of stack-machines, the resulting Turing categories, and complexity classes. This will involve a study of the interplay between traced monoidal structure and computation. We will explore the idea of stack machines qua programming languages, investigate the expressive power, and tie this to complexity theory. We will also consider questions such as the following: can we characterize Turing categories arising from stack machines? Is there an initial such category? How does this structure relate to other categorical structures associated with computability?

Background reading:

- J.R.B. Cockett and P.J.W. Hofstra. Introduction to Turing categories. APAL, Vol 156, pp. 183-209, 2008.
- J.R.B. Cockett, P.J.W. Hofstra and P. Hrubes. Total maps of Turing categories. ENTCS (Proc. of MFPS XXX), pp. 129-146, 2014.
- A. Joyal, R. Street and D. Verity. Traced monoidal categories. Mat. Proc. Cam. Phil. Soc. 3, pp. 447-468, 1996.

#### Bartosz Milewski

Title: Traversal optics and profunctors

Description: In functional programming, optics are ways to zoom into a specific part of a given data type and mutate it. Optics come in many flavors such as lenses and prisms and there is a well-studied categorical viewpoint, known as profunctor optics. Of all the optic types, only the traversal has resisted a derivation from first principles into a profunctor description. This project aims to do just this.

Background reading:

- Bartosz Milewski. Profunctor optics, categorical view.
- Craig Pastro, Ross Street. Doubles for monoidal categories.

#### Mehrnoosh Sadrzadeh

Title: Formal and experimental methods to reason about dialogue and discourse using categorical models of vector spaces

Description: Distributional semantics argues that meanings of words can be represented by the frequency of their co-occurrences in context. A model extending distributional semantics from words to sentences has a categorical interpretation via Lambek’s syntactic calculus or pregroups. In this project, we intend to further extend this model to reason about dialogue and discourse utterances where people interrupt each other, there are references that need to be resolved, disfluencies, pauses, and corrections. Additionally, we would like to design experiments and run toy models to verify predictions of the developed models.

Background reading:

- Gerhard Jager (1998): A multi-modal analysis of anaphora and ellipsis. University of Pennsylvania Working Papers in Linguistics 5(2), p. 2.
- Matthew Purver, Ronnie Cann, and Ruth Kempson. Grammars as parsers: meeting the dialogue challenge. Research on Language and Computation, 4(2-3):289–326, 2006.

#### David Spivak

Title: Toward a mathematical foundation for autopoiesis

Description: An autopoietic organization—anything from a living animal to a political party to a football team—is a system that is responsible for adapting and changing itself, so as to persist as events unfold. We want to develop mathematical abstractions that are suitable to found a scientific study of autopoietic organizations. To do this, we’ll begin by using behavioral mereology and graphical logic to frame a discussion of autopoeisis, most of all what it is and how it can be best conceived. We do not expect to complete this ambitious objective; we hope only to make progress toward it.

Background reading:

- Brendan Fong, David Jaz Myers, David Spivak. Behavioral mereology.
- Brendan Fong, David Spivak. Graphical regular logic.
- Luhmann.
*Organization and Decision*, CUP. (Preface)

### School structure

All of the participants will be divided up into groups corresponding to the projects. A group will consist of several students, a senior researcher, and a TA. Between January and June, we will have a reading course devoted to building the background necessary to meaningfully participate in the projects. Specifically, two weeks are devoted to each paper from the reading list. During this two week period, everybody will read the paper and contribute to discussion in a private online chat forum. There will be a TA serving as a domain expert and moderating this discussion. In the middle of the two week period, the group corresponding to the paper will give a presentation via video conference. At the end of the two week period, this group will compose a blog entry on this background reading that will be posted to the n-category cafe.

After all of the papers have been presented, there will be a two-week visit to Oxford University, 15–26 July 2019. The second week is solely for participants of the ACT2019 School. Groups will work together on research projects, led by the senior researchers.

The first week of this visit is the ACT2019 Conference, where the wider applied category theory community will arrive to share new ideas and results. It is not part of the school, but there is a great deal of overlap and participation is very much encouraged. The school should prepare students to be able to follow the conference presentations to a reasonable degree.

### To apply

To apply please send the following to act2019school@gmail.com by January 30th, 2019:

- Your CV
- A document with:
- An explanation of any relevant background you have in category theory or any of the specific projects areas
- The date you completed or expect to complete your Ph.D and a one-sentence summary of its subject matter.

- Order of project preference
- To what extent can you commit to coming to Oxford (availability of funding is uncertain at this time)
- A brief statement (~300 words) on why you are interested in the ACT2019 School. Some prompts:
- how can this school contribute to your research goals?
- how can this school help in your career?

Also have sent on your behalf to act2019school@gmail.com a brief letter of recommendation confirming any of the following:

- your background
- ACT2019 School’s relevance to your research/career
- your research experience

### Questions?

For more information, contact either

– Daniel Cicala. cicala (at) math (dot) ucr (dot) edu

– Jules Hedges. julian (dot) hedges (at) cs (dot) ox (dot) ac (dot) uk