Adjoint School 2022

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 February 4th! 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:

About the Adjoint School.

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: Span(Graph), introduced by Katis, Sabadini and Walters as a categorical algebra for automata with interfaces, provides, in a very intuitive way, a compositional description of hierarchical networks of interacting components with fixed topology. The algebra also provides a calculus of connectors, with an elegant description of signal broadcasting. In particular, the operations of “parallel with communication” (that allows components to evolve simultaneously, like connected gears), and “non-sequential feedback” (not considered in Kleene’s algebra for classical automata) are fundamental in modelling complex distributed systems such as biological systems. Similarly, the dual algebra Cospan(Graph) allows us to compose systems sequentially. Hence, the combined algebra CospanSpan(Graph), which extends Kleene’s algebra for classical automata, is a general algebra for reconfigurable networks of interacting components. Still, some very interesting aspects and possible applications of this model deserve a better understanding:

• How can timed actions and probability be combined in CospanSpan(Graph)?

• If not, can we describe time-varying probability in a compositional setting?

• Which is the possible role of “parallel with communication” in understanding causality?

Readings:

• L. de Francesco Albasini, N. Sabadini, and R.F.C. Walters, The compositional construction of Markov processes II.

• A. Cherubini, N. Sabadini, and R.F.C. Walters, Timing in the Cospan-Span model.

Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description: Fox’s theorem provides a bridge between structures defined by universal properties (products in a category) and structures specified by algebraic means (comonoids in a symmetric monoidal category). Such a theorem has recently received a renewed interest as the algebraic structures allows for reasoning in terms of string diagrams. While the universal properties underlying logical theories have been extensively studied in categorical logic, their algebraic counterparts have been the objects of fewer investigations. This raises a natural question: can we capture the universal content of logical theories algebraically? In other words, what are the ‘Fox theorems’ for logic? In this project, we attempt to answer to this question by taking as starting point Cartesian bicategories which serves as algebraic setting for regular logic.

Readings:

• Aurelio Carboni and R. F. C. Walters, Cartesian bicategories I.

• Filippo Bonchi, Jens Seeber and Pawel Sobocinski, Graphical conjunctive queries.

• Filippo Bonchi, Dusko Pavlovic and Pawel Sobocinski, Functorial semantics for relational theories.

3 Responses to Adjoint School 2022

  1. John Baez says:

    Pawel Sobocinski had a lot of good things to say about this school on Twitter, especially topics 3 and 4 (Sabadini and Bonchi). Let me just quote him:

    The applications for the Applied Category Theory 2022 Adjoint School https://t.co/80SqBZHyHx are open! This initiative has been running for several years and has been a massive success – lots of new collaborations, projects, publications. (1/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Successful candidates are assigned to one of four groups, each led by a mentor. The groups are given a bunch of papers for background reading, and the groups meet online weekly to discuss the papers amongst themselves and with the mentor. (2/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    This leads up to the research week where the groups come together and work intensively to identify the most promising directions of research, applications, difficult problems, etc. This process has sometimes led to great new research. (3/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    It’s a great opportunity for early career researchers to establish new collaborations, get exposed to areas of the field that they are unfamiliar with, and — most importantly — work on cool problems! (4/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    If you’re a PhD student, a postdoc or just passionate about research and want to learn about ACT, please consider applying. In previous years we have had brilliant candidates with almost no previous experience of category theory. Passion and drive are way more important. (5/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    The 2022 topics:
    S. Breiner and J. Moeller – Compositional Thermodynamics
    P. North – Fuzzy Type Theory for Opinion Dynamics
    N. Sabadini – A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)
    F. Bonchi – Algebraic Structures in Logic and Relations (6/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Let me focus on the last two. Nicoletta Sabadini has authored some of my favourite papers and her work in collaboration with RFC Walters has been hugely influential on my thinking and my work. They were doing ACT 20 years before the current crop! (7/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    The work on Span(Graph) and the work that followed it is, in my opinion, hugely important and under-appreciated. In the 90s and 00s the field of process calculi was exploding, with several different calculi of processes proposed (e.g. pi-calculus, ambient calculus, ..) (8/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    The field became less active in the late 00s and the 10s… Let me play the amateur historian of science and theorise about why this happened.
    A process calculus involves many engineering decisions: designing the syntax, the operational semantics, the equivalence, etc. (9/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Ultimately all these small decisions make each calculus a pretty unique snowflake; a mathematical universe with its own techniques, its own problems, its own community. The process calculus community was splintered into micro specialists and overall progress slowed (10/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    What can we learn from this? It is important to have some underlying mathematics to inform your design decisions. Functional programmers have this in the Curry-Howard-Lambek correspondence, another example is the amazing PL work stemming from HoTT. (11/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Nicoletta and Bob identified Span(Graph) as providing an algebra of communicating automata. Span(Graph) has a lot of mathematical structure, for example it is a free thing! This work has continued to capture more expressive models (probabilistic, stochastic, etc.). (12/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    So if you’re interested in how to understand models of computation in a compositional way, I encourage you to join Nicoletta’s group!

