Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the **Adjoint School**.

You can apply to be a student at the 2022 Adjoint School *now*, and applications are due January 29th! Go here:

• 2022 Adjoint School: application.

The school will be run online from February to June, 2022, and then—coronavirus permitting—there will be in-person research at the University of Strathclyde in Glasgow, Scotland the week of July 11 – 15, 2022. This is also the location of the applied category theory conference ACT2022.

The 2022 Adjoint School is organized by Angeline Aguinaldo, Elena Di Lavore, Sophie Libkind, and David Jaz Myers. You can read more about how it works here:

There are four topics to work on, and you can see descriptions of them below.

### Who should apply?

Anyone, from anywhere in the world, 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, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

Also check out our inclusivity statement.

### Topic 1: Compositional Thermodynamics

**Mentors:** Spencer Breiner and Joe Moeller

**TA:** Owen Lynch

**Description:** Thermodynamics is the study of the relationships between heat, energy, work, and matter. In category theory, we model flows in physical systems using string diagrams, allowing us to formalize physical axioms as diagrammatic equations. The goal of this project is to establish such a compositional framework for thermodynamical networks. A first goal will be to formalize the laws of thermodynamics in categorical terms. Depending on the background and interest of the participants, further topics may include the Carnot and Otto engines, more realistic modeling for real-world systems, and software implementation within the AlgebraicJulia library.

**Readings**:

• John C. Baez, Owen Lynch, and Joe Moeller, Compositional thermostatics.

• F. William Lawvere, State categories, closed categories and the existence of semi-continuous entropy functions.

### Topic 2: Fuzzy Type Theory for Opinion Dynamics

**Mentor:** Paige North

**TA:** Hans Reiss

**Description:** When working in type theory (or most logics), one is interested in proving propositions by constructing witnesses to their incontrovertible truth. In the real world, however, we can often only hope to understand how likely something is to be true, and we look for evidence that something is true. For example, when a doctor is trying to determine if a patient has a certain condition, they might ask certain questions and perform certain tests, each of which constitutes a piece of evidence that the patient does or does not have that condition. This suggests that a fuzzy version of type theory might be appropriate for capturing and analyzing real-world situations. In this project, we will explore the space of fuzzy type theories which can be used to reason about the fuzzy propositions of disease and similar dynamics.

**Readings:**

• Daniel R. Grayson, An introduction to univalent foundations for mathematicians.

• Jakob Hansen and Robert Ghrist, Opinion dynamics on discourse sheaves.

### Topic 3: A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)

**Mentor:** Nicoletta Sabadini

**TA:** Mario Román

Description coming soon!

### Topic 4: Algebraic Structures in Logic and Relations

**Mentor:** Filippo Bonchi

Description coming soon!