    Now a few words about Filippo’s project. (13/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Filippo and I collaborate for 10 years now; in 2012 I spent an amazing sabbatical in Lyon. Our research journey was kind of crazy; we just kept on stumbling on new mind-blowing stuff to discover and understand. I’ve lost track of how many papers we’ve coauthored. (14/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Our work evolved when we visited Dusko Pavlovic in 2016. We realised that our various graphical theories fit into a general framework of functorial semantics of relational algebras: algebraic gadgets that take their models in the category of relations rather than sets. (15/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    This leads to a “2-dimensional” version of relational algebra, based on the work of Bob Walters and Aurelio Carboni on Cartesian Bicategories. Relational algebra is an old topic, going back to Peirce, but usually understood with the treatement by Tarski in the 40s. (16/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    The importance of relational algebra and its various flavours and fragments is difficult to overstate. It is used all over the place in computer science; for example, it can be seen as the mathematical foundation of database theory. (17/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    So why the “diagrammatic approach” based on cartesian bicategories? First, similarly to the work of Nicoletta, it connects the syntax of this new relational algebra with an underlying, rich mathematical world. The “engineering decisions” are better justified. (18/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Second, there are intriguing open problems to explore: e.g., various negative results about existence of finite axiomatisations fail: we can find finite axiomatisations as symmetric monoidal theories of things that have no first order order finite axiomatisations (19/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Third, the diagrammatic calculus is super intuitive. It is similar to existential graphs of Peirce, and various diagrammatic formalisms used in database theory and practice. This is, therefore, another example of taking diagrams seriously as mathematical objects! (20/?)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    Filippo will take you the the limits of what we understand about this stuff. I’m pretty confident that his project will lead to cool new results.

    Nicoletta’s project is just as exciting. I have also heard great things also about the other two projects. (21/22)

    — Pawel Sobocinski (@PawSob) January 3, 2022

    So, go ahead and apply!

    As I said before, you don’t need to know category theory: we just want your passion, drive and loads of curiosity! (22/22)

    — Pawel Sobocinski (@PawSob) January 3, 2022

  2. John Baez says:

    9 days until applications are due! Nicoletta Sabadini has come out with her project description:

    Description: Span(Graph), introduced by Katis, Sabadini and Walters as a categorical algebra for automata with interfaces, provides, in a very intuitive way, a compositional description of hierarchical networks of interacting components with fixed topology. The algebra also provides a calculus of connectors, with an elegant description of signal broadcasting. In particular, the operations of “parallel with communication” (that allows components to evolve simultaneously, like connected gears), and “non-sequential feedback” (not considered in Kleene’s algebra for classical automata) are fundamental in modelling complex distributed systems such as biological systems. Similarly, the dual algebra Cospan(Graph) allows us to compose systems sequentially. Hence, the combined algebra CospanSpan(Graph), which extends Kleene’s algebra for classical automata, is a general algebra for reconfigurable networks of interacting components. Still, some very interesting aspects and possible applications of this model deserve a better understanding:

    • How can timed actions and probability be combined in CospanSpan(Graph)?

    • If not, can we describe time-varying probability in a compositional setting?

    • Which is the possible role of “parallel with communication” in understanding causality?

    Readings:

    • L. de Francesco Albasini, N. Sabadini, and R.F.C. Walters, The compositional construction of Markov processes II.

    • A. Cherubini, N. Sabadini, and R.F.C. Walters, Timing in the Cospan-Span model.

  3. John Baez says:

    The fourth mentor for the Adjoint School of applied category theory, Filippo Bonchi, has finally announced his course.

    Due to the delay, the deadline for applying—and choosing which course you want!—has been pushed back to February 4. Apply here:

    http://adjointschool.com/apply.html

    Topic 4: Algebraic Structures in Logic and Relations

    Mentor: Filippo Bonchi

    Description: Fox’s theorem provides a bridge between structures defined by universal properties (products in a category) and structures specified by algebraic means (comonoids in a symmetric monoidal category). Such a theorem has recently received a renewed interest as the algebraic structures allows for reasoning in terms of string diagrams. While the universal properties underlying logical theories have been extensively studied in categorical logic, their algebraic counterparts have been the objects of fewer investigations. This raises a natural question: can we capture the universal content of logical theories algebraically? In other words, what are the ‘Fox theorems’ for logic? In this project, we attempt to answer to this question by taking as starting point Cartesian bicategories which serves as algebraic setting for regular logic.

    Readings:

    • Aurelio Carboni and R. F. C. Walters, Cartesian bicategories I.

    • Filippo Bonchi, Jens Seeber and Pawel Sobocinski, Graphical conjunctive queries.

    • Filippo Bonchi, Dusko Pavlovic and Pawel Sobocinski, Functorial semantics for relational theories.

